TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Telangana TSBIEĀ TS Inter 2nd Year Chemistry Study Material Lesson 12(a) Alcohols, Phenols, and Ethers Textbook Questions and Answers.

TS Inter 2nd Year Chemistry Study Material Lesson 12(a) Alcohols, Phenols, and Ethers

Very Short Answer Questions (2 Marks)

Question 1.
Explain why propanol has higher boiling point than that of the hydrocarbon butane.
Answer:
Propanol and butane are of comparable molecular mass. However, the boiling point of propanol is higher than that of butane, due to the presence of intermolecular hydrogen bonding.

Question 2.
Alcohols are comparatively more soluble in water than hydrocarbons of comparable molecular masses. Explain this fact.
Answer:
Alcohols are comparatively more soluble in water than hydrocarbons of comparable molecular masses, due to their ability to form hydrogen bonds with water molecules.

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 3.
Give the structures and 1UPAC names of monohydric phenols of molecular formula, C7H8O.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 1
Common Name:
o – Cresol
m – Cresol
p – Creol

IUPAC Name:
2 – Methylphenol
3 – Methylphenol
4 – Methylphenol

Question 4.
Give the reagents used for the preparation of phenol from chlorobenzene.
Answer:

  1. NaOH
  2. HCl.

Question 5.
Preparation of ethers by acid dehydration of secondary or tertiary alcohols is not a suitable method. Give reason.
Answer:
This is because elimination competes over substitution and as a consequence, alkenes are easily formed.

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 6.
Write the mechanism of the reaction of HI with methoxymethane.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 2

Question 7.
Name the reagents used in the following reactions.

  1. Oxidation of primary alcohol to carboxylic acid.
  2. Oxidation of primary alcohol to aldehyde.

Answer:

  1. Acidified potassium permanganate.
  2. Pyridinium chlorochromate (PCC) or CrO3 in anhydrous medium.

Question 8.
Write the equations for the following reactions.
i) Bromination of phenol to 2, 4, 6-tribromophenoI.
ii) Benzyl alcohol to benzoic acid.
Answer:
i) A white precipitate of 2,4,6 – tribromophenol is formed when phenol is treated with bromine water.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 3
ii) Benzyl alcohol is oxidised to benzoic acid by acidified KMn04 or acidic solution of sodium dichromate.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 4

Question 9.
Identify the reactant needed to form t-butyl alcohol from acetone.
Answer:
Methyl magnesium halide (Grignard reagent) is needed to form t-butyl alcohol from acetone.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 5

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 10.
Write the structures for the following compounds.
i) Ethoxyethane
ii) Ethoxybutane
iii) Phenoxyethane
Answer:
i) C2H5OC2H5 – Ethoxyethane
ii) CH3CH2CH2CH2OCH2CH3 – Ethoxybutane
iii) C2H5OC6H5 – Phenoxyethane

Short Answer Questions (4 Marks)

Question 11.
Draw the structures of all isomeric alcohols of molecular formula C5H12O and give their IUPAC names and classify them as primary, secondary and tertiary alcohols.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 6

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 12.
While separating a mixture of ortho and para nitrophenols by steam distillation, name the isomer which will be steam volatile. Give reason.
Answer:
O-nitrophenol will be steam volatile due to intramolecular hydrogen bonding while p-nitro- phenol will be less volatile due to intermolecular hydrogen bonding which causes association of molecules.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 7

Question 13.
Give the equations for the preparation of phenol from Cumene.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 8

Question 14.
Write the mechanism of hydration of ethene to yield ethanol.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 9
Mechanism : The mechanism of the reaction involves three steps.
Step 1: Protonation of the alkene (ethene) to form carbonation by electrophilic attack of H3O+.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 10
Step 2 : Nucleophilic attack of water on carbocation.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 11
Step 3 : Deprotonation to form an alcohol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 12

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 15.
Explain the acidic nature of phenols and compare with that of alcohols.
Answer:
The reactions of phenol with metals (e.g. : Na, Al) and NaOH indicate its acidic nature. The hydroxyl group, in phenol is directly attached to the sp2 hybridised carbon of benzene ring which acts as an electron withdrawing group. The charge distribution in phenol molecule as depicted in its resonance structures, causes the oxygen of – OH group to be positive.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 13
The reaction of phenol with NaOH solution indicates that phenols are stronger acids than alcohols and water. Let us compare the acidic nature of phenol with that of alcohol.
The ionisation of an alcohol and a phenol takes place as shown below.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 14
Owing to the higher electronegativity of sp2 hybridised carbon of phenol to which – OH is attached, electron density decreases on oxygen. This increases the polarity of – OH bond and results in an increase in ionisation of phenols than that of alcohols.

The relative stabilities of alkoxide and phenoxide ions should be considered. In alkoxide ion, the negative charge is localised on oxygen whereas in phenoxide ion, the charge is delocalised. The delocalisation of negative charge makes phenoxide ion more stable and favours the ionisation of phenol. There is delocalisation in unionised phenol also but the resonance structures have charge separation. Hence, phenol molecule is less stable them phenoxide ion.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 15

Question 16.
Write the products formed by the reduction and oxidation of phenol. [IPE 14]
Answer:
Reduction of phenol: Phenol is converted to benzene when heated with zinc dust.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 16
Oxidation : Phenol gives benzoquinone on oxidation with chromic acid.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 17
In the presence of air, phenols are slowly oxidised to dark coloured mixtures containing quinones.

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 17.
Ethanol with H2SO4, at 443K forms ethane while at 413 K it forms ethoxy ethane. Explain the mechanism.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 18
Step 2 : Formation of carbocation.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 19
Mechanism : The formation of ether is a nucleophilic bimolecular reaction (SN2) involving the attack of alcohol molecule on a protonated alcohol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 20

Question 18.
Account for the statement: Alcohols boil at highest temperature than hydrocarbons and ethers of comparable molecular masses.
Answer:
The higher boiling points of alcohols are mainly due to the presence of inter molecular hydrogen bonding in them which is lacking in ethers and hydrocarbons.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 21

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 19.
Explain why in anisole electrophilic substitution takes place at ortho and para positions and not at meta position.
Answer:
In benzene derivatives the electrophile is most likely to react at the position of highest electron density. The methoxy group ih anisole is an electron releasing group. It increases the relative electron density at ortho and para positions and hence electrophilic substitution takes place at these positions and not at the meta position.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 22

Question 20.
Write the products of the following reactions:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 23
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 24

Long Answer Questions (8 Marks)

Question 21.
Write the IUPAC names of the following compounds:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 25
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 26

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 22.
Write structures of the compounds whose IUPAC names are as follows: [IPE 14]
i) 2- Methyl butan-1- ol
ii) 1-Phenylpropan-2-ol
iii) 3, 5-Dimethythexane- 1, 3, 5- triol
iv) 2, 3- Dlethylphenol
v) 1 – Ethoxypropane
vi) 2- Ethoxy -3- methylpentane
vii) Cyclohexylmethanol
viii) 3- Chloromethylpentan – 1 – ol
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 27
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 28

Question 23.
Write the equations for the preparation of phenol using benzene, Cone. H2SO4 and NaOH.
Answer:
Benzene is sulphonated with oleum and benzene sulphonic acid so formed is converted to sodium phenoxide on heating with molten sodium hydroxide. Acidification of the sodium salt gives phenol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 29

Question 24.
Illustrate hydroboration-oxidation reaction with a suitable example.
Answer:
Diborane B2H6 [or (BH3)2] reacts with aikenes to give trialkyl boranes as addition product. This is oxidised to alcohol by hydrogen peroxide in the presence of aqueous sodium hydroxide.
Ex : Propene gives Propan-1-ol on hydroboration – oxidation reaction.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 30

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 25.
Write the IUPAC names of the following compounds.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 31
Answer:
i) 2-methyl phenol
ii) 4 – methyl phenol
iii) 2, 5 – dimethyl phenol
iv) 2, 6 – dimethyl phenol

Question 26.
How will you synthesise :
i) 1 – Phenylethanol from a suitable alkene ?
ii) Cyclohexylmethanol using an alkyl halide by an SN2 reaction ?
iii) Pentan-1-ol using a suitable alkyl halide ?
Answer:
i) Styrene on acid catalysed hydration gives 1-phenyl ethanol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 32
ii) Cyclohexyl chloromethane reacts with aqueous sodium hydroxide solution to give cyclohexyl methanol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 33
iii) Pentan-1-ol is obtained by the reaction of 1-chloropentane with aqueous NaOH solution.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 34

Question 27.
Explain why –
i) Ortho nitrophenol is more acidic than ortho methoxyphenol.
ii) OH group attached to benzene ring activates it towards electrophilic substitution.
Answer:
i) Electron withdrawing groups enhance the acidic strength of phenol. This effect is more pronounced if these groups are present in ortho and para positions. It is due to the effective delocalisation of negative charge in phenoxide ion. On the other hand electron releasing
‘ groups do not favour the formation of phenoxide ion resulting in decrease in acid strength.

Nitro group is an electron withdrawing group. It increases the acidic strength of phenol. Methoxy group is an electron releasing group. It decreases the acidic strength of phenol. Hence o-nitrophenol is more acidic than orthomethoxyphenol.

ii) The – OH group attached to the benzene ring activates it towards electrophilic substitution. Further, it directs the incoming group to ortho and para positions in the ring as these positions become electron rich due to the resonance effect caused by – OH group.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 35

Question 28.
With a suitable example write equations for the following:
i) Kolbe’s reaction.
ii) Reimer-Tiemann reaction.
iii) Willamson’s ether synthesis.
Answer:
i) Kolbe’s synthesis: When sodium salt of phenol (sodium phenoxide) is heated with carbon dioxide under pressure, a carboxylic group is introduced in the aromatic ring preferably at ortho position with respect to phenolic group. The sodium salt of o-hydroxybenzoic acid (sodium salicylate) formed, when treated with acid yields salicylic acid.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 36

ii) Reimer-Tiemann reaction : When phenol is heated with chloroform in the presence of sodium hydroxide at 60°C, a – CHO group is introduced at the ortho position with respect to the phenolic group. This reaction is known as Reimer-Tiemann reaction.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 37

iii) Willamson’s ether synthesis: It is an important method for the preparation of symmetrical and unsymmetrical ethers. In this method, an alkyl halide is allowed to react with sodium alkoxide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 38
Ethers containing substituted alkyl groups (2° or 3°) may also be prepared by this method. The reaction involves SN2 attack of an alkoxide ion on primary alkyl halide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 39
In case of secondary and tertiary alkyl halides, elimination competes over substitution. If a tertiary alkyl halide is used, an alkene is the only reaction product and no ether is formed.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 40

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 29.
How are the following conversions carried out ?
i) Benzyl chloride to Benzyl alcohol
ii) Ethyl magnesium bromide to Propan-1-ol
iii) 2-butanone to 2-butanol
Answer:
i) Benzyl chloride is converted to benzyl alcohol by hydrolysis with aqueous sodium hydroxide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 41

ii) Ethyl magnesium bromide reacts with formaldehyde to form an adduct. Hydrolysis of the adduct yields an propan-1-ol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 42

iii) 2-butanone bn reduction with sodium borohydride (NaBH4) gives 2-butanol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 43

Question 30.
Write the names of the reagents and equations for the preparation of the following ethers by Williamson’s synthesis:
i) 1-Propoxypropane
ii) Ethoxybenzene
iii) 2-Methoxy-2-methylpropane
iv) 1-Methoxyethane
Answer:
i) Sodium propoxide and propyl bromide.
CH3CH2CH2ONa + CH3CH2CH2 – Br → CH3CH2CH2OCH2CH2CH3 + NaBr

ii) Chlorobenzene and sodium ethoxide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 44

iii) Sodium tertiary butoxide and methylbromide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 45

iv) Sodium ethoxide and methyl bromide.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 46

Question 31.
How is 1-propoxypropane synthesized from propan-1-ol ? Write mechanism of this reaction.
Answer:
1-rpropoxypropane is synthesised from propan-1-ol by dehydration in the presence of sulphuric acid.
Mechanism :
The formation of 1-propoxypropane is an SN2 reaction involving the attack of alcohol molecule on a protonated alcohol.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 48
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 47

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 32.
Explain the fact that in aryl alkyl ethers the alkoxy group activates the benzene ring towards electrophilic substitution.
Answer:
Alkoxy group TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 49 is an electron releasing group. When attached to benzene ring alkoxy group activates the ring towards electrophilic substitution. Further, it directs the incoming group to ortho and para positions in the benzene ring as these positions become electron rich due to the resonance effect caused by the – OR group.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 50

Question 33.
Write equations of the below given reactions :
i) Alkylation of anisole
ii) Nitration of anisole
iii) Friedel-Crafts acylation of anisole
Answer:
i) Alkylation of anisole: Anisole undergoes Friedel-Crafts alkylation reaction with alkyl halide in the presence of anhydrous AlCl3 as catalyst. The alkyl group is introduced in the ortho and para positions.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 51

ii) Nitration of anisole : Anisole on nitration with a mixture of concentrated H2SO4 and HNO3 yields a mixture of ortho and para nitroanisole.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 52

iii) Friedel-Crafts acylation of anisole : Anisole undergoes Friedel-Crafts acylation with acyl halide in the presence of anhydrous AlCl3.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 53
The acyl group is introduced in the ortho and para positions.

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 34.
Show how you would synthesize Hie following alcohols from appropriate aikenes ?
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 54
Answer:
By acid – catalysed hydration
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 55

Question 35.
Explain why phenol with bromine water forms 2, 4, 6-tribromophenol while on reaction with bromine in CS2 at low temperatures forms para-bromophenol as the major product.
Answer:
The hydroxyl group (- OH) is a very powerful activating substituent, and electrophilic substitution in phenols occurs faster, and under milder conditions, than in benzene.

Bromination of phenol occurs readily even in the absence of a catalyst at low temperature; Substitution occurs primarily at the para position to the hydroxyl group. When the para position is blocked, ortho substitution is observed. The reaction is carried out in a non-polar solvent CS2 or ClCH2CH2Cl. In polar solvents such as water it is difficult to limit the bromination of phenols to monobromination. With bromine water all three positions that are ortho or para to the hydroxyl undergo rapid substitution.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 56

Intext Questions – Answers

Question 1.
Classify the following as primary, secondary and teritary alcohols :
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 57
Answer:
i) Primary alcohol
ii) Primary alcohol
iii) Primary alcohol
iv) Secondary alcohol
v) Secondary alcohol
vi) Tertiary alcohol

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 2.
Identify allylic alcohols in the above examples.
Answer:
In the examples given above (under question 1), (ii) and (vi) are allylic alcohols.

Question 3.
Name the following compounds according to IUPAC system.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 58
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 59

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 4.
Show how the following alcohols are prepared by the reaction of a suitable Grignard reagent on methanal ?
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 60
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 61

Question 5.
Write structures of the products of the following reactions.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 62
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 63

Question 6.
Write the structures of the major products expected from the following reactions :
a) Mononitration of 3-methylphenol
b) Dinitration of 3-methylphenol
c) Mononitration of phenyl methanoate.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 64

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 7.
Give structures of the products you would expect when each of the following alcohol reacts with
a) HCl – ZnCl2,
b) HBr and
c) SOCl2.
i) Butan-1-ol
ii) 2-Methylbutan-2-ol
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 65
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 66
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 67

Question 8.
Predict the major product of acid catalysed dehydration of
i) 1-methylcyclohexanol and
ii) butan-1-ol
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 68
A mixture of but-l-ene and but-2-ene will be obtained. The 1 ° carbocation formed as intermediate will undergo rearrangement to give a more stable 2° carbocation. Loss of proton results in the mixture of but-1-ene and but-2-ene. However, but-l-ene will be the major product.

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 9.
Ortho and para nitrophenols are more acidic titan phenol. Draw the resonance structures of the corresponding phenoxide ions.
Answer:
Electron delocalization in o-nitrophenoxide ion.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 69
Electron delocalization in p-nitrophenoxide ion.
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 70

Question 10.
Write the equations involved in the following reactions:
i) Reimer – Tiemann reaction
ii) Kolbe’s reaction
Answer:
i) Reimer-Tiemann reaction :
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 71
ii) Kolbe’s reaction
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 72

Question 11.
Write the reactions of Williamson synthesis of 2-ethoxy-3-methylpentane starting from ethanol and 3-methylpentan-2-ol
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 73

TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers

Question 12.
Which of the following is an appropriate set of reactants for the preparation of 1-methoxy- 4-nitrobenzene and why ?
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 74
Answer:
Set (ii) is appropriate.

Question 13.
Predict the products of the following reactions:
i) CH3-CH2-CH2-O-CHS + HBr →
Answer:
CH3 – CH2 – CH2 – OH + CH3Br

ii)
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 75
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 76

iii)
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 77+
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 78

iv)
TS Inter 2nd Year Chemistry Study Material Chapter 12(a) Alcohols, Phenols, and Ethers 77
Answer:
(CH3)3C – I + C2H5OH

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Students must practice these Maths 2A Important Questions TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type to help strengthen their preparations for exams.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Questions 1.
Find the 6th term in \(\left(\frac{2 x}{3}+\frac{3 y}{2}\right)^9\). [May ’13, AP – Mar. 2019]
Solution:
Given \(\left(\frac{2 x}{3}+\frac{3 y}{2}\right)^9\)
Here, x = \(\frac{2 x}{3}\); a = \(\frac{3 y}{2}\); n = 9
r + 1 = 6
⇒ r = 5
The general term in the expansion of (x + a)n is given by
Tr + 1 = \({ }^n C_r\) . xn – r . ar
The 6th term in the expansion of \(\left(\frac{2 x}{3}+\frac{3 y}{2}\right)^9\) is
T5 + 1 = \({ }^9 C_5\left(\frac{2 x}{3}\right)^{9-5}\left(\frac{3 y}{2}\right)^5\)
T6 = \({ }^9 \mathrm{C}_5 \cdot\left(\frac{2 \mathrm{x}}{3}\right)^4 \cdot\left(\frac{3 \mathrm{y}}{2}\right)^5\)
T6 = 126 . \(\frac{2^4 \cdot x^4}{3^4} \cdot \frac{3^5 y^5}{2^5}\)
T6 = 126 . \(\frac{3 x^4 y^5}{2}\)
T6 = 189 . x4y5

Question 2.
Find the 3rd term from the end in the expansion of \(\left(x^{-\frac{2}{3}}-\frac{3}{x^2}\right)^8\).
Solution:
Given \(\left(x^{-\frac{2}{3}}-\frac{3}{x^2}\right)^8\)
Here, x = \(\mathrm{x}^{\frac{-2}{3}}\); a = \(\frac{-3}{x^2}\); n = 8
r + 1 = 7
⇒ r = 6
The general term th the expansion of (x + a)n is
Tr + 1 = \({ }^n C_r\) xn – r ar.
The 7th term in the expansion of \(\left(x^{\frac{-2}{3}}-\frac{3}{x^2}\right)^8\) is
T6 + 1 = \({ }^8 C_6\left(x^{\frac{-2}{3}}\right)^{8-6} \cdot\left(\frac{-3}{x^2}\right)^6\)
T7 = 28 \(\left(x^{\frac{-2}{3}}\right)^2\left(\frac{3}{x^2}\right)^6\)
= 28 . \(x^{-4} \cdot \frac{3^6}{x^{12}}\)
= 28 . 36 . \(\frac{-40}{x^3}\)
The 3rd term from the end is 28 . 36 . \(x^{\frac{-40}{3}}\).

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 3.
Find the 4th term from the end ”n the expansion of (2a + 5b)8.
Solution:
Given (2a + 5b)8
Here x = 2a, a = 5b, n = 8.
The expansion has 9 terms so that, the fourth term from the end is 6th term from the beginning, in the expansion of (2a + 5b)8.
r + 1 = 6
⇒ r = 5
The general term in the expansion of (x + a)n is Tr + 1 = \({ }^n \mathrm{C}_r\) xn – r ar.
The r term in the expansion of (2a + 5b)8 is
T5 + 1 = \({ }^8 C_5\) (2a)8 – 5 (5b)5
= \({ }^8 C_5\) (2a)3 (5b)6
= \({ }^8 C_5\) . 23 . a3 . 53 . b5
∓ The 4th term from the end is \({ }^8 C_5\) . 23 . 55 . a3 . b5.

Question 4.
Find the middle term (s) in the expansion of \(\left(\frac{3 x}{7}-2 y\right)^{10}\) [March’12, May ā€˜10]
Solution:
Given \(\left(\frac{3 x}{7}-2 y\right)^{10}\)
Here x = \(\frac{3 x}{7}\); a = – 2y, n = 10
Since, n = 10 is even then
Middle term = \(\frac{n}{2}\) + 1 = \(\frac{10}{2}\) + 1
= 5 + 1 = 6th term.
r + 1 = 6
⇒ r = 5
The general term in the expansion of \(\left(\frac{3 x}{7}-2 y\right)^{10}\) is
Tr + 1 = \({ }^n C_r\) xn – r ar
= \({ }^{10} C_5\left(\frac{3 x}{7}\right)^{10-5}(-2 y)^5\)
6th term of \(\left(\frac{3 x}{7}-2 y\right)^{10}\) is
T5 + 1 = \({ }^{10} C_5\left(\frac{3 x}{7}\right)^5(-2 y)^5\)
= \({ }^{-10} C_5 \frac{3^5}{7^5}\) 25 . x5 . y5.

Question 5.
Find the middle term(s) in \(\left(4 a+\frac{3 b}{2}\right)^{11}\).
Solution:
Given \(\left(4 a+\frac{3 b}{2}\right)^{11}\)
Here x = 4a, a = \(\frac{3}{2}\)b; n = 11
Since, n = 11 is odd,
Middle terms = \(\frac{\mathrm{n}+1}{2}, \frac{\mathrm{n}+3}{2}\) = 6, 7 terms
6th term:
r + 1 = 6
⇒ r = 5
The general term in this expansion ”s
Tr + 1 = \({ }^n C_r\) xn – r . ar
∓ 6th term of given expansion is
T5 + 1 = \({ }^{11} \mathrm{C}_5\) (4a)11 – 5 (\(\frac{3}{2}\)b)5
T6 = \({ }^{11} \mathrm{C}_5\) (4a)6 (\(\frac{3}{2}\)b)5
= \({ }^{11} C_5 \frac{4^6 \cdot 3^5}{2^5} \cdot a^6 \cdot b^5\)
T6 = \({ }^{11} C_5 \frac{2^{12} \cdot 3^5}{2^5} \cdot a^6 b^5\)
= \({ }^{11} \mathrm{C}_5\) . 27 . 35 . a6 . b5

7th term:
r + 1 = 7
⇒ r = 6
The general term in this expansion is
Tr + 1 = \({ }^n C_r\) xn – r . ar
∓ 7th term of given expansion is
T6 + 1 = \({ }^{11} C_5(4 a)^{11-6}\left(\frac{3}{2} b\right)^6\)
T7 = \({ }^{11} C_6(4 a)^5\left(\frac{3}{2} b\right)^6\)
= \({ }^{11} C_6 \frac{4^5 \cdot 3^6}{2^6}\) . a5 . b6
T7 = \({ }^{11} \mathrm{C}_6 \frac{2^{10} \cdot 3^6}{2^6}\) . a5 . b6
= \({ }^{11} \mathrm{C}_6\) 24 . 36 . a5 . b6.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 6.
Find the coefficient of x-7 in \(\left(\frac{2 x^2}{3}-\frac{5}{4 x^5}\right)^7\).
Solution:
Given \(\left(\frac{2 x^2}{3}-\frac{5}{4 x^5}\right)^7\).

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type 1

Question 7.
Find the coefficients of x9 and x10 in the expansion of (2x2 – \(\frac{1}{x}\))20
Solution:
Given (2x2 – \(\frac{1}{x}\))20
Here x = 2x2; a = \(-\frac{1}{x}\); n = 20
Now, the general term in this expansion is
Tr + 1 = \({ }^n C_r\) xn – r . ar
= \({ }^{20} \mathrm{C}_{\mathrm{r}}\) (2x2)20-r (\(-\frac{1}{x}\))r
= \({ }^{20} \mathrm{C}_{\mathrm{r}}\) 220-r x40-2r (- 1)r x– r
= \({ }^{20} \mathrm{C}_{\mathrm{r}}\) 220-r (- 1)r x40-3r …………(1)
I) To find the coefficient of x9:
put 40 – 3r = 9
⇒ 3r = 31
⇒ r = \(\frac{31}{3}\)
Since, r is a positive integer, this is not possible.
This means, that the expansion of \(\left(2 x^2-\frac{1}{x}\right)^{20}\) does not possess x9 term.
This means that,
The coeff. of x9 in the exp. of \(\left(2 x^2-\frac{1}{x}\right)^{20}\) is zero.

ii) To find the coefficient of x10:
Put 40 – 3r = 10 of substituting r = 10 in equation (1), we get
T10 + 1 = \({ }^{20} \mathrm{C}_{10}\) 220-10 (- 1)10 x40-30
T11 = \({ }^{20} \mathrm{C}_{10}\) . 1 . x10
= \({ }^{20} \mathrm{C}_{10}\) 210 x10
The coeff. of \(\left(2 x^2-\frac{1}{x}\right)^{20}\) in the expansion of is \({ }^{20} \mathrm{C}_{10}\) 210.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 8.
Find the term independent of x in the expansion of \(\left(\frac{3}{\sqrt[3]{x}}+5 \sqrt{x}\right)^{25}\).
Solution:
Given, \(\left(\frac{3}{\sqrt[3]{x}}+5 \sqrt{x}\right)^{25}\)
Here, x = \(\frac{3}{\sqrt[3]{x}}\), a = 5√x, n = 25
The general term in this expansion is
Tr + 1 = \({ }^n C_r\) xn-r ar
= \({ }^{25} C_r\left(\frac{3}{\sqrt[3]{x}}\right)^{25-r}(5 \sqrt{x})^r\)
= \({ }^{25} \mathrm{C}_{\mathrm{r}} 3^{25-\mathrm{r}} \mathrm{x}^{\frac{-25+\mathrm{r}}{3}} \cdot 5^{\mathrm{r}} \cdot \mathrm{x}^{\mathrm{r} / 2}\)
= \({ }^{25} \mathrm{C}_{\mathrm{r}} 3^{25-\mathrm{r}} 5^{\mathrm{r}} \times \frac{-25+\mathrm{r}}{3}+\frac{\mathrm{r}}{2}\) ………………..(1)

To find the term independent of x:
i.e., the coeff. of x0 put \(\) = 0
⇒ – 50 + 2r + 3r = 0
⇒ 5r = 50
⇒ r = 10
Substitute r = 10 in equation (1),
T10+1 = \({ }^{25} \mathrm{C}_{10} 3^{15} 5^{10} \mathrm{x}^{\frac{-25+10}{3}+\frac{1}{10}}\)
T11 = \({ }^{25} \mathrm{C}_{10}\) . 315 . 510 . x0
∓ The term independent of x in the given expansion is
T11 = \({ }^{25} \mathrm{C}_{10}\) . 315 . 510 . x0

Question 9.
Find largest binomial coefficients in the expansion of (1 + x)24.
Solution:
Given (1 + x)24
Here, n = 24, an even integer.
∓ The largest binomial coefficient is \({ }^{\mathrm{n}} \mathrm{C}_{\left(\frac{\mathrm{n}}{2}\right)}={ }^{24} \mathrm{C}_{12}\).

Question 10.
If \({ }^{22} \mathrm{C}_{\mathrm{r}}\) is the largest binomial coefficient in the expansion of (1 + x)22 find the value of \({ }^{13} \mathrm{C}_{\mathrm{r}}\). [TS & AP – May 2015; May ’11] [AP. Mar. 2016]
Solution:
Given, (1 + x)22
Here n = 22 an even integer.
∓ The Largest binomial coefficient = \({ }^{\mathrm{n}} \mathrm{C}_{\left(\frac{\mathrm{n}}{2}\right)}={ }^{22} \mathrm{C}_{\left(\frac{22}{2}\right)}={ }^{22} \mathrm{C}_{11}\)
Given that the largest binomial coeff. = \({ }^{22} \mathrm{C}_{11}\)
[∵ \({ }^n C_r={ }^n C_s\)
⇒ n = r + s (or) r = s]
⇒ \({ }^{22} \mathrm{C}_{\mathrm{r}}={ }^{22} \mathrm{C}_{11}\)
⇒ r = 11
Now, \({ }^{13} C_r={ }^{13} C_{11}={ }^{13} C_2\)
= \(\frac{13 \times 12}{2 \times 1}\) = 78.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 11.
If n is a positive integer, then prove that \(\sum_{r=1}^n \mathbf{r} \cdot C_r\) = n . 2n – 1. [May ’97]
Solution:
We know that,
(1 + x)n = C0 + C1x + C2x2 + C3x3 ± …………… + Cnxn ……………(1)
Now differentiating (1) on both sides with respect to ā€˜x’ we get
n(1 + x)n – 1 = 0 + C1(1) + C2(2x) + C3(3x2) + …………….. + Cn (nxn-1)
n(1 + x)n-1 = C1 + 2x . C2 + 3x2 . C3 + nxn – 1Cn
Now, put x = 1 we get
n(2)n – 1 = C1 + 2(1) C2 + 3(1)2C3 + …………… + (n) (1)n – 1 Cn
n2n – 1 = C1 + 2C2 + 3C3 + …………. + nCn
C1 + 2C2 + 3C3 + …………… nCn = n . 2n – 1
\(\sum_{r=1}^n\) r . Cr = n . 2n – 1

Question 12.
If the coefficients of (2r + 4)th, (3r + 4)th terms in the expansion of (1 + x)21 are equal, find ā€˜r’. [TS – Mar, 2015]
Solution:
Given (1 + x)21
The general term in the expansionof (x + a)n is
Tr + 1 = nCr xn – r . ar.
(2r + 4)th term in the expansion of (1 + x)21 is
T(2r + 3) + 1 = \({ }^{21} \mathrm{C}_{2 \mathrm{r}+3}\) (1)21 – (r + 3) (x)2r + 3
T2r + 4 = \({ }^{21} \mathrm{C}_{2 \mathrm{r}+3}\) x2r + 3
The coefficient of (2 + 4)th term is \({ }^{21} C_{2 r+3}\)
(3r + 4)th term in the expansion of (4 + x)21 is .
T(3r + 3) + 1 = \({ }^{21} \mathrm{C}_{3 \mathrm{r}+3}\) (1)21 – (3r + 3) x3r + 3
T3r + 4 = \({ }^{21} \mathrm{C}_{3 \mathrm{r}+3}\) x3r + 3
The coeff. of (3r + 4)th term is \({ }^{21} \mathrm{C}_{3 \mathrm{r}+3}\)
Given that, two coeff. are equal.
\({ }^{21} \mathrm{C}_{2 \mathrm{r}+3}={ }^{21} \mathrm{C}_{3 \mathrm{r}+3}\)
\({ }^n C_r={ }^n C_s\)
⇒ n = r + s (or) r = s
n = r + s
21 = 21 + 3 + 3r + 3
5r = 15
⇒ r = 3

r = s
2r + 3 = 3r + 3
⇒ r = 0.

Question 13.
Prove that C0 + 2 . C1 + 4 . C2 + 8 . C3 + …………. + 2n . Cn = 3n. [AP – Mar. ’15; May ’07] [TS – Mar. ’18]
Solution:
We know that,
(1 + x)n = C0 + C1x + C2x2 + C3x3 + ………….. + Cnxn …………….(1)
put x = 2 in the equation (1) we get,
(1 + 2)n = C0 + C1(2) + C2(2)2 + C3(2)3 + …………….. + Cn.(2)n
∓ C0 + 2 . C1 + 4 . C2 + 8 . C3 + …………… + 2n Cn = 3n .

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 14.
Prove that C0 + 3 . C1 + 32 . C2 + ………….. + 3n . Cn = 4
Solution:
We know that,
(1 + x)n = C0 + C1x + C2x2 + C3x3 + ………….. + Cnxn …………….(1)
Put x = 3 in equation (1), we get
(1 + 3)n = C0 + C1 (3) + C2 (3)2 + C3 (3)3 + ………………… + Cn . 3n
C0 + 3 . C1 + 32 . C2 + 33 . C3 + …………… + 3n Cn = 4n.

Question 15.
Prove that \(\frac{\mathrm{C}_1}{\mathrm{C}_0}+2 \cdot \frac{\mathrm{C}_2}{\mathrm{C}_1}+3 \cdot \frac{\mathrm{C}_3}{\mathrm{C}_2}+\ldots \ldots\) + \(\mathbf{n} \cdot \frac{\mathbf{C}_{\mathbf{n}}}{\mathbf{C}_{\mathbf{n}-\mathbf{1}}}=\frac{(\mathbf{n}+\mathbf{1})}{2}\)
[March ’88]
Solution:
L.H.S:
\(\frac{\mathrm{C}_1}{\mathrm{C}_0}+2 \cdot \frac{\mathrm{C}_2}{\mathrm{C}_1}+3 \cdot \frac{\mathrm{C}_3}{\mathrm{C}_2}+\ldots \ldots+\mathrm{n} \cdot \frac{\mathrm{C}_{\mathrm{n}}}{\mathrm{C}_{\mathrm{n}-1}}\)
= \(\frac{{ }^n C_1}{{ }^n C_0}+2 \cdot \frac{{ }^n C_2}{{ }^n C_1}+3 \cdot \frac{{ }^n C_3}{{ }^n C_2}+\ldots . .+n \cdot \frac{{ }^n C_n}{{ }^n C_{n-1}}\)
= \(\frac{\mathrm{n}}{1}+2 \cdot \frac{\frac{\mathrm{n}(\mathrm{n}-1)}{1 \cdot 2}}{\frac{\mathrm{n}}{1}}+3 \cdot \frac{\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{1 \cdot 2 \cdot 3}}{\frac{\mathrm{n}(\mathrm{n}-1)}{1 \cdot 2}}+\ldots \ldots+\mathrm{n} \cdot \frac{1}{5}\)

= n + (n – 1) + (n – 2) + …………… + 1
= 1 + 2 + 3 + ………….. + n
= Σn = \(\frac{n(n+1)}{2}\) = R.H.S

Question 16.
Find the number of terms in the expansion of (2x + 3y + z)7. [May ’14, March ’14, ’13] [TS – Mar. 2019]
Solution:
Given (2x + 3y + z)7
Here, n = 7
∓ Number of terms in the expansion of
(2x + 3y + z)7 = \(\frac{(n+1)(n+2)}{2}\)
= \(\frac{(7+1)(7+2)}{2}=\frac{8 \times 9}{2}\) = 36.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 17.
Find the number of terms with non-zero coefficients in (4x – 7y) + (4x + 7y) .
Solution:

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type 2

Question 18.
Find the sum of last 20 coefficients in the expansion of (1 + x)39.
Solution:
The last 20 coefficients in the expansion of (1 + x)39 are \(\).
We know that

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type 3

Question 19.
If A and B are coefficients of xn in the expansion of (1 + x)2n and (1 + x)2n-1 respectively, then find the value of \(\frac{A}{B}\).
Solution:
Coefficient of xn in the expansion of (1 + x)2n is \({ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}\)
Coefficient of xn in the expansion of (1 + x)2n-1 is \({ }^{2 n-1} C_n\)
∓ A = \({ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}\) and B = \({ }^{2 n-1} C_n\)
∓ \(\frac{A}{B}=\frac{{ }^{2 n} C_n}{2 n-1}=\frac{\frac{2 n !}{n ! n !}}{\frac{(2 n-1) !}{(n-1) ! \cdot n !}}\)
= \(\frac{2 n !}{(2 n-1) ! n !} \cdot(n-1) !=\frac{2 n}{n}\)
⇒ \(\frac{\mathrm{A}}{\mathrm{B}}\) = 2.

Question 20.
Find the set E of x for which the binomial expansion (2 + 3x)-2/3. [March ā€˜11, ā€˜06] [TS – Mar. 2016]
Solution:
Given, (2 + 3x)-2/3 = 2-2/3 (1 + \(\frac{3}{2}\)x)-2/3
The binomial expansion of (2 + 3x)-2/3 is valid when
\(\left|\frac{3 x}{2}\right|<1 \Rightarrow|x|<\frac{2}{3}\)
⇒ x ∈ (\(\left.\frac{-2}{3}, \frac{2}{3}\right)\)).

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 21.
Find the set E of x for which the binomial expansion of (3 – 4x)3/4. [AP-Mar. 2017] [May ’12]
Solution:
Given (3 – 4x)3/4 = \(3^{\frac{3}{4}}\left(1-\frac{4 x}{3}\right)^{\frac{3}{4}}\)
The binomial expansion of (3 – 4x)3/4 is valid
when \(\left|\frac{-4 x}{3}\right|<1 \Rightarrow\left|\frac{4 x}{3}\right|<1\)
x < \(\frac{3}{4}\)
x ∈ (\(\frac{-3}{4}\), \(\frac{3}{4}\))
E = (\(\frac{-3}{4}\), \(\frac{3}{4}\))

Question 22.
Find the 7th term of \(\left(1-\frac{x^2}{3}\right)^{-4}\).
Solution:
Given \(\left(1-\frac{x^2}{3}\right)^{-4}\)
Comparing this with (1 + x)n
where x = , n = – 4,
r + 1 = 7
⇒ r = 6
The general term in the expression of (1 + x)n is
Tr + 1 = \(\frac{n(n-1) \ldots \ldots(n-r+1)}{5 n}\) . x
= \(\frac{(-4)(-4-1)(-4-2)(-4-3)(-4-4)(-4-5)}{6 !}\) \(\left(\frac{-x^2}{3}\right)^6\)
= \(\frac{(-4)(-5)(-6)(-7)(-8)(-9)}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot \frac{x^{12}}{3^6}\)
= 84 . \(\frac{28}{243}\) x12.

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 23.
Find the set E of the values of x for which, the binomial expansion (2 + 5x)-1/2 is valid. [TS – Mar. 2017]
Solution:
Given (2 + 5x)-1/2
The binomial expansion of (2 + 5x)-1/2 is valid when
|\(\frac{5 x}{2}\)| < 1
|x| < \(\frac{2}{5}\)
x ∈ (- \(\frac{2}{5}\), \(\frac{2}{5}\)).

Question 24.
Find the 7th term in the expansion of \(\left(\frac{4}{x^3}+\frac{x^2}{2}\right)^{14}\). [AP – May 2016]
Solution:
\({ }^{14} C_6 \cdot \frac{2^{10}}{x^{12}}\)

Question 25.
Find the 5th term in the expansion of (3x – 4y)7.
Solution:
241920 x3 y4.

Question 26.
Find the middle term(s) in \(\left(\frac{3}{a^3}+5 a^4\right)^{20}\).
Solution:
\({ }^{20} \mathrm{C}_{10}\) (15)10 (a)10.

Question 27.
Find the middle term(s) in the expansion of (4x2 + 5x3)17.
Solution:
\({ }^{17} \mathrm{C}_{9}\) 48 . 59 . x43

Question 28.
Find the coefficient of x11 in (2x2 + \(\frac{3}{x^3}\))13
Solution:
\({ }^{13} \mathrm{C}_3\) 210 . 33

Question 29.
Find the term independent of x in the expansion of \(\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^2}\right)^{10}\).
Solution:
\(\frac{5}{4}\)

Question 30.
Find the largst binomial coefficients in the expansion of (1 + x)19.
Solution:
\({ }^{19} \mathrm{C}_9 ;{ }^{19} \mathrm{C}_{10}\)

Question 31.
If the coefficients of (2x + 4)th, (r – 2)th terms in the expansion of (1 + x)21 are equal find ā€˜r’.
Solution:
6

TS Inter Second Year Maths 2A Binomial Theorem Important Questions Very Short Answer Type

Question 32.
Find the set E of x for which the binomial expansion (7 + 3x)-5. [May ’09]
Solution:
\(\left(\frac{-7}{3}, \frac{7}{3}\right)\)

Question 33.
Find the set E of x for which the binomial expansion (7- 4x)-5 is valid.
Solution:
\(\left(\frac{-7}{4}, \frac{7}{4}\right)\)

Question 34.
Find the 6th term of \(\left(1+\frac{x}{2}\right)^{-5}\).
Solution:
\(\frac{-63}{16}\) x5.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Telangana TSBIEĀ TS Inter 2nd Year Chemistry Study Material 13th Lesson Organic Compounds Containing Nitrogen Textbook Questions and Answers.

TS Inter 2nd Year Chemistry Study Material 13th Lesson Organic Compounds Containing Nitrogen

Very Short Answer Questions (2 Marks)

Question 1.
Write the IUPAC names of the following compounds and classify them into primary, secondary and tertiary amines. [TS 15]
(i) (CH3)3C NH2
(ii) CH3(CH2)2 NH2
(iii) (CH3 CH2)2 NCH3
Answer:
(i) 1,1- dimethylethanamine. It is a primary amine.
(ii) Propan -1 – amine. It is a primary amine.
(iii) N – methyl – N – ethylethanamine. It is a tertiary amine.

Question 2.
Explain why ethylamine is more soluble in water whereas aniline is not soluble.
Answer:
Ethylamine is soluble in water because it can form hydrogen bonds with water mole-cules. The phenyl group in aniline is bulky and hydrophobic. It opposes the formation of hydrogen bonds with water molecules. Hence, aniline is insoluble in water.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 3.
Why aniline does not undergo Friedel – Craft’s reaction ?
Answer:
Aniline does not undergo Friedel – Craft’s reaction (alkylation and acylation)’due to salt formation with AlCl3, the Lewis acid, which is used as a catalyst. Because of this, nitrogen of aniline acquires positive charge and hence acts as a strong deactivation group for further reaction.
C6H5 NH2 + AlCl3 → [C6H5NH2]+ [AlCl3]

Question 4.
Gabriel phthalimide synthesis exclusively forms primary amines only. Explain.
Answer:
Because there is only one hydrogen bonded to the nitrogen of pathalimide, only one alkyl group can be placed on the nitrogen. This means that the gabriel synthesis can be used only for the preparation of primary amines.

Question 5.
Arrange the following bases in decreasing order of pKb values.
C2H5NH2, C2H5NHCH3, (C2H5)2 NH and C6H5NH2.
Answer:
C6H5NH2 > C2H5NHCH3 > C2H5NH2 > (C2H5)2 NH
pKb Values: 9.38 9.30 3.29 3.00

Question 6.
Arrange the following bases in increasing order of their basic strength. Aniline, p – nitro- aniline and p – toluidine.
Answer:
p – nitro Aniline < aniline < p – toluidine.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 7.
Write equations for carbylamine reaction of any one aliphatic amine. [TS Mar. 19; (IPE 14)]
Answer:
A primary amine, for example, n – butyla-mine forms foul smelling isocyanide or carbylamine when heated with chloroform and alcoholic potassium hydroxide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 1

Question 8.
Give structures of A, B and C in the following reaction.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 2
Answer:
A is C6H5CN Phenylcyanide or Benzonitrile
B is C6H5COOH Benzoic acid
C is C6H5CONH2 Benzamide

Question 9.
Accomplish the following conversions : [IPE 14]
i) Benzoic acid to Benzamide
ii) Aniline to p – Bromoaniline
Answer:
i) Benzoic acid reacts with ammonia to give ammonium benzoate which on fur-ther heating at temperature gives benza-mide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 3

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

ii) The amino group of aniline is protected by acetylation. Acetanilide, so obtained, is reacted with bromine in acetic acid followed by hydrolysis to get p – bromoaniline.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 4

Question 10.
Why cannot aromatic primary amines be prepared by Gabriel phthalimide synthesis ?
Answer:
Aromatic primary amines cannot be prepared by Gabriel phthalimide synthesis because arylhalides do not undergo nucleophilic substitution with the anion formed by phthalimide.

Short Answer Questions (4 Marks)

Question 11.
Write the IUPAC names of the following compounds.
i) CH3CH2NH CH2CH2CH3
ii) PhCH2 CN
iii)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 5
iv)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 6
Answer:
i) N – Ethylpropanamine
ii) Phenylethanenitrile or Benzylcyanide
iii) 3 – Bromoaniline or 3 – Bromobenzenamine
iv) 4 – Bromophenyl methylisocyanide

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 12.
Give one chemical test to distinguish between the following pairs of compounds.
i) Methylamine and dimethylamine
ii) Aniline and N – methylaniline
iii) Ethylamine and aniline
Answer:
i) Methylamine on heating with chloroform and alcoholic potassium hydroxide forms foul smelling isocyanide or carbylamine. Dimethylamine does not give this test.

ii) Aniline (a primary amine) gives positive carbylamine test N – methylaniline (a secondary amine) does not give this test. Thus when aniline is heated with chloroform and alcoholic KOH forms fouls melling phenylisocyanide or carbylamine.

iii) Ethylamine reacts with nitrous acid to give nitrogen gas and ethylalcohol. Aniline reacts with nitrous acid (NaNO2 + HCl) at low temperatures (0 – 5°C) to form diazonium salt.

Question 13.
Account for the following :
i) pKb of aniline is more than that of methylamine.
ii) Reduction of alkylcyanide forms primary amine whereas alkylisocyanide forms secondary amine.
Answer:
i) In aniline the -NH2 group is directly attached to the benzene ring. It results in the unshared electron pair on nitrogen atom to be in conjugation with the benzene ring and thus making it less available for protonation. Such a situation is absent in methylamine. Hence PKb value of aniline is greater than that of methylamine.

ii) In, alkylcyanides, the alkylgroup is linked to the carbon of the cyanide ion while in isocyanides the alkyl group is linked to the nitrogen of cyanide.
R – C ≔ N R – N ≔ C
Hence, alkylcyanide gives primary amine whereas alkyl isocyanide gives secondary amine on reduction.

Question 14.
How do you prepare the following?
i) N, N – Dimethyipropanamine from ammonia
ii) Propanamine from chloroethane
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 7

Question 15.
Compare the basicity of the following in gaseous and in aqueous state and arrange them in increasing order of basicity. [TS 15[
CH3NH2, (CH3)2NH, (CH3)3N and NH3
Answer:
The alkyl (methyl) group has electron – releasing inductive effect. It pushes the electrons towards nitrogen and makes the electron pair more available for sharing with the proton of the acid. Hence, in the gaseous state the basicity of the amines follows the order.
(CH3)3N > (CH3)2NH > CH3NH2 > NH3
In the aqueous state the basicity of the amines depends upon inductive effect, solvation effect and steric hindrance of the alkyl group. Hence, the order basic strength of the methyl substituted amine in the aqueous solution is
(CH3)2NH > CH3NH2 > (CH3)3N > NH3

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 16.
How do you carryout the following conversions ?
i) N – Ethylamine to N, N – Diethyl propanamine
ii) Aniline to p – Aminobenzene sulphonamide
Answer:
i) N – ethylamine is first reacted with propylchloride to convert it to N – ethyl propanamine. It is then reacted with ethyl chloride to convert it to N, N – Diethyl propanamine.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 8
ii) Aniline is converted to acetanilide by reaction with acetylchloride. Acetanilide is treated with chlorosulphonic acid and the product on reaction with ammonia followed by hydrolysis gives sulphanilamide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 9

Question 17.
Explain with a suitable example how benzene sulphonylchloride can distinguish primary, secondary and tertiary amines.
Answer:
Benzene sulphonylchloride (Hinsberg reagent) reacts with a primary amine, ethylamine, to give N – ethylbenzene sulphonyl amide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 10
This compound contains hydrogen attached to nitrogen. It is acidic in nature and hence it is soluble in sodium hydroxide solution.

When benzene sulphonyl chloride reacts with a secondary amine, for example diethyl amine, to give N, N – diethyl benzene sulphonamide. It does not contain hydrogen attached to nitrogen. It is not acidic and hence it is insoluble in sodium hydroxide solution.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 11
Tertiary amines do not react with Hinsberg reagent.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 18.
Write the reactions of
i) aromatic and
ii) aliphatic primary amines with nitrous acid.
Answer:
Aromatic primary amines react with nitrous acid at low temperatures (0 – 5°C) to form diazonium salts.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 12
Aliphatic primary amines react with nitrous acid to form alcohol and nitrogen gas.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 13

Question 19.
Explain why amines are less acidic than alcohols of comparable molecular masses.
Answer:
The acidic character of alcohols is due to the polar nature of O – H bond. The polarity of the N – H bond in amines is less than that of the O – H bond in alcohols of comparable molecular mass. Hence amines are less acidic than alcohols of comparable molecular mass.

Question 20.
How do you prepare Ethyl cyanide and Ethyl isocyanide from a common alkyl halide. [IPE 14]
Answer:
Ethyl chloride reacts with ethanolic potassium cyanide to form ethyl cyanide as the major product. However, when ethyl chloride reacts with ethanolic silver cyanide ethyl isocyanide will be the major product.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 14

Long Answer Questions (8 Marks)

Question 21.
An aromatic compound ‘A’ on treatment with aqueous ammonia and heating forms compound ‘B’ which on heating with Br2 and KOH forms compound ‘C’ of molecular formula C6H7N. Write the structures and IUPAC names of compounds A, B and C.
Answer:
The final product ‘C’ with molecular formula C6H7N is Aniline. The sequence of reactions can be explained as follows.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 15
Compound A is Benzoic acid. On treatment with aqueous ammonia gives ammonium benzoate which on heating gives Benzamide (B). Benzamide on heating with bromine and potassium hydroxide (Hofmann hypobromite reaction) gives Aniline (C) with molecular formula C6H7N.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 22.
Complete the following conversations.
i) CH3NC + HgO → ?
ii) ? + 2H2O → CH3NH2 + HCOOH
iii) CH3CN + C2H5 Mg Br → ? TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 16
iv) CH3 CH2 NH2 + CHCl3 + KOH TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 17 ?
v) TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 18
Answer:
i) HgO is a mild oxidising agent. It converts isocyanides to isocyanates. Thus, methyl isocyanide is converted to methyl isocyanate.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 19

ii) Isocyanides on hydrolysis give amines and formic acid.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 20

iii) Ethyl magnesium bromide adds on to methylcyanide or acetonitrile to give an addition product which on hydrolysis forms acetone.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 21

iv) Primary amine on heating with chloroform and alcoholic KOH gives foul smelling isocyanide or carbylamine.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 22

v) Aniline on treatment with bromine water gives a white precipitate of 2,4,6 – tribromoaniline.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 23

Question 23.
i) Write the structures of different isomeric amines corresponding to the molecular formula C9H13N.
ii) What reducing agents can bring about reduction of nitrobenzene ?
iii) Write the product formed when benzyl chloride is reacted with ammonia followed by treatment with methyl and ethyl chlorides.
Answer:
i)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 24
ii) The following reagents can bring about reduction of nitrobenzene.
a) Sn / HCl (or)Fe/HCl
b)Zn/NaOH
c) Zn / NH4Cl/ H2O
d)H2/Pd

iii)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 25

Question 24.
i) Identify the amide and cyanide which on reduction with appropriate reducing agent give n – butylamine.
ii) Write the mechanism of Hoffmann bromamide reaction.
Answer:
i) Butanamide, CH3CH2CH2CONH2 on reduction with lithium aluminium hydride yields n – butylamine
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 26
n – propylcyanide on reduction LiAlH4 or Na(Hg)/C2H5OH gives n – Butylamine.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 27

 

ii) Hoffmann bromamide reaction mechanism :
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 28

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 25.
How do you make the following conversions ?
i) Chlorophenylmethane to phenylacetic acid
ii) Chlorophenylmethane to 2 – phenylethanamine
Answer:
i)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 29
Chlorophenyl methane is reacted with potassium cyanide and converted to benzylcyanide which on hydrolysis gives phenylacetic acid.

ii)
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 30
Chlorophenylmethane is reacted with KCN and converted to benzylcyanide which on reduction gives 2 – phenylethanamine.

Question 26.
Identify the starting amide which gives p – methyl aniline on reaction with bromine and sodium hydroxide and write all the steps involved in the reaction.
Answer:
p – methylbenzamide on reaction with bromine and sodium hydroxide (Hoffmann bromamide degradation reaction) gives p – methylaniline.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 31
The following steps are involved in the reaction.
2NaOH + Br2 → NaBr + NaOBr + H2O
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 32

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 27.
Explain wiy the order of basicity methylamine, N, N – dimethylamine and N, N, N – trimethylamine changes in gaseous and aqueous medium.
Answer:
In the gaseous state the basicity of the methyl substituted amines follows the order.
N, N, N – trimethylamine > N, N – dimethylamine > methylamine

It can be explained as follows. The methyl group has electron releasing nature. It pushes electrons towards nitrogen and thus makes the unshared electron pair on nitrogen more available for sharing with the proton of the acid.

Thus the methyl substituted ammonium gets stabilised due to to the dispersal of the positive charge by the +1 effect of thecilkyl group. Thus the basic nature of the methyl substituted amines increases with the increase in the number of methyl groups. This trend is followed in the gaseous phase.

In the aqueous phase, the substituted ammonium cations get stabilised not only by electron releasing effect of the methyl group but also by solvation with water molecules. Another factor that decides the basic strength of the alkylamines in aqueous state is steric hindrance of the alkyl groups:

Hence, due to the presence of two electron releasing methyl groups attached to the nitrogen atom, dimethylamine is a stronger base than methylamine. If so, trimethyl amine having three methyl groups attached to nitrogen should be expected to be more basic them dimethyl amine. But actually trimethylamine is considerably less basic than dimethyl amine.

Why so ? In methyl amine and dimethylamine the ‘electronic effect’ increases the basic strength of the amine. However, in trimethylamine the over crowding of the three methyl groups attached to nitrogen causes the ‘steric effect’ – to dominate over the ‘electronic effect’. This steric effect retards the protonation of nitrogen which results in an appreciably lower basic strength of trimethylamine. Hence the basic strength of the amines in the aqueous phase follows the order :
(CH3)2 NH > CH3NH2 > (CH3)3N > NH3.

Question 28.
Write the equations involved in the reaction of Nitrous acid with ethylamine and aniline.
Answerw:
Ethyl amine reacts with nitrous acid to form ethyl diazonium salt which being unstable liberates nitrogen gas and forms ethyl alcohol.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 33
Aniline reacts with nitrous acid at low temperatures (0 – 5°C) to form diazonium salt.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 34

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 29.
Explain with equations how methylamine, N, N – dimethylamine and N, N, N-trimethylamine react with benzene sulphonyl chloride and how this reaction is useful to separate these amines.
Answer:
Methylamine reacts with benzene suphonyl chloride to give N – methyl benzene sulphonamide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 35
The hydrogen attached to nitrogen in sulphonamide is strongly acidic due to the presence of strong electron withdrawing sulphonyl group. Hence, it is soluble in alkali, say NaOH solution. N, N – dimethylamine reacts with benzene sulphonyl chloride to give N, N – dimethylbenzene sulphonair; le. Since this compound does not contain any hydrogen atom attached to nitrogen atom it is not soluble in alkali.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 36
N, N, N – trimethylamine does not react with benzene sulphonyl chloride. This property of these three methylamines reacting with benzene sulphonylchloride in a different manner is used for their separation from a mixture.

Question 30.
Explain why aniline in strong acidic medium gives a mixture of Nitro anilines and what steps need to be take to prepare selectively p-nitroaniline.
Answer:
In strongly acidic medium, aniline is protonated to form the anilinium ion which is meta directing. That is why besides the ortho and para derivatives, significant amount of meta derivative is also formed.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 37
However, by protecting the -NH2 group by acetylation reaction with acetic anhydride, the nitration reaction can be controlled and the p-nitro derivative can be obtained as the major product.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 38

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 31.
i) Account for the stability of aromatic diazonium ions when compared to aliphatic diazonium ions.
ii) Write the equations showing the conversion of aniline diazonium chloride to
a) Chlorobenzene, b) Iodobenzene and c) Bromobenzene.
Answer:
i) The relative stability of aromatic diazonium ions can be ascribed to the fact that its structure is a resonance hybrid of the canonical forms involving the participation of the benzene ring.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 39
The hybrid structure shows that: a) benzene ring is deactivated to attack of electrophiles, (b) the C -N bond acquires some double bond character and becomes stronger. Alkyl diazonium ions cannot exhibit such resonance and hence C – N bond in them is weak. That is why they are unstable relative to their aromatic counterparts.

ii) a) Aqueous solution of benzene diazonium chloride when heated with cuprous chloride gives chlorobenzene.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 40
b) Iodobenzene is formed when benzene diazonium chloride solution is treated with potassium iodide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 41
c) Bromobenzene is formed when benzenediazonium chloride solution is treated with hydrobromic acid in the presence of copper power.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 42

Question 32.
Complete the following conversions:
Aniline to i) Fluorobenzene iQ Cyanobenzene iif) Benzene and iv) Phenol
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 43
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 44

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 33.
Explain the following name reactions : [AP 16, 15]
i) Sandmeyer reaction
ii) Gatterman reaction.
Answer:
i) Sandmeyer reaction: The diazonium group of a diazonium salt can be replaced by chlorine (-Cl) or bromine (-Br) by heating the aqueous solution of the diazonium salt with cuprous chloride or cuprous bromide. This reaction is called Sandmeyer reaction.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 45

ii) Gatterman reaction : It is a modification of Sandmeyer reaction. The diazonium group is replaced by – Cl or – Br when the diazonium salt solution is treated with the corresponding halogen acid in the presence of copper powder.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 46

Question 34.
Write the steps involved in the coupling of Benzene diazoniumchloride with aniline and phenol.
Answer:D
iazonium salts react with aromatic amines and phenols to give azocompounds having the general formula Ar – N = N – Ar. The reaction is known as coupling reaction. The coupling of benzene diazonium chloride with phenols is carried out in mild alkaline solution and with amines in weakly acidic medium.
Coupling with aniline :
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 47
Coupling with Phenol :
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 48

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 35.
Write the equations involved in the conversion of acetamide and propanaldehydeoxime to methyl cyanide and ethyl cyanide respectively.
Answer:
Acetamide is converted to methyl cyanide by heating it with benzene sulphonyl chloride in pyridine at 70°C.
CH3 CO NH2 + C6H5SO2Cl TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 49 > CH3CN + C6H5SO3H + HCl
Propanaldehydeoxime is converted to ethylcyanide by dehydrating with acetic anhydride.
CH3 – CH2 – CH = NOH + (CH3CO)2O → CH3 – CH2 – CN + 2CH3COOH

Intext Questions – Answers

Question 1.
Classify the following amines as primary, secondary or tertiary.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 50
iii) (C2H5)2 CHNH2
iv) (C2H5)2 NH
Answer:
i) and
iii) are primary amines
ii) is a tertiary amine
iv) is a secondary amine

Question 2.
i) Write structures of different isomeric amines to the molecular formula, C4H11N
ii) Write IUPAC names of all the isomers.
iii) What types of isomerism is exhibited by different types of amines ?.
Answer:
Molecular formula C4H11N
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 51

ii) IUPAC names ;
a) Butan-1-amine
b) 2-methyl propanamine
c) 2-methyl-propan-2-amine
d) N-methyl propan-1-amine
e) N-ethyl ethanamine
f) N-methyl-1-methylethanamine
g) N, N-Dimethylmethanamine

iii) Primary amines (a), (b) and (c) exhibit chain isomerism.
Secondary amines (a), (e) and (b) exhibit metamerism.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 3.
How will you convert
i) Benzene into aniline
ii) Benzene into N, N – dimethylaniline
iii) Cl – (CH2)4 – Cl into hexan -1, 6 – diamine ?
Answer:
i) Benzene is first converted into nitrobenzene by nitration. Nitrobenzene on reduction with tin and hydrohloric acid gives aniline.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 52

ii) Benzene is converted into aniline by nitration followed by reduction. Aniline on heating with excess of methyliodide gives N, N – Dimethylaniline.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 53

iii) Cl – (CH2)4 – Cl is converted to NC – (CH2)4 – CN by reacting with ethanolic potassium cyanide. NC – (CH2)4 – CN on reduction with LiAlH4 or sodium and alcohol gives H2N (CH2)6 NH2.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 54

Question 4.
Arrange the following in increasing order of their basic strength :
i) C2H5NH2, C6H5NH2, NH3,C6H5CH2 NH2 and (C2H5)2NH
ii) C2H5NH2, (C2H5)2 NH, (C2H5)3N, C6H5NH2
iii) CH3NH2, (CH3)2 NH, (CH3)3N, C6H5NH2, C6H5CH2NH2
Answer:
i) C6H5NH2 < NH3 < C6H5CH2NH2 < C2H5NH2 < (C2H5)2 NH
ii) C6H5NH2 < C2H5NH2 < (C2H5)3N < (C2H5)2 NH
iii) C6H5NH2 < C6H5CH2NH2 < (CH3)3 N < CH3NH2 < (CH3)2 NH

Question 5.
Complete the following acid – base reactions and name the products.
i) CH3CH2CH2NH2 + HCl →
ii) (C2H5)3N + HCl →
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 55

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 6.
Write reactions of the final alkylation product of aniline with excess of methyl iodide in the presence Of sodium carbonate solution.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 56
Thus, when aniline is treated with an excess of methyl iodide in a basic medium, the final product obtained is N, N-dimethylaniline.

Question 7.
Write chemical reaction of aniline with benzoyl chloride and write the name of the product obtained.
Answer:
Aniline reacts with benzoylchloride to give benzanilide.
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 57

Question 8.
Write structures of different isomers corresponding to the molecular formula, C3H9N. Write IUPAC names of the isomers which will liberate nitrogen gas on treatment with nitrous acid.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 58
Propanamine (a) and 1- methylethenamine (b) being aliphatic primary amines liberate nitrogen gas on treatment with nitrous acid.

TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen

Question 9.
Convert
i) 3-Methylaniline into 3-nitrotoluene
ii) Aniline into 1, 3, 5 – tribromobenzene.
Answer:
TS Inter 2nd Year Chemistry Study Material Chapter 13 Organic Compounds Containing Nitrogen 59

TS Inter 1st Year Maths 1B Applications of Derivatives Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 10 Applications of Derivatives will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Applications of Derivatives Formulas

→ If y = f(x) is a differentiable function of x and Ī”x is a small change in ā€˜x’ then the

  • actual change in y is Ī”y = f (x + Ī”x) – f(x)
  • the differential of y is dy = f'(x) Ī”x

→ The approximate value of f(x) in a neighbourhood of Ī”x is f(x + Ī”x) – f (x) + f'(x) Ī”x.

→ If error in x of y = f(x) is Ī”x then

  • Ī”y is the approximate error in y.
  • \(\frac{\Delta \mathrm{y}}{\mathrm{y}}\) is called the relative error in v and
  • \(\frac{\Delta \mathrm{y}}{\mathrm{y}}\) Ɨ 100 is the percentage error in y.

→ The slope of the curve y = f(x) at the point P(x1, y1) is \(\left(\frac{d y}{d x}\right)_{\left(x_1 \cdot y_1\right)}\) = m = f'(x1).

→ If Īø is the angle between the curves at y = f(x) and y = g(x) at the point of intersection P(x1, y1) then tan Īø = \(\frac{m_1-m_2}{1+m_1 m_2}\) where m1 = f'(x1) and m2 = g'(x)
If m1 = m2, then the two curves touch each other at (x1, y1) and if m1m2 = – 1, the two curves are said to be orthogonal.

TS Inter 1st Year Maths 1B Applications of Derivatives Formulas

→ If m = \(\left(\frac{d y}{d x}\right)_{\left.i x_1, y_1\right)}\) = f'(x,) is the slope of the curve at the point P(x1, y1) on y = f(x) then

  • The length of the tangent to the curve at P is \(\frac{y_1 \sqrt{1+\left[f^{\prime}\left(x_1\right)\right]^2}}{f^{\prime}\left(x_1\right)}\)
  • The length of the normal to the curve at P is y1\(\sqrt{1+\left[f\left(x_1\right)\right]^2}\)
  • The length of the subtangent to the curve at P = \(\left|\frac{y_1}{f^{\prime}\left(x_1\right)}\right|\)
  • The length of the subnormal to the curve at P is |y1f(x1)|.

→ The rate of change of the function y = f(x) with respect to ‘t’ is \(\frac{d y}{d x}\)

→ If s = f(t) is the functional relation between the distance ā€˜s’ and time ā€˜t’, then the velocity of the body at time ā€˜t’ is \(\frac{d s}{d x}\) = v and the acceleration of the body at time ā€˜t’ is \(\frac{d^2 s}{d t^2}=\frac{d v}{d t}\)

→ If a function ‘f’ is increasing and differentiable at a’ ⇔ f'(a) > 0.

  • A differentiable function is said to be decreasing at ā€˜a’ ⇔ f'(a) < 0.
  • A differentiable function is said to be stationary at ā€˜a’ ⇔ f'(a) = 0.

→ A differentiable function f(x) in the interval which has f'(x) and f”(x) at ā€˜a’ and if

  • f’(a) = 0, f”(a) < 0, then f(a) has local maxima.
  • f'(a) = 0. f”(a) > 0. then f(a) has local minima.

→ Rolle’s Mean Value Theorem : If a function ā€˜f defined over [a, b] is such that

  • f is continuous over [a, b]
  • f is differentiable on (a. b)
  • f(a) = f(b). Then ∃ a point c ∈ (a, b) such that f'(c) = 0.

→ Lagrange’s Mean Value Theorem : If a function f is defined over [a, b] is such that

  • f is continuous over [a, b] .
  • f is differentiable over (a. b) then ∃ a point c ∈ (a, b) such that f’(c) = \(\frac{f(b)-f(a)}{b-a}\)

TS Inter 1st Year Maths 1B Applications of Derivatives Formulas

→ Mensuration fundamentals:
1. If r is the radius, x is the diameter, P is the perimeter and A is the area of the circle then

  • A = Ļ€r or A = \(\frac{\pi x^2}{4}\).
  • P = 2Ļ€r = Ļ€x.

2. If ā€˜r’ is the radius, l is the length of the arc and 0 is the angle then

  • Area A = \(\frac{1}{2}\) lr = \(\frac{1}{2}\) r2Īø
  • Perimeter P = l + 2r = r (Īø + 2)

3. If r is the radius, h is the height of the cylinder then

  • Lateral surface area = 2Ļ€rh
  • Total surface area S = 2Ļ€rh + 2Ļ€r2
  • Volume V = Ļ€r2h

4. If r is the radius, l is the slant height, h is the height and α is the vertical angle of the cone, then

  • l2 = r2 + h2
  • Lateral surface area = Ļ€rl
  • Total surface area S = Ļ€rl + Ļ€r2
  • Volume V = \(\frac{1}{3}\)Ļ€r2H

5. If L is the length, T is the period of oscillation of a simple pendulum and g is the acceleration due to gravity then T = 2Ļ€\(\sqrt{\frac{l}{g}}\).

6. If r is the radius of sphere then

  • Surface area = S = 4 Ļ€r2
  • Volume V = \(\frac{4}{3}\)Ļ€r3

7. Let x be the side of a cube then surface area of the cube is 6x2 and volume of the cube is x3.

TS Inter 1st Year Maths 1B Differentiation Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 9 Differentiation will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Differentiation Formulas

→ Formula for finding derivative f'(x) of a function y = f(x) using the definition is
\(\frac{\mathrm{dy}}{\mathrm{dx}}\) = f'(x) = \({Lt}_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
Derivative of a function at a point ‘a’ f'(a) = \({Lt}_{x \rightarrow a}\left[\frac{f(x)-f(a)}{x-a}\right]\)

→ \(\frac{\mathrm{d}}{\mathrm{dx}}\)(u ± v) = \(\frac{d u}{d x} \pm \frac{d v}{d x}\)

→ \(\frac{d}{d x}\)(uv) = u.\(\frac{d v}{d x}\) + v.\(\frac{d u}{d x}\)

→ \(\frac{d}{d x}\) (uvw) = uv \(\frac{d}{d x}\)(w) + uw\(\frac{d}{d x}\)(v) + vw\(\frac{d}{d x}\)(u)

→ \(\frac{d}{d x}\left(\frac{u}{v}\right)=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^2}\)

→ \(\frac{d}{d x}\)(xn) = n.xn-1

→ \(\frac{d}{d x}\left(\frac{1}{x^n}\right)=\frac{-n}{x^{n+1}}\)

TS Inter 1st Year Maths 1B Differentiation Formulas

→ \(\frac{d}{d x}\)(log x) = \(\frac{1}{x}\), \(\frac{d}{d x}\)(loga x) = loga e

→ \(\frac{d}{d x}\)(ex) = ex, \(\frac{d}{d x}\)(ax) = ax loge a

→ \(\frac{d}{d x}\)(sin x) = cos x

→ \(\frac{d}{d x}\)(cos x) = -sin x

→ \(\frac{d}{d x}\)(tan x) = sec2 x

→ \(\frac{d}{d x}\)(cot x) = -cosec2 x

→ \(\frac{d}{d x}\)(sec x) = sec x tan x

→ \(\frac{d}{d x}\)(cosec x) = -cosec x cot x

→ \(\frac{d}{d x}\)(sin-1x) = \(\frac{1}{\sqrt{1-x^2}}\)

→ \(\frac{d}{d x}\)(cos-1x) = \(-\frac{1}{\sqrt{1-x^2}}\)

→ \(\frac{d}{d x}\)(tan-1x) = \(\frac{1}{1+x^2}\)

→ \(\frac{d}{d x}\)(cot-1x) = \(-\frac{1}{1+x^2}\)

→ \(\frac{d}{d x}\)(sec-1x) = \(\frac{1}{|x| \sqrt{x^2-1}}\)

→ \(\frac{d}{d x}\)(cosec-1x) = \(-\frac{1}{|x| \sqrt{x^2-1}}\)

→ \(\frac{d}{d x}\)(sinh-1x) = \(\frac{1}{\sqrt{1+x^2}}\)

→ \(\frac{d}{d x}\)(cosh-1x) = \(\frac{1}{\sqrt{x^2-1}}\)

TS Inter 1st Year Maths 1B Differentiation Formulas

→ \(\frac{d}{d x}\)(tanh-1x) = \(\frac{-1}{1-x^2}\)

→ \(\frac{d}{d x}\)(coth-1x) = \(\frac{1}{1-x^2}\)

→ \(\frac{d}{d x}\)(sech-1x) = \(-\frac{1}{|x| \sqrt{1-x^2}}\)

→ \(\frac{d}{d x}\)(cosech-1x) = \(\frac{1}{|x| \sqrt{x^2+1}}\)

→ Logarithmic differentiation: If y = f(x)g(x) > then log y = g(x) log f(x)
⇒ \(\frac{1}{y} \frac{d y}{d x}\) = g(x).\(\frac{1}{f(x)}\)f'(x) + log[f(x)]g'(x)

→ Derivative of one function w.r.t. another function: If y = f(x); z = g(x) then \(\frac{d y}{d z}=\frac{d f}{d g}=\frac{f^{\prime}(x)}{g^{\prime}(x)}\), It is called as chain rule.

→ Parametric differentiation: If x = f(t), y = g(t) then \(\frac{d y}{d x}=\frac{d y}{\frac{d t}{d t}}=\frac{g^{\prime}(t)}{f^{\prime}(t)}\), Itis called as chain rule.
\(\frac{d^2 y}{d x^2}=\frac{d}{d x}\left(\frac{d y}{d x}\right)=\left[\frac{d}{d t}\left(\frac{d y}{d x}\right)\right]\left(\frac{d t}{d x}\right)\)

TS Inter 1st Year Maths 1B Limits and Continuity Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 8 Limits and Continuity will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Limits and Continuity Formulas

→ If a variable x approaches a value a’ from the left i.e., through values just smaller than ’a’ than the limit of f defined is called the left limit of f(x) denoted by \(\lim _{x \rightarrow a^{-}}\)f(x)
\(\lim _{x \rightarrow a^{-}}\)f(x)= \(\lim _{h \rightarrow 0^{+}}\)f(a – h) = \(\lim _{x \rightarrow 0}\)f(a – x) (∵ x → a ⇒ x < a)

→ If x approaches a’ from the right i.e., through the values just greater than ‘a’ then the limit of f defined is called the right limit of f(x) denoted by \(\lim _{x \rightarrow a^{+}}\)(x).
\(\lim _{x \rightarrow a^{+}}\) f(x)= \(\lim _{h \rightarrow 0^{+}}\) f(a + h)= \(\lim _{x \rightarrow 0}\)f(a + x) (∵ x → a+ ⇒ x > a)

→ Suppose f is defined in a deleted neighbourhood of ‘a’ and l e R then
\(\lim _{x \rightarrow a}\)f(x) = l ⇒ \(\lim _{x \rightarrow a^{+}}\)f(x) = \(\lim _{x \rightarrow a^{-}}\)f(x) = l

TS Inter 1st Year Maths 1B Limits and Continuity Formulas

→ Standard limits:

  • \(\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}\) = nan-1 and \(\lim _{x \rightarrow a}\left(\frac{x^m-a^m}{x^n-a^n}\right)=\frac{m}{n}\)am-n
  • \(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)\) = 1, \(\lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)\) = 1
  • \(\lim _{x \rightarrow 0}\left(\frac{a^x-1}{x}\right)\) = logea
  • \(\lim _{x \rightarrow 0}\)(1 + x)\(\frac{1}{x}\) = e and \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^x\) = e
  • \(\lim _{x \rightarrow 0}\left(\frac{e^x-1}{x}\right)\) = 1

Note:
For finding \(\lim _{x \rightarrow a}\)f(x), first verify f(a). If this is in indeterminate form like \(\frac{0}{0}, \frac{\infty}{\infty}\) etc., then reduce the given limit into standard form or rationalise numerator or denominator or factorise according to the problem.

TS Inter 1st Year Maths 1B The Plane Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 7 The Plane will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B The Plane Formulas

→ A plane is a proper subset of R’* which has atleast three non-collinear points and is such that for any two points in it. the line joining them also lies in it.

→ The general equation of a plane in the first degree equation in x, y, z given by ax + by + cz + d = 0. the coefficients a, b, c represent direction ratios of normal to the plane.

→ The equation of a plane passing through (x1, y1, z1) and perpendicular to the line with direction ratios a, b, c is a (x – x1) + b (y – y1) + c (z – z1) = 0.

→ Normal form of the plane is lx + my + nz – p where /. rn. n are direction cosine’s of normal and p is the perpendicular distance from origin to the plane.

→ The perpendicular distance from (0, 0, 0) to ax + by + cz t d = 0 is \(\frac{|d|}{\sqrt{a^2+b^2+c^2}\)

→ The perpendicular distance from A (x1, y1, z1) to the plane ax + by + cz + d = 0 is \(\frac{\left|a x_1+b y_1+c z_1+d\right|}{\sqrt{a^2+b^2+c^2}}\)

TS Inter 1st Year Maths 1B The Plane Formulas

→ The distance between parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is \(\frac{\left|d_1-d_2\right|}{\sqrt{a^2+b^2+c^2}}\)

→ The equation of plane with x. y. z intercepts a. b. c is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\) = 1.

→ The equation of the plane passing through 3 non-collinear points A (x1, y1 z1). B (x2, y2, z2) and C (x3, y3 z3) is \(\left|\begin{array}{ccc}
x-x_1 & y-y_1 & z-z_1 \\
x_2-x_1 & y_2-y_1 & z_2-z_1 \\
x_3-x_1 & y_3-y_1 & z_3-z_1
\end{array}\right|\) = 0

→ If Īø is the angle between planes a1x + b1y + c1z – d1 = 0 and a2x + b2y + c2z + d2 = 0 then cos Īø = \(\)

→ The planes a1x + b1y + c1z + d1 = 0 and a2x + b2y – c2z + d = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) and perpendicular if a1a2 + b1b2 + c1c2 = 0.

TS Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 6 Direction Cosines and Direction Ratios will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

→ If a line makes angles a, [3. y with the coordinate axes then cos α, cos β, cos γ are called ‘the direction cosines of the lines denoted by l, m, n.
The relation between l, in. n is l2 + m2 + n2 = 1

→ An ordered triple of numbers proportional to the direction cosines of a line are called as direction ratios of the line.

→ If a, b, c are the dirrc!ion ratios of a ray then the direction cosine are given by \(\left(\frac{a}{\sqrt{a^2+b^2}+c^2} \cdot \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2}+c^2}\right)\)

→ Direction ratios of the line joining A (x1, y1, z2) and B (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1) (or) (x1 – x2, y1 – y2, z1 – z2)

→ Direction cosines of the above line = \(\left(\frac{x_2-x_1}{A B}, \frac{y_2-y_1}{A B}, \frac{z_2-z_1}{A B}\right)\)

TS Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

→ If Īø is the angle between two lines with direction ratio’s (a1, b1, c1) and (a2, b2, c2) then
cos Īø = \(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{\left(a_1^2+b_1^2+c_1^2\right)\left(a_2^2+b_2^2+c_2^2\right)}}\)

→ If the above lines are perpendicular then a1a2 + b1b2 + c1c2 = 0.

→ In terms of direction cosine’s cos Īø = l1l2 + m1m2 + n1n2, and for perpendicular lines l1l2 + m1m2 + n1n2 = 0.

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Students must practice these Maths 2B Important Questions TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type to help strengthen their preparations for exams.

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 1.
Find ∫(1 – x)(4 – 3x)(3 + 2x) dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q1

Question 2.
Evaluate \(\int \frac{2 x^3-3 x+5}{2 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q2

Question 3.
Evaluate \(\int \frac{x^2+3 x-1}{2 x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q3

Question 4.
Evaluate \(\int \frac{(3 x+1)^2}{2 x} d x\). [(AP) May ’18, ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q4

Question 5.
Evaluate \(\int\left(x+\frac{4}{1+x^2}\right) d x\). [(TS) May ’15]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q5

Question 6.
Evaluate \(\int\left[\frac{1}{\sqrt{1-x^2}}+\frac{2}{\sqrt{1+x^2}}\right] d x\). [May ’11]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q6

Question 7.
Evaluate \(\int\left(x+\frac{1}{x}\right)^3 d x\), x > 0. [May ’12]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q7

Question 8.
Evaluate \(\int \frac{x^2+1}{x^4+1} d x\) on R. [May ’14]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q8

Question 9.
Evaluate \(\int\left(\frac{x^6-1}{1+x^2}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q9
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q9.1

Question 10.
Evaluate \(\int \frac{\left(a^x-b^x\right)^2}{a^x b^x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q10

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 11.
Evaluate \(\int \sec ^2 x cosec^2 x d x\). [(TS) May ’18; Mar. ’16; (AP) May ’17]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q11
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q11.1

Question 12.
Evaluate \(\int \frac{1+\sin ^2 x}{1+\cos 2 x} d x\). [Mar. ’06, May ’95]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q12

Question 13.
Evaluate \(\int \frac{1+\cos ^2 x}{1-\cos 2 x} d x\). [Mar. ’19 (TS); ’13]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q13

Question 14.
Find \(\int \sqrt{1+\sin 2 x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q14

Question 15.
Evaluate \(\int \sqrt{1-\sin 2 x} d x\). [(TS) May ’17]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q15

Question 16.
Evaluate \(\int \sqrt{1-\cos 2 x} d x\). [Mar. ’09, May ’06]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q16

Question 17.
Evaluate \(\int \frac{d x}{\sqrt{1+5 x}}\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q17

Question 18.
Find \(\int \frac{x}{1+x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q18

Question 19.
Evaluate \(\int \frac{e^x}{e^x+1} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q19

Question 20.
Evaluate \(\int \frac{1}{x \log x[\log (\log x)]} d x\). [Mar. ’19 (TS); ’11]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q20
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q20.1

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 21.
Evaluate \(\int \frac{1}{x \log x} d x\). [May ’99, ’94]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q21

Question 22.
Evaluate \(\int \frac{1-\tan x}{1+\tan x} d x\).
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q22

Question 23.
Evaluate \(\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x\). [May & Mar. ’98]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q23

Question 24.
Evaluate \(\int\left(1-\frac{1}{x^2}\right) e^{\left(x+\frac{1}{x}\right)} d x\). [Mar. ’12]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q24
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q24.1

Question 25.
Evaluate \(\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x\). [(AP) Mar. ’19; May ’17 (TS) Mar. ’17; May ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q25

Question 26.
Evaluate \(\int \frac{\cot (\log x)}{x} d x\). [Mar. ’05]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q26

Question 27.
Evaluate \(\int \frac{(1+\log x)^n}{x} d x\). [(AP) May ’18]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q27

Question 28.
Evaluate \(\int \frac{\log (1+x)}{1+x} d x\). [(TS) Mar. ’15]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q28

Question 29.
Evaluate \(\int \frac{\sin ^4 x}{\cos ^6 x} d x\). [Mar. ’11]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q29

Question 30.
Evaluate \(\int \frac{{cosec}^2 x}{(a+b \cot x)^5} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q30

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 31.
Evaluate \(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^2}} d x\). [(AP) May ’15]
Solution:
Put sin-1x = t2 then
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q31

Question 32.
Evaluate \(\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^2} d x\). [Mar. ’18, ’15 (AP); May ’13]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q32

Question 33.
Evaluate \(\int \frac{\mathrm{e}^{\tan ^{-1} x}}{1+\mathrm{x}^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q33

Question 34.
Evaluate ∫sin mx cos nx dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q34
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q34.1

Question 35.
Evaluate ∫cos mx cos nx dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q35

Question 36.
Evaluate ∫sin mx sin nx dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q36

Question 37.
Evaluate ∫cos x cos 3x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q37
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q37.1

Question 38
Evaluate \(\int \frac{d x}{\sin x+\sqrt{3} \cos x}\). [May ’12]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q38

Question 39.
Evaluate \(\int \frac{\mathbf{x}^2}{\sqrt{1-x^6}} d \mathbf{x}\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q39

Question 40.
Evaluate \(\int \frac{2 x^3}{1+x^8} d x\). [May ’08]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q40

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 41.
Evaluate \(\int \frac{x^8}{1+x^{18}} d x\). [(TS) May ’19; (AP) Mar. ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q41

Question 42.
Evaluate \(\int \frac{x^5}{1+x^{12}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q42

Question 43.
Evaluate \(\int \frac{3 x^2}{1+x^6} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q43

Question 44.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\). [May ’09, ’02]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q44

Question 45.
Find \(\int \frac{1}{(x+3) \sqrt{x+2}} d x\). [May ’14, ’12; Mar. ’14]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q45

Question 46.
Evaluate ∫sec x log(sec x + tan x) dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q46
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q46.1

Question 47.
Evaluate \(\int \frac{\sec ^2 x}{\sqrt{16+\tan ^2 x}} d x\). [May ’08, ’07]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q47

Question 48.
Evaluate \(\int \frac{d x}{x^2-81}\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q48

Question 49.
Evaluate \(\int \frac{3}{\sqrt{9 x^2-1}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q49

Question 50.
Evaluate \(\int \frac{d x}{(x+1)(x+2)}\). [Mar. ’19(AP); Mar. ’14, ’12, May ’11]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q50

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 51.
Evaluate \(\int \frac{d x}{\sqrt{x^2+2 x+10}}\). [May ’06]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q51

Question 52.
Evaluate ∫log x dx. [Mar. ’99, May ’10, ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q52

Question 53.
Evaluate ∫x log x dx. [(AP) Mar. ’20; May ’94]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q53

Question 54.
Evaluate \(\int \sin ^{-1} x d x\). [Mar. ’00, May ’05]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q54
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q54.1

Question 55.
Evaluate \(\int x \tan ^{-1} x d x\). [Mar. ’05, May ’02]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q55

Question 56.
Evaluate \(\int e^x \cos x d x\). [May ’15 (AP)]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q56
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q56.1

Question 57.
Evaluate ∫ex (sec x + sec x tan x) dx. [(AP) May ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q57

Question 58.
Evaluate ∫(tan x + log sec x) ex dx. [May ’18, ’15 (TS); Mar. ’08, May ’07]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q58

Question 59.
Evaluate \(\int e^x\left(\frac{1+x \log x}{x}\right) d x\). [(TS) May ’19; Mar. ’18, ’15 (AP); Mar. ’13]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q59
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q59.1

Question 60.
Evaluate \(\int \frac{e^x(x+1)}{(x+2)^2} d x\). [May ’09, ’98]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q60

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 61.
Evaluate \(\int e^x\left[\tan ^{-1} x+\frac{1}{1+x^2}\right] d x\). [May ’94]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q61

Question 62.
Evaluate \(\int e^x\left(\tan x+\sec ^2 x\right) d x\). [Mar. ’06, ’00, ’92]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q62

Question 63.
Evaluate ∫ex (sin x + cos x) dx. [(AP) Mar. ’17]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q63

Question 64.
Evaluate \(\int \frac{1}{1+\cos x} d x\). [Mar. ’15 (TS)]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q64

Question 65.
Evaluate \(\int \frac{1}{\cosh x+\sinh x} d x\). [(AP) May ’19, ’16 ; Mar. ’17 (TS)]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L1 Q65

Question 66.
Evaluate \(\int\left(1-x^2\right)^3 d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q1

Question 67.
Evaluate \(\int\left(\frac{1}{1-x^2}+\frac{1}{1+x^2}\right) d x\). [(TS) May ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q2

Question 68.
Evaluate \(\int \frac{1-\cos 2 x}{1+\cos 2 x} d x\). [May ’02]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q3

Question 69.
Evaluate \(\int \sqrt{1+\cos 2 x} d x\). [Mar. ’94]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q4

Question 70.
Evaluate \(\int \frac{1}{x \sqrt{x}} d \mathbf{x}\). [May ’92]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q5

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 71.
Evaluate ∫2x√x dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q6

Question 72.
Evaluate ∫x3 (4 + x2)2 dx. [May ’82]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q7

Question 73.
Evaluate \(\int \frac{1-x^4}{1-x} d x\). [Mar. ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q8

Question 74.
Evaluate \(\int \frac{x^4}{x^2+1} d x\). [May ’98, ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q9

Question 75.
Evaluate \(\int e^{2 \log x} d x\). [May ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q10

Question 76.
Evaluate ∫5x dx. [May ’93]
Solution:
\(\int 5^x d x=\frac{5^x}{\log 5}+c\)

Question 77.
Evaluate \(\int \frac{\cos ^2 x}{1-\sin x} d x\). [Mar. ’01, May ’93]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q12

Question 78.
Evaluate \(\int \frac{\cos x+\sin x}{\sqrt{1+\sin 2 x}} d x\). [Mar. ’98, May ’92]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q13

Question 79.
Find \(\int \frac{x^3}{\sqrt{x+1}} d x\). [Mar. ’00]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q14

Question 80.
Find \(\int \frac{x^2}{\sqrt{x+5}} d x\). [May ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q15
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q15.1

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 81.
Find \(\int \frac{2 x+3}{\sqrt{4 x+3}} d x\). [May ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q16

Question 82.
Evaluate ∫x3 sin x4 dx. [Mar. ’01]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q17

Question 83.
Evaluate ∫ex sin(ex) dx. [(AP) Mar. ’17]
Solution:
Put ex = t then ex dx = dt
Now ∫ex sin(ex) dx = ∫sin t dt
= -cos t + c
= -cos(ex) + c

Question 84.
Evaluate ∫2x sin(x2 + 1) dx.
Solution:
Put x2 + 1 = t then 2x dx = dt
Now ∫2x sin(x2 + 1) dx = ∫sin t dt
= -cos t + c
= -cos(x2 + 1) + c

Question 85.
Evaluate \(\int \frac{x}{1+x^4} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q20

Question 86.
Evaluate \(\int \frac{(\log x)^2}{x} d x\). [(AP) May ’18]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q21

Question 87.
Evaluate ∫sec(tan x) sec2x dx.
Solution:
Put tan x = t then sec2x dx = dt
Now ∫sec(tan x) sec2x dx = ∫sec t dt
= log|sec t + tan t| + c
= log|sec(tan x) + tan(tan x)| + c

Question 88.
Evaluate \(\int \frac{\sin (\log x)}{x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q23

Question 89.
Evaluate \(\int \frac{\cos (\log x)}{x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q24

Question 90.
Evaluate \(\int \frac{3 x+7}{3 x^2+14 x-5} d x\). [Mar. ’00]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q25

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 91.
Evaluate \(\int \frac{3 \cos 3 x-2 \sin 2 x}{\cos 2 x+\sin 3 x} d x\). [May ’93]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q26

Question 92.
Find \(\int \frac{6 x}{3 x^2-2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q27

Question 93.
Evaluate \(\int \frac{\left(\sin ^{-1} x\right)^2}{\sqrt{1-x^2}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q28

Question 94.
Evaluate \(\int \frac{{cosec}^2 x}{(1+\cot x)^2} d x\). [Mar. ’01]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q29

Question 95.
Evaluate ∫cos3x sin x dx. [(TS) Mar. ’18]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q30

Question 96.
Evaluate \(\int \sqrt[3]{\sin x} \cdot \cos x d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q31

Question 97.
Evaluate \(\int \sqrt{\sin x} \cdot \cos x d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q32

Question 98.
Evaluate \(\int {cosec}^2 x \cdot \sqrt{\cot x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q33

Question 99.
Evaluate \(\int \frac{d x}{4-9 x^2}\). [May ’98]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q34

Question 100.
Evaluate \(\int \frac{1}{\sqrt{1-4 x^2}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q35

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 101.
Evaluate \(\int \frac{d x}{\sqrt{25+x^2}}\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q36

Question 102.
Evaluate \(\int \frac{1}{e^x+e^{-x}} d \mathbf{x}\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q37

Question 103.
Evaluate \(\int \frac{1}{x^2+6 x+10} d x\). [Mar. ’98, May ’93]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q38

Question 104.
Evaluate \(\int \frac{1}{\sqrt{x^2-3}} d x\). [May ’92]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q39

Question 105.
Evaluate \(\int \frac{1}{x^2-4} d x\). [May ’94]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q40

Question 106.
Evaluate \(\int \sqrt{x^2+4} d x\). [May ’93]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q41

Question 107.
Evaluate \(\int \sqrt{4 x^2+9} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q42

Question 108.
Evaluate \(\int \sqrt{16-25 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q43

Question 109.
Evaluate ∫x ex dx. [Mar. ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q44

Question 110.
Evaluate ∫tan-1x dx. [May ’98]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q45

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 111.
Evaluate ∫ex (cos x – sin x) dx. [May ’99, ’95, Mar. ’99]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q46

Question 112.
Evaluate \(\int \frac{x e^x}{(x+1)^2} d x\). [May ’14, ’98, ’94: Mar. ’05]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q47

Question 113.
Evaluate \(\int \frac{1}{\sqrt{2 x-3 x^2+1}} d x\). [May ’08]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q48
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q48.1

Question 114.
Evaluate ∫cot2x dx.
Solution:
∫cot2x dx = ∫(cosec2x – 1) dx
= ∫cosec2x dx – ∫1 dx
= -cot x – x + c

Question 115.
Evaluate \(\int e^{\log \left(1+\tan ^2 x\right)} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q50

Question 116.
Evaluate \(\int \frac{\sin ^2 x}{1+\cos 2 x} d x\). [(TS) Mar. ’20]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q51

Question 117.
Evaluate ∫sin2x dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q52

Question 118.
Evaluate \(\int \frac{x}{1+x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q53

Question 119.
Evaluate \(\int \frac{\sin x}{\sin (a+x)} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q54

Question 120.
Evaluate \(\int 2 x e^{x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q55

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 121.
Evaluate \(\int \frac{1}{1+\sin 2 x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q56

Question 122.
Evaluate ∫tan4x sec2x dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q57

Question 123.
Evaluate \(\int \frac{2 x+3}{\sqrt{x^2+3 x-4}} d x\)
Solution:
Put x2 + 3x – 4 = t2 then (2x + 3) dx = 2t dt
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q58

Question 124.
Evaluate ∫sin3x dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q59

Question 125.
Evaluate ∫cos3x dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q60

Question 126.
Evaluate \(\int \frac{d x}{1+e^x}\)
Solution:
Put 1 + ex = t then ex = t – 1
⇒ ex dx = dt
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q61

Question 127.
Evaluate \(\int e^x\left[\frac{1-\sin x}{1-\cos x}\right] d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q62

Question 128.
Evaluate ∫x sin2x dx. [Mar. ’02]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q63
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q63.1

Question 129.
Evaluate \(\int \frac{e^x(x+2)}{(x+3)^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q64

Question 130.
Evaluate \(\int \frac{e^{\mathbf{x}}}{e^x+1} d x\). [Mar. ’18 (TS)]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q65

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 131.
Evaluate \(\int\left(x^3-\cos x+\frac{4}{\sqrt{x^2+1}}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L2 Q66

Question 132.
Evaluate ∫(x3 – 2x2 + 3) dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q1

Question 133.
Find ∫2x7 dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q2

Question 134.
Evaluate ∫(1 – 2x3) x2 dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q3

Question 135.
Evaluate \(\int \sqrt[3]{2 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q4

Question 136.
Evaluate \(\int\left(1+\frac{2}{x}-\frac{3}{x^2}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q5

Question 137.
Evaluate \(\int \frac{2 x^3-3 x+5}{2 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q6
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q6.1

Question 138.
Evaluate \(\int\left(\frac{3}{\sqrt{x}}-\frac{2}{x}+\frac{1}{3 x^2}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q7

Question 139.
Evaluate \(\int \frac{1-\sqrt{x}}{x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q8

Question 140.
Evaluate \(\int\left(\frac{2 x-1}{3 \sqrt{x}}\right)^2 d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q9

Question 141.
Evaluate \(\int\left(e^x-\frac{1}{x}+\frac{2}{\sqrt{x^2-1}}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q10

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 142.
Evaluate \(\int\left(\frac{1}{\sqrt{x}}+\frac{2}{\sqrt{x^2-1}}-\frac{3}{2 x^2}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q11

Question 143.
Evaluate ∫(sec2x – cos x + x2) dx.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q12

Question 144.
Evaluate \(\int\left(\sec x \tan x+\frac{3}{x}-4\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q13

Question 145.
Evaluate \(\int\left(\cosh x+\frac{1}{\sqrt{x^2+1}}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q14

Question 146.
Evaluate \(\int\left(\sin h x+\frac{1}{\left(x^2-1\right)^{1 / 2}}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q15

Question 147.
Evaluate \(\int(3 x-1)^{1 / 2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q16

Question 148.
Evaluate ∫e2x dx
Solution:
\(\int \mathrm{e}^{2 \mathrm{x}} \mathrm{dx}=\frac{\mathrm{e}^{2 \mathrm{x}}}{2}+\mathrm{c}\)

Question 149.
Evaluate \(\int \frac{e^{\log x}}{x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q18

Question 150.
Evaluate ∫sin 7x dx
Solution:
∫sin 7x dx = \(\frac{-\cos 7 x}{7}+c\)

Question 151.
Evaluate ∫(3x2 – 4)x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q20

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 152.
\(\int \frac{1}{7 x+3} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q21

Question 153.
Evaluate \(\int \frac{1}{1+(2 x+1)} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q22
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q22.1

Question 154.
Evaluate ∫ex cot ex dx
Solution:
Put ex = t ⇒ ex dx = dt
∓ ∫ex cot ex dx = ∫cot t dt
= log|sin t| + c
= log|sin ex| + c

Question 155.
Evaluate ∫cot hx dx
Solution:
∫cot hx dx = \(\int \frac{\cos h x}{\sin h x} d x\)
= log|sin hx| + c

Question 156.
Evaluate \(\int \frac{\cos x}{(1+\sin x)^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q25

Question 157.
Evaluate ∫cos4x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q26
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q26.1

Question 158.
Evaluate \(\int \frac{x}{\sqrt{1-x}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q27

Question 159.
Evaluate \(\int x \sqrt{4 x+3} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q28

Question 160.
Evaluate \(\int \frac{x^2}{\sqrt{1-x^2}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q29

Question 161.
Evaluate \(\int \frac{\sin \theta}{\sqrt{2-\cos ^2 \theta}} d \theta\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q30

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 162.
Evaluate \(\int \frac{3}{\sqrt{9 x^2-1}} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q31
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q31.1

Question 163.
Evaluate \(\int \frac{1}{1+4 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q32

Question 164.
Evaluate \(\int \frac{1}{8+2 x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q33

Question 165.
Evaluate \(\int\left(\sqrt{x}-\frac{2}{1-x^2}\right) d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q34
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q34.1

Question 166.
Evaluate \(\int \sqrt{9 x^2-25} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q35

Question 167.
Evaluate \(\int \frac{1}{e^x-1} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q36

Question 168.
Evaluate \(\int \frac{1}{1-\cot x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q37
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q37.1

Question 169.
Evaluate \(\int \frac{1}{1+\tan x} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q38

Question 170.
Evaluate ∫ex (1 + x2) dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q39

Question 171.
Evaluate ∫cot-1x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q40

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 172.
Evaluate ∫sec-1x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q41

Question 173.
Evaluate ∫cosec-1x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q42
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q42.1

Question 174.
Evaluate ∫x2 cos x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q43

Question 175.
Evaluate ∫x sec2x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q44

Question 176.
Evaluate \(\int \frac{\log x}{x^2} d x\)
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q45
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q45.1

Question 177.
Evaluate ∫(log x)2 dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q46

Question 178.
Evaluate ∫xn log x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q47

Question 179.
Evaluate ∫log(1 + x)2 dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q48

Question 180.
Evaluate ∫√x log x dx. [(TS) Mar. ’16]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q49

Question 181.
Evaluate \(\int \mathbf{e}^{\sqrt{\mathbf{x}}} d \mathbf{x}\). [(AP) May ’19]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q50
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q50.1

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 182.
Evaluate ∫cos√x dx. [(TS) May ’17]
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q51

Question 183.
Evaluate ∫x cot2x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q52
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q52.1

Question 184.
Evaluate ∫x2 e-3x dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q53

Question 185.
Evaluate ∫x3 eax dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q54

Question 186.
Evaluate ∫cos (log x) dx
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q55

Question 187.
If f(x) is a differentiable function, then prove that \(\int \frac{f^{\prime}(x)}{f(x)} d x\) = log|f(x)| + c.
Solution:
Let f(x) = t
⇒ f'(x) dx = dt
∓ \(\int \frac{f^{\prime}(x)}{f(x)} d x=\int \frac{1}{t} d t\)
= log|t| + c
= log|f(x)| + c

Question 188.
If f(x) is a differentiable function and n ≠ -1, then prove that \(\int[f(x)]_{f^{\prime}(x)}^n \cdot d x=\frac{[f(x)]^{n+1}}{n+1}+c\)
Solution:
Put f(x) = t
⇒ f'(x) dx = dt
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q57

Question 189.
Prove that ∫tan x dx = log|sec x| + c.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q58

Question 190.
Prove that ∫cot x dx = log|sin x| + c.
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q59

Question 191.
Prove that ∫sec x dx = log|sec x + tan x| + c = \(\log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right|\) + c
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q60
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q60.1

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type

Question 192.
Prove that ∫cosec x dx = log|cosec x – cot x| + c = \(\log \left|\tan \frac{x}{2}\right|\) + c
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q61
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q61.1

Question 193.
Prove that ∫ex [f(x) + f'(x)] dx = ex f(x) + c
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q62

Question 194.
If \(I_n=\int x^n \cdot e^{-x} d x\), then prove that \(I_n=-x^n \cdot e^{-x}+n I_{n-1}\).
Solution:
TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type L3 Q63

TS Inter 1st Year Maths 1B Three Dimensional Coordinates Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 5 Three Dimensional Coordinates will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Three Dimensional Coordinates Formulas

→ Perpendicular distances front the point P(x, y, z ) to yz, zx and xy planes are |x|, |y|, |z|.

→ The distance between points A (x1, y1, z1), B (x2, y2, z2) is AB = \(\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2+\left(z_1-z_2\right)^2}\)

→ The coordinates of a point which divides A = (x1, y1, z1) and B = (x2, y2, z2) internally in the ratio m1 m2 is = \(\left(\frac{m_1 x_2+m_2 x_1}{m_1+m_2}, \frac{m_1 y_2+m_2 y_1}{m_1+m_2}, \frac{m_1 z_2+m_2 z_1}{m_1+m_2}\right)\)

TS Inter 1st Year Maths 1B Three Dimensional Coordinates Formulas

→ Coordinates of midpoint of a line segment AB joining
A = (x1, y1, z1) and B = (x2, y2, z2) is = \(\left(\frac{\mathrm{x}_1+\mathrm{x}_2}{2}, \frac{\mathrm{y}_1+\mathrm{y}_2}{2}, \frac{\mathrm{z}_1+\mathrm{z}_2}{2}\right)\).

→ The centroid of the triangle formed by the points A (x1, y1, z1) , B (x2, y2, z2), C(x3, y3, z3) is
G = \(\left(\frac{\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3}{3}, \frac{\mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3}{3}, \frac{\mathrm{z}_1+\mathrm{z}_2+\mathrm{z}_3}{3}\right)\)

→ The centroid of the tetrahedron formed by (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) is
G = \(\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4}\right)\)

TS Inter 1st Year Maths 1B Pair of Straight Lines Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 4 Pair of Straight Lines will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B Pair of Straight Lines Formulas

→ If a b and h are not all zero then the equation H ≔ ax2 + 2hxy + by2 = 0 represents a pair of straight lines if and only if h2 ≄ ab.

→ If ax2 + 2hxy + by2 = 0 represent a pair of lines passing through the origin then the sum of the slopes of lines is \(\frac{-2h}{b}\) and product of the slopes is \(\frac{a}{b}\).
i.e.., if ax2 + 2hxy + by2 = (y – m1x) (y – m2x) then m1 + m2 = \(\frac{-2h}{b}\) and m1 m2 = \(\frac{a}{b}\).

→ If Īø is the angle between the lines represented by ax2 + 2hxy + 2 = 0 then
cos Īø = \(\frac{a+b}{\sqrt{(a-b)^2+4 h^2}}\) and tan Īø = \(\frac{2 \sqrt{\mathrm{h}^2-a b}}{a+b}\)

  • If h2 = ab then ax2 + 2hxy + by2 = 0 represents coincident or parallel lines.
  • ax2 + 2hxy + by2 = 0 represents a pair of perpendicular lines ⇔ a + b = 0 i.e., coefficient of x2 + coefficient of y2 = 0.

→ (i) The equation of pair < >f lines passing! Iirough origin and perpendicular to ax2 + 2hxy + by2 = 0 is bx2 – 2hxy + ay2 = 0,
(ii) The equation of pair of lines passing through (x1, y1) and perpendicular to ax2 + 2hxy + by2 = 0 is b(x – x1)2 – 2h (x – x1) (y – y1) – a(y – y1)2 = 0.
(iii) The equation of pair of lines passing through (x1, y1) and parallel to ax2 + 2hxy + by2 = 0 is a(x – x1)2+ 2h (x – x1) (y – y1) + b(y – y1)2 = 0.

→ The equation of bisectors of angles between the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 is \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}\) = \(\frac{(a_2 x+b_2 y+c_2)}{\sqrt{a_2^2+b_2^2}}\)

TS Inter 1st Year Maths 1B Pair of Straight Lines Formulas

→ The equation to the pair of bisectors of angles between the pair of lines ax2 + 2hxy + by2 = 0 is h(x2 – y2) – (a – b)xy

→ The area of the triangle formed by ax2 + 2hxy + by2 = 0 and lx + my + n = 0 is \(\frac{n^2 \sqrt{h^2-a b}}{\left|a m^2-2 h l m+b l^2\right|}\)

→ The product of the perpendiculars from (α, β) to the pair of lines ax2 + 2hxy + by2 = 0 is \(\frac{\left|a \alpha^2+2 h \alpha \beta+b \beta^2\right|}{\sqrt{(a-b)^2+4 h^2}}\)

→ The line ax + by + c – 0 and pair of lines (ax + by)2 – 3(bx – ay)2 = 0 form an equilateral triangle and the area is \(\frac{c^2}{\sqrt{3}\left(a^2+b^2\right)}\) units

→ If S = ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent the equation of pair of lines then

  • Ī” = abc + 2fgh – af2 – bg2 – ch2 = 0
  • h2 ≄ ab, g2 ≄ ac, f2 ≄ be

→ The point of intersection of the pair of lines S ≔ 0 is \(\left(\frac{h f-b g}{a b-h^2}, \frac{g h-a f}{a b-h^2}\right)\)

→ If S ≔ ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent a pair of parallel lines then

  • h2 = ab
  • bg2 = af2
  • distance between them is 2\(\sqrt{\frac{g^2-a c}{a(a+b)}}\) (or) 2\(\sqrt{\frac{f^2-b c}{a(a+b)}}\)

→ The equation to the pair of lines joinmg the ongin to the points of intersection of the curve ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 and the line lx + my + n = 0 is obtained by homogenisation ax2 + 2hxy + by2 + 2gx\(\left(\frac{l x+m y}{-n}\right)\) + 2fy\(\left(\frac{l x+m y}{-n}\right)\) + c\(\left(\frac{l x+m y}{-n}\right)^2\) = 0

TS Inter 1st Year Maths 1B The Straight Lines Formulas

Learning these TS Inter 1st Year Maths 1B Formulas Chapter 3 The Straight Lines will help students to solve mathematical problems quickly.

TS Inter 1st Year Maths 1B The Straight Lines Formulas

→ The equation of a horizontal line which is parallel to X – axis and at a distance of k’ from X – axis and lying above X – axis is given by y = k.

→ Similarly, y = -k is the equation of the horizontal line which is at a distance of k from X – axis and lying below X -axis.

→ The equation of X – axis is y = 0.

→ The equation of a vertical line which is parallel to Y – axis and at a distance of k from Y – axis and lying left of Y – axis is x = k.

→ Similarly, x = -k is the equation of the vertical line which is at a distance of k units from Y – axis and lying right of Y – axis is x = -k.

→ Equation of Y- axis is x = 0.

→ If a non vertical straight line L makes an angle Īø with X – axis measured anti-clockwise from the positive direction of the X – axis then tan Īø is called the slope or gradient of the line L denoted by ‘m’.

TS Inter 1st Year Maths 1B The Straight Lines Formulas

→ Slope of horizontal line is 0 since tan 0 – 0 and slope of vertical line is not defined.

→ If m1, m2, are slopes of two lines and Īø is called the angle between them then tan Īø = \(\left(\frac{m_1-m_2}{1+m_1 m_2}\right)\)

→ If two lines are parallel then slopes are equal, m1 = m2, and if two lines are perpendicular then m1. m1 = -1.

→ Equation of a line passing through (x1; y1) with slope m’ is y – y1 = m (x – x1).

→ Equation of a line passing through origin with slope in is y = mx.

→ Equation of a line passing through the points A (x1, y1) and B (x2, y2) is \(\frac{y-y_1}{y_1-y_2}=\frac{x-x_1}{x_1-x_2}\)

→ Equation of a line with Y – intercept ‘c’ and slope m is y = mx + c.

→ Equation of a line in intercept form is \(\frac{x}{a}+\frac{y}{b}\) = 1.

→ Reduction of a straight line ax + by + c = 0 in intercept form is \(\frac{x}{-\left(\frac{c}{a}\right)}+\frac{y}{-\left(\frac{c}{b}\right)}\) = 1

→ Area of the triangle formed by the line ax + by + c = 0 with coordinate axes is \(\frac{c^2}{2|a b|}\).

→ Equation of a line in normal form or perpendicular form is x cos α + y sin α = p where p is the length of the perpendicular from origin to line and a. is the angle made by the perpendicular with + ve X – axis.

→ Reduction of the equation ax + by + c = 0 of a line to the normal form is \(\pm\left(\frac{a}{\sqrt{a^2+b^2}}\right) x+\left(\pm \frac{b}{\sqrt{a^2+b^2}}\right)=\frac{\pm c}{\sqrt{a^2+b^2}}\)

→ Perpendicular distance from (x1; y1) to the line ax + by + c = 0 is \(\frac{\left|a x_1+b y_1+c\right|}{\sqrt{a^2+b^2}}\)

→ Perpendicular distance from origin to the line ax + by + c = 0 is points A (x1, y1) and B (x2, y2) is \(\frac{|c|}{\sqrt{a^2+b^2}}\)

→ The ratio in which the line L = ax + by + c = 0 (ab ≠ 0) divides the line segment AB joining points A(x1, y1) and B(x2, y2) is \(-\left(\frac{a x_1+b y_1+c}{a x_2+b y_2+c}\right)=-\frac{L_{11}}{L_{22}}\)
If L11 and L22 are having same sign or opposite sign then the points on same side or opposite sides of the line L = 0.

→ If (h, k) is the foot of the perpendicular from (x1, y1) to the line ax + by + c = 0. then \(\frac{h-x_1}{a}=\frac{k-y_1}{b}=-\left(\frac{a x_1+b y_1+c}{a^2+b^2}\right)\)

→ If (h, k) is the image of the point (x1, y1) with respect to the line ax + by + c = 0, then \(\frac{h-x_1}{a}=\frac{k-y_1}{b}=-2\left(\frac{a x_1+b y_1+c}{a^2+b^2}\right)\)

→ The point of intersection of lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is \(\left(\frac{b_1 c_2-b_2 c_1}{a_1 b_2-a_2 b_1}, \frac{c_1 a_2-a_1 c_2}{a_1 b_2-a_2 b_1}\right)\)

TS Inter 1st Year Maths 1B The Straight Lines Formulas

→ If angle between lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is 0 where (0 ≤ Īø ≤ Ļ€), then
cos Īø = \(\frac{a_1 a_2+b_1 b_2}{\sqrt{a_1^2+b_1^2} \sqrt{a_2^2+b_2^2}}\)
sin Īø = \(\frac{a_1 b_2-a_2 b_1}{\sqrt{a_1^2+b_1^2} \sqrt{a_2^2+b_2^2}}\)
and tan Īø = \(\frac{a_1 b_2-a_2 b_1}{a_1 a_2+b_1 b_2}\)

  • Lines are perpendicular ⇔ a1a2 + b1b2 = 0
  • Lines are parallel ⇔ \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\)

→ The equation of a line passing through (x1, y1) and parallel to the line ax + by + c = 0 is a (x – x1) – b (y – y1) = 0.

→ The equation of a line passing through (x1, y1) and perpendicular to ax + by + c = 0 is b(x – x1) – a(y – y1) = 0.

→ If a1x + b1y + c1 = 0. a2x + b2y + c2 = 0, and a3x + b3y + c3 = 0 represent three lines, no two of which are parallel, then a necessary and sufficient condition for these lines to be concurrent is Ī£a1(b2c3 – b3c2) = 0 (0r) \(\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|\) = 0

→ The distance between parallel lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is \(\frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}}\)