MCQ Questions

Students are advised to practice the NCERT MCQ Questions for Class 12 Chemistry Chapter 15 Polymers with Answers Pdf free download is available here. MCQ Questions for Class 12 Chemistry with Answers are prepared as per the Latest Exam Pattern. Students can solve these Polymers Class 12 MCQs Questions with Answers and assess their preparation level.

Polymers Class 12 MCQs Questions with Answers

Solving the Polymers Multiple Choice Questions of Class 12 Chemistry Chapter 15 MCQ can be of extreme help as you will be aware of all the concepts. These MCQ Questions on Polymers Class 12 with answers pave for a quick revision of the Chapter thereby helping you to enhance subject knowledge. Have a glance at the MCQ of Chapter 15 Chemistry Class 12 and cross-check your answers during preparation.

Question 1.
Polystyrene is a/an
(a) addition polymer
(b) thermoplastic polymer
(c) both (a) and (b)
(d) none of these.

Answer

Answer: (c) both (a) and (b)

---

Question 2.
Cellulose is a
(a) natural polymer
(b) semi-synthetic polymer
(c) synthetic polymer
(d) none of these.

Answer

Answer: (a) natural polymer

---

Question 3.
Buna-S is a/an
(a) addition polymer
(b) condensation polymer
(c) both (a) and (b)
(d) none of these.

Answer

Answer: (a) addition polymer

---

Question 4.
Bakelite is a
(a) addition polymer
(b) thermoplastic polymer
(c) elastomer polymer
(d) thermosetting polymer.

Answer

Answer: (d) thermosetting polymer.

---

Question 5.
Which of the following is called a polyamide?
(a) Rayon
(b) Nylon
(c) Terylene
(d) Bakelite

Answer

Answer: (b) Nylon

---

Question 6.
Which is a condensation polymer?
(a) PVC
(b) Teflon
(c) Bakelite
(d) None of these

Answer

Answer: (c) Bakelite

---

Question 7.
Bakelite is a product of the reaction between
(a) Formaldehyde and NaOH
(b) Aniline and Urea
(c) Phenol and Methanol
(d) Phenol and Formaldehyde.

Answer

Answer: (d) Phenol and Formaldehyde.

---

Question 8.
Vulcanisation makes rubber
(a) more elastic
(b) soluble in inorganic solvent
(c) crystalline
(d) none of these.

Answer

Answer: (a) more elastic

---

Question 9.
Neoprene is a polymer of
(a) Chloroprene
(b) Isoprene
(c) Styrene
(d) Ethene.

Answer

Answer: (a) Chloroprene

---

Question 10.
The monomer unit of PVC is
(a) vinyl chloride
(b) ethylene
(c) chloroprene
(d) acrylonitrile.

Answer

Answer: (a) vinyl chloride

---

Question 11.
Which of the following is not applicable to Nylon-6, 6?
(a) Synthetic polymer
(b) Fibre
(c) Addition polymer
(d) Condensation polymer

Answer

Answer: (c) Addition polymer

---

Question 12.
In the following, thermosetting polymer is
(a) Bakelite
(b) Polythene
(c) Polyester
(d) Buna -N.

Answer

Answer: (a) Bakelite

---

Question 13.
Monomer of teflon is
(a) CF₂ = CF₂
(b) CH₂ = CHCN
(c) CH₂ = CH₂
(d) C₆H₅CH = CH₂

Answer

Answer: (a) CF₂ = CF₂

---

Question 14.
The incorrect statement about LDP is:
(a) It is obtained through the free radical addition of ethene.
(b) It consists of linear molecules.
(c) It is obtained by the H-atom abstraction.
(d) Peroxide is used as an initiator.

Answer

Answer: (b) It consists of linear molecules.

---

Assertion and Reason Type Questions

The questions given below consist of an assertion and a reason. Use the following key to choose the appropriate answer.
(a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is correct, Reason (R) is wrong statement.
(d) Assertion (A) is wrong, but Reason (R) is correct statement.

Question 15.
Assertion: Rayon is a semi-synthetic polymer and is taken as a better choice than cotton fabric.
Reason: Mechanical and aesthetic properties of cellulose can be improved by acetylation.

Answer

Answer: (a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).

---

Question 16.
Assertion: Most of the synthetic polymers are not biodegradable.
Reason: Polymerisation process induces toxic character in organic molecules.

Answer

Answer: (c) Assertion (A) is correct, Reason (R) is wrong statement.

---

Question 17.
Assertion: Olefinic monomers undergo addition polymerisation.
Reason: Polymerisation of vinyl chloride is initiated by peroxides/persulphates.

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Question 18.
Assertion: Polyamides are best used as fibres because of high tensile strength.
Reason: Strong intermolecular forces (like hydrogen bonding within polyamides) lead to close packing of chains and increase the crystalline character, hence, provide high tensile strength to polymers.

Answer

Answer: (a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).

---

Question 19.
Assertion: For making rubber synthetically, isoprene molecules are polymerised.
Reason: Neoprene (a polymer of chloroprene) is a synthetic rubber.

Answer

Answer: (d) Assertion (A) is wrong, but Reason (R) is correct statement.

---

Question 20.
Assertion: Network polymers are thermosetting.
Reason: Network polymers have high molecular mass.

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Question 21.
Assertion: Polytetrafluoroethene is used in making non-stick cookwares.
Reason: Fluorine has highest electronegativity

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Hope the information shared regarding the NCERT MCQ Questions for Class 12 Chemistry Chapter 15 Polymers with Answers Pdf free download is true and genuine as far as our knowledge is concerned. If you feel any information regarding CBSE Class 12 Chemistry Polymers MCQs Multiple Choice Questions with Answers is missing do reach us and we will look into it and add it accordingly.

Students are advised to practice the NCERT MCQ Questions for Class 12 Chemistry Chapter 16 Chemistry in Everyday Life with Answers Pdf free download is available here. MCQ Questions for Class 12 Chemistry with Answers are prepared as per the Latest Exam Pattern. Students can solve these Chemistry in Everyday Life Class 12 MCQs Questions with Answers and assess their preparation level.

Chemistry in Everyday Life Class 12 MCQs Questions with Answers

Solving the Chemistry in Everyday Life Multiple Choice Questions of Class 12 Chemistry Chapter 16 MCQ can be of extreme help as you will be aware of all the concepts. These MCQ Questions on Chemistry in Everyday Life Class 12 with answers pave for a quick revision of the Chapter thereby helping you to enhance subject knowledge. Have a glance at the MCQ of Chapter 16 Chemistry Class 12 and cross-check your answers during preparation.

Question 1.
Antipyretics are medicinal compounds which
(a) lower body temperature
(b) relieve pain
(c) control malaria
(d) kill microorganisms.

Answer

Answer: (a) lower body temperature

---

Question 2.
0.2% solution of phenol is an
(a) antibiotic
(b) antiseptic
(c) disinfectant
(d) analgesic

Answer

Answer: (b) antiseptic

---

Question 3.
Which of the following is an analgesic?
(a) Ranitidine
(b) Aspirin
(c) Penicillin
(c) None of these

Answer

Answer: (b) Aspirin

---

Question 4.
Aspirin is an
(a) antipyretic
(b) antibiotics
(c) antiseptic
(d) None of these

Answer

Answer: (a) antipyretic

---

Question 5.
Acetyl salicylic acid is used as
(a) an antiseptic
(b) an antibiotic
(c) an analgesic
(d) a pesticide

Answer

Answer: (c) an analgesic

---

Question 6.
Which is used as a preservative to protect processed food?
(a) Sodium sulphate
(b) Saccharin
(c) Sodium bicarbonate
(d) Sodium metabisulphite

Answer

Answer: (d) Sodium metabisulphite

---

Question 7.
Dettol is used as
(a) disinfectant
(b) antiseptic
(c) analgesic
(d) anti-allergic

Answer

Answer: (b) antiseptic

---

Question 8.
Penicillin is
(a) Hormone
(b) Antibiotic
(c) Antiseptic
(d) Lipid

Answer

Answer: (b) Antibiotic

---

Question 9.
Paracetamol is
(a) antiseptic
(b) analgesic
(c) antiseptic and analgesic
(d) antibiotic

Answer

Answer: (b) analgesic

---

Question 10.
Which of the following is used as artificial sweetener?
(a) Saccharin
(b) Aspirin
(c) Omeprazole
(d) Pheniramine

Answer

Answer: (a) Saccharin

---

Question 11.
Which is not a tranquilizer?
(a) Luminal
(b) Seconal
(c) Valium
(d) Bithional

Answer

Answer: (d) Bithional

---

Question 12.
Which of the following artificial sweeteners is methyl ester of a dipeptide?
(a) Aspartame
(b) Sucralose
(c) Saccharine
(d) Alitame

Answer

Answer: (a) Aspartame

---

Question 13.
Which of the following can be used as an antacid?
(a) Ranitidine
(b) Histamine
(c) Equanil
(d) Aspirin

Answer

Answer: (a) Ranitidine

---

Question 14.
The class of drugs used for the treatment of cut or wound is.
(a) Tranquilizers
(b) Antiseptics
(c) Antihistamines
(d) Antipyretic

Answer

Answer: (b) Antiseptics

---

Assertion and Reason Type Questions

The questions given below consist of an assertion and a reason. Use the following key to choose the appropriate answer.
(a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is correct, Reason (R) is wrong statement.
(d) Assertion (A) is wrong, but Reason (R) is correct statement.

Question 15.
Assertion: Detergents are preferred to soaps for washing purposes.
Reason: Detergents having branched chain hydrocarbon are non-biodegradable.

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Question 16.
Assertion: Aspirin can cause ulcer in stomach when taken empty stomach.
Reason: Aspirin gets hydrolysed to salicylic acid in stomach where pH is 2.

Answer

Answer: (a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).

---

Question 17.
Assertion: Combinations of progesterone and estrogen are used as antifertility drugs.
Reason: These control the pregnancy.

Answer

Answer: (a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).

---

Question 18.
Assertion: Certain narcotics are used as analgesics.
Reason: Narcotics lower the body temperature in high fever.

Answer

Answer: (c) Assertion (A) is correct, Reason (R) is wrong statement.

---

Question 19.
Assertion: Penicillin has a narrow spectrum.
Reason: Acid gastritis is common ailment associated with digestion.

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Question 20.
Assertion: Pheniramine is used as an antihistamine.
Reason: Antimicrobials are used for treatment of gastric ulcers

Answer

Answer: (c) Assertion (A) is correct, Reason (R) is wrong statement.

---

Question 21.
Assertion: A 0.2% solution of phenol is an antiseptic while 1% solution is a disinfectant.
Reason: Disinfectants kill microorganisms but are harmful to human tissues.

Answer

Answer: (b) Both Assertion (A) and Reason (R) are correct statements, but Reason (R) is not the correct explanation of the Assertion (A).

---

Question 22.
Assertion: Sodium-2-dodecyl benzene
sulphonate is a biodegradable detergent. Reason: Detergents having highly branched chains are non-biodegradable.

Answer

Answer: (d) Assertion (A) is wrong, but Reason (R) is correct statement.

---

Question 23.
Assertion: Saccharin is an artificial sweetener.
Reason: It has a high calorific value.

Answer

Answer: (c) Assertion (A) is correct, Reason (R) is wrong statement.

---

Question 24.
Assertion: Aspartame is used as artificial sweetener in cold drinks.
Reason: Aspartame is stable under cold conditions.

Answer

Answer: (a) Both Assertion (A) and Reason (R) are correct statements, and Reason (R) is the correct explanation of the Assertion (A).

---

Hope the information shared regarding the NCERT MCQ Questions for Class 12 Chemistry Chapter 16 Chemistry in Everyday Life with Answers Pdf free download is true and genuine as far as our knowledge is concerned. If you feel any information regarding CBSE Class 12 Chemistry Chemistry in Everyday Life MCQs Multiple Choice Questions with Answers is missing do reach us and we will look into it and add it accordingly.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 1 Relations and Functions with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Relations and Functions Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 1 Relations and Functions Objective Questions.

Relations and Functions Class 12 MCQs Questions with Answers

Students are advised to solve the Relations and Functions Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Relations and Functions Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Relations and Functions Class 12 with answers provided with detailed solutions by looking below.

Question 1.
Let R be the relation in the set (1, 2, 3, 4}, given by:
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.
Then:
(a) R is reflexive and symmetric but not transitive
(b) R is reflexive and transitive but not symmetric
(c) R is symmetric and transitive but not reflexive
(d) R is an equivalence relation.

Answer

Answer: (b) R is reflexive and transitive but not symmetric

---

Question 2.
Let R be the relation in the set N given by : R = {(a, b): a = b – 2, b > 6}. Then:
(a) (2, 4) ∈ R
(b) (3, 8) ∈ R
(c) (6, 8) ∈ R
(d) (8, 7) ∈ R.

Answer

Answer: (c) (6, 8) ∈ R

---

Question 3.
Let A = {1, 2, 3}. Then number of relations containing {1, 2} and {1, 3}, which are reflexive and symmetric but not transitive is:
(a) 1
(b) 2
(c) 3
(d) 4.

Answer

Answer: (a) 1

---

Question 4.
Let A = (1, 2, 3). Then the number of equivalence relations containing (1, 2) is
(a) 1
(b) 2
(c) 3
(d) 4.

Answer

Answer: (b) 2

---

Question 5.
Let f: R → R be defined as f(x) = x⁴. Then
(a) f is one-one onto
(b) f is many-one onto
(c) f is one-one but not onto
(d) f is neither one-one nor onto.

Answer

Answer: (d) f is neither one-one nor onto.

---

Question 6.
Let f : R → R be defined as f(x) = 3x. Then
(a) f is one-one onto
(b) f is many-one onto
(c) f is one-one but not onto
(d) f is neither one-one nor onto.

Answer

Answer: (a) f is one-one onto

---

Question 7.
If f: R → R be given by f(x) = (3 – x³)^1/3, then fof (x) is
(a) x^1/3
(b) x³
(c) x
(d) 3 – x³.

Answer

Answer: (c) x

---

Question 8.
Let f: R – {-\(\frac { 4 }{3}\)} → R be a function defined as: f(x) = \(\frac { 4x }{3x+4}\), x ≠ –\(\frac { 4 }{3}\). The inverse of f is map g : Range f → R -{-\(\frac { 4 }{3}\)} given by
(a) g(y) = \(\frac { 3y }{3-4y}\)
(b) g(y) = \(\frac { 4y }{4-3y}\)
(c) g(y) = \(\frac { 4y }{3-4y}\)
(d) g(y) = \(\frac { 3y }{4-3y}\)

Answer

Answer: (b) g(y) = \(\frac { 4y }{4-3y}\)

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Question 9.
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is
(a) Reflexive and symmetric
(b) Transitive and symmetric
(c) Equivalence
(d) Reflexive, transitive but not symmetric.

Answer

Answer: (b) Transitive and symmetric

---

Question 10.
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is:
(a) 144
(b) 12
(c) 24
(d) 64

Answer

Answer: (c) 24

---

Question 11.
Let f: R → R be defined by f(x) = sin x and g : R → R be defined by g(x) = x², then fog is
(a) x² sin x
(b) (sin x)²
(c) sin x²
(d) \(\frac { sin x }{x^2}\)

Answer

Answer: (c) sin x²

---

Question 12.
Let f: R → R be defined by f(x) = x² + 1. Then pre-images of 17 and – 3 respectively, are
(a) ø, {4,-4}
(b) {3, -3}, ø
(c) {4, -4}, ø
(d) {4, -4}, {2,-2}.

Answer

Answer: (c) {4, -4}, $

---

Question 13.
Let f: R → R be defined by
f(x)= \(\left\{\begin{array}{lr}
2 x ; & x>3 \\
x^{2} ; & 1<x<3 \\
3 x ; & x \leq 1
\end{array}\right.\)
(a) 9
(b) 14
(c) 5
(d) None of these.

Answer

Answer: (a) 9

---

Question 14.
The domain of the function f (x) = \(\frac { 1 }{\sqrt{|x|-x}}\) is
(a) (-∞, ∞)
(b) (0, ∞)
(c) (-∞, 0)
(d) (-∞, ∞) – {0}.

Answer

Answer: (c) (-∞, 0)
Hint:
f(x) = \(\frac { 1 }{\sqrt{|x|-x}}\)
f(x) is defined if |x| – x > 0
if |x| > x
if x < 0.
Hence, D_f = (-∞, 0).

---

Question 15.
If a ∈ R and the equation
-3(x – [x] )2 + 2 (x – [x]) + a² = 0,
where [x] denotes the greatest integer (≤ x) has no integral solution, then all possible values of a lie in the interval:
(a) (1, 2)
(b) (-2, -1)
(c) (-∞, -2) ∪(2, ∞)
(d) (-1, 0) ∪ (0, 1).

Answer

Answer: (d) (-1, 0) ∪ (0, 1).
Hint:
Put x – [x] = t.
Then – 3t² + 2t + a² = 0
⇒ a² = 3t² – 2t.
For non-integral solutions, 0 < a² < 1.
Hence, as (- 1, 0) ∪ (0, 1).

---

Fill in the Blanks

Question 1.
Let A = {1, 2, 3}. Then the number of equivalence relations containing (1, 2) is ………………

Answer

Answer: 2.

---

Question 2.
If A = {0, 1, 3}, then the number of relations on A is ………………..

Answer

Answer: 9.

---

Question 3.
A bijective function is both ……………….. and ………………

Answer

Answer: one-one, onto.

---

Question 4.
Let R be a relation defined on A = { 1, 2, 3) by R = {(1, 3), (3, 1), (2, 2)}, R is ………………..

Answer

Answer: symmetric.

---

Question 5.
If f be the greatest integer function defined as f(x) = [x] and g be the modulus function defined as g(x) = |x|, then the value of gof (\(\frac { -5 }{4}\)) is ………………..

Answer

Answer: 2.
Hint:
gof (\(\frac { -5 }{4}\)) = g (f(\(\frac { -5 }{4}\)))
= g(-2) = |-2| = 2.

---

Question 6.
If f : R → R is defined by 3x + 4, then f(f(x)) is …………….

Answer

Answer: 9x + 16.
Hint:
f(f (x)) = f(3x + 4)
= 3f(x) + 4
= 3 (3x + 4) + 4
= 9x + 16.

---

Question 7.
If f(x) = e^x and g(x) = log x, then gof is ……………….

Answer

Answer: x.
Hint:
gof (x) = g(f(x) = g (e^x)
= log e^x = x log e = x(1) = x.

---

Question 8.
The domain of the function f (x) = \(\frac { x }{|x|}\) is ………………..

Answer

Answer: R – {0}.
Hint:
f is defined for all x ∈ R except at x = 0.

---

Question 9.
Let f: R – {-\(\frac { 4 }{3}\)} → R be a function defined as: f(x) = \(\frac { 4x }{3x+4}\), x ≠ –\(\frac { 4 }{3}\)
Then inverse of f is map g : Range f → R – {-\(\frac { 4 }{3}\)} given by …………………

Answer

Answer: g(y) = \(\frac { 4x }{3x+4}\)
Hint:
Let y = f(x)= \(\frac { 4x }{3x+4}\)
⇒ 3xy + 4y = 4x
⇒ (4 – 3y)x = 4y
⇒ x = \(\frac { 4y }{4-3y}\)

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We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 1 Relations and Functions with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Relations and Functions MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 3 Matrices with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Matrices Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 3 Matrices Objective Questions.

Matrices Class 12 MCQs Questions with Answers

Students are advised to solve the Matrices Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Matrices Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Matrices Class 12 with answers provided with detailed solutions by looking below.

Question 1.
If A = [a_ij]_{m × n} is a square matrix, if:
(a) m < n
(b) m > n
(c) m = n
(d) None of these.

Answer

Answer: (c) m = n

---

Question 2.
Which of the given values of x and y make the following pair of matrices equal:
\(\left[\begin{array}{cc}
3 x+7 & 5 \\
y+1 & 2-3 x
\end{array}\right]\) \(\left[\begin{array}{lc}
0 & y-2 \\
8 & 4
\end{array}\right]\)
(a) x = –\(\frac { 1 }{3}\), y = 7
(b) Not possible to find
(c) y = 7, x = –\(\frac { 2 }{3}\)
(d) x = –\(\frac { 1 }{3}\), y = –\(\frac { 2 }{3}\)

Answer

Answer: (b) Not possible to find

---

Question 3.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512.

Answer

Answer: (d) 512.

---

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × 1, 2 × p, n × 3 and p × k respectively. Now answer the following (4-5):
Question 4.
The restrictions on n, k and p so that PY + WY will be defined are
(a) k = 3, p = n
(b) k is arbitrary, p = 2
(c) p is arbitrary
(d) k = 2,p = 3.

Answer

Answer: (a) k = 3, p = n

---

Question 5.
If n =p, then the order of the matrix 7X – 5Z is:
(a) p × 2
(b) 2 × n
(c) n × 3
(d) p × n.

Answer

Answer: (b) 2 × n

---

Question 6.
If A, B are symmetric matrices of same order, then AB – BA is a
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix.

Answer

Answer: (a) Skew-symmetric matrix

---

Question 7.
If A = \(\left[\begin{array}{l}
\cos \alpha-\sin \alpha \\
\sin \alpha \cos \alpha
\end{array}\right]\) then A + A’ = I, the value of α is
(a) \(\frac { π }{6}\)
(b) \(\frac { π }{3}\)
(c) π
(d) \(\frac { 3π }{2}\)

Answer

Answer: (a) \(\frac { π }{6}\)

---

Question 8.
Matrices A and B will be inverse of each other only if:
(a) AB = BA
(b) AB – BA = O
(c) AB = O, BA = I
(d) AB = BA = I.

Answer

Answer: (d) AB = BA = I.

---

Question 9.
If A = \(\left[\begin{array}{lr}
\alpha & \beta \\
\gamma & -\alpha
\end{array}\right]\) is such that A² = I, then
(a) 1 + α² + ßγ = 0
(b) 1 – α² + ßγ = 0
(c) 1 – α² – ßγ = 0
(d) 1 + α² – ßγ = 0

Answer

Answer: (c) 1 – α² – ßγ = 0

---

Question 10.
If a matrix is both symmetric and skew- symmetric matrix, then:
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a square matrix
(d) None of these.

Answer

Answer: (b) A is a zero matrix

---

Question 11.
If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to :
(a) A
(b) I – A
(c) I
(d) 3A.

Answer

Answer: (c) I

---

Question 12.
The matrix A = \(\left[\begin{array}{lll}
0 & 0 & 5 \\
0 & 5 & 0 \\
5 & 0 & 0
\end{array}\right]\) is a
(a) scalar matrix
(b) diagonal matrix
(c) unit matrix
(d) square matrix.

Answer

Answer: (d) square matrix.

---

Question 13.
If matrix A = [a_ij]_2×2
where a_ij = 1 if i ≠ j = 0 if i = j,
(a) I
(b) A
(c) O
(d) None of these

Answer

Answer: (a) I

---

Question 14.

then A – B is equal to
(a) I
(b) O
(c) 2I

Answer

Answer: (c) 2I

---

Question 15.
If A is a matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n.

Answer

Answer: (d) m × n.

---

Question 16.
For any two matrices A and B, we have :
(a) AB = BA
(b) AB ≠ BA
(c) AB = 0
(d) None of these.

Answer

Answer: (d) None of these.

---

Question 17.
The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is:
(a) less than 4
(b) 5
(c) 6
(d) at least 7.

Answer

Answer: (d) at least 7.
Hint:
First row with exactly one zero total number of cases = 6.
First row with 2 zeroes, we get more cases.
∴ Total no. of non-singular matrices = 7.

---

Question 18.
If A is an 3 × 3 non-singular matrix such that AA’ = A’A and B = A^-1 A’, then BB’ equals
(a) I
(b) B^-1
(c) (B^-1)’
(d) I + B.

Answer

Answer: (a) I
B = A^-1A’
⇒ AB = A’
ABB’ = A’B’
⇒ ABB’ = (BA)’ = (A^-1 A’A)’
= (A^-1 AA’)’ = (A’)’ = A
BB’ = I.

---

Fill in the Blanks

Question 1.
If \(\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]\) \(\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(\left[\begin{array}{l}
5 \\
4
\end{array}\right]\) then value of y is ………………..

Answer

Answer: 2.

---

Question 2.
If A is an m × n matrix such that AB and BA are defind, then B is of order …………….

Answer

Answer: n × m.

---

Question 3.
A diagonal matrices is said to be ……………… if its diagonal elements are equal (other than unity).

Answer

Answer: scalar.

---

Question 4.
For a 2 × 2 matrix, A = [a_ij], whose elements are given by a_ij = \(\frac { i }{j}\), then a₁₂ = ………………..

Answer

Answer: \(\frac {1}{2}\)

---

Question 5.
If \(\left[\begin{array}{cc}
a-b & 2 a+c \\
2 a-b & 3 c+d
\end{array}\right]\) = \(\left[\begin{array}{cc}
-1 & 5 \\
0 & 13
\end{array}\right]\)

Answer

Answer: 1

---

Question 6.
If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the ……………….

Answer

Answer: inverse of A.

---

Question 7.
If A is of order m × n and B is also of order m × n, then (A + B) is a matrix of order ………………..

Answer

Answer: m × n.

---

Question 8.
If [5 x 1] \(\left[\begin{array}{l}
4 \\
2 \\
7
\end{array}\right]\) = 35 then x = ………………

Answer

Answer: 4

---

Question 9.
For any matrix A, A – A’ is …………….. matrix.

Answer

Answer: Skew-symmetric.

---

Question 10.
If A = \(\left[\begin{array}{cc}
2 & -2 \\
-2 & 2
\end{array}\right]\) and A² = λA, then λ = ………………..

Answer

Answer: 4

---

We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 3 Matrices with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Matrices MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 4 Determinants with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Determinants Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 4 Determinants Objective Questions.

Determinants Class 12 MCQs Questions with Answers

Students are advised to solve the Determinants Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Determinants Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Determinants Class 12 with answers provided with detailed solutions by looking below.

Question 1.
If \(\left|\begin{array}{rr}
x & 2 \\
18 & x
\end{array}\right|\) = \(\left|\begin{array}{rr}
6 & 2 \\
18 & 6
\end{array}\right|\), then x is equal to
(a) 6
(b) ±6
(c) -6
(d) 6, 6

Answer

Answer: (a) 6

---

Question 2.
Let A be a square matrix of order 3 × 3. Then |kA| is equal to
(a) k|A|
(b) k²|A|
(c) k³|A|
(d) 3k|A|

Answer

Answer: (c) k³|A|

---

Question 3.
Which of the following is correct?
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
(c) Determinant is a number associated to a square matrix
(d) None of these.

Answer

Answer: (c) Determinant is a number associated to a square matrix

---

Question 4.
If area of triangle is 35 sq. units with vertices (2, -6), (5, 4) and (k, 4). Then k is
(a) 12
(b) -2
(c) -12, -2
(d) 12, -2.

Answer

Answer: (d) 12, -2.

---

Question 5.
If A = \(\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]\) and A_ij is co-factors of a_ij, then A is given by
(a) a₁₁A₃₁ +a₁₂A₃₂ + a₁₃A₃₃
(b) a₁₁A₁₁ + a₁₂A₂₁ + a₁₃A₃₃
(c) a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃
(d) a₁₁A₁₁ + a₂₁A₂₁ + a₃₁A₃₁

Answer

Answer: (d) a₁₁A₁₁ + a₂₁A₂₁ + a₃₁A₃₁

---

Question 6.
Let A be a non-singular matrix of order 3 × 3. Then |adj. A| is equal to
(a) |A|
(b) |A|²
(c) |A|³
(d) 3|A|

Answer

Answer: (b) |A|²

---

Question 7.
If A is any square matrix of order 3 x 3 such that |a| = 3, then the value of |adj. A| is?
(a) 3
(b) \(\frac { 1 }{3}\)
(c) 9
(d) 27

Answer

Answer: (c) 9

---

Question 8.
If A is an invertible matrix of order 2, then det (A^-1) is equal to
(a) det (A)
(b) \(\frac { 1 }{det(A)}\)
(c) 1
(d) 0

Answer

Answer: (b) \(\frac { 1 }{det(A)}\)

---

Question 9.
If a, b, c are in A.P., then determinant

(a) 0
(b) 1
(c) x
(d) 2x

Answer

Answer: (a) 0

---

Question 10.
If x, y, z are non-zero real numbers, then the inverse of matrix A = \(\left[\begin{array}{lll}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z
\end{array}\right]\) is

Answer

Answer:

---

Question 11.
Let A =
where 0 ≤ θ ≤ 2π then
(a) Det (A) = 0
(b) Det (A) ∈ (2, ∞)
(c) Det (A) ∈ (2, 4)
(d) Det (A) ∈ [2, 4]

Answer

Answer: (d) Det (A) ∈ [2, 4]

---

Question 12.
If \(\left|\begin{array}{rr}
2x & 5 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{rr}
6 & -2 \\
7 & 3
\end{array}\right|\), then value of ‘x’ is
(a) 3
(b) ±3
(c) ±6
(d) 6

Answer

Answer: (c) ±6

---

Question 13.
Let Δ = \(\left|\begin{array}{lll}
\mathbf{A} x & x^{2} & 1 \\
B y & y^{2} & 1 \\
C z & z^{2} & 1
\end{array}\right|\) and Δ₁ = \(\left|\begin{array}{rrr}
\mathbf{A} & \mathbf{B} & \mathbf{C} \\
\boldsymbol{x} & \boldsymbol{y} & z \\
z \boldsymbol{y} & z x & x y
\end{array}\right|\), then
(a) Δ₁ = -Δ
(b) Δ ≠ Δ₁
(c) Δ – Δ₁ = 0
(d) None of these

Answer

Answer: (c) Δ – Δ₁ = 0

---

Question 14.
If x, y ∈R, then the determinant:

lies in the interval
(a) [-√2, √2]
(b) [-1, 1]
(c) [√2, 1]
(d) [-1, √2]

Answer

Answer: (a) [-√2, √2]

---

Question 15.
The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of ‘k’ will be:
(a) 9
(b) 3
(c) -9
(d) 6.

Answer

Answer: (b) 3

---

Question 16.
If A, B and C are angles of a triangle, then the determinant:
\(\left|\begin{array}{ccc}
-1 & \cos C & \cos B \\
\cos C & -1 & \cos A \\
\cos B & \cos A & -1
\end{array}\right|\) is equal to
(a) 0
(b) -1
(c) 1
(d) None of these.

Answer

Answer: (a) 0

---

Question 17.
Let A be a square matrix all of whose entries are integers. Then which of the following is true?
(a) If det A = ± 1, then A^-1 need not exist
(b) If det A = ± 1, then A^-1 exists but all entries are not necessarily integers.
(c) If det A ≠ ± 1, then A^-1 exists and all its entries are non-integers
(d) If det A = ± 1, then A^-1 exists and all its entries are integers.

Answer

Answer: (d) If det A = ± 1, then A^-1 exists and all its entries are integers.
Hint:
Since each entry of A is an integer,
∴ co-factor of each entry is also an integer.
Hence, each entry of the adjoint is an integer.
Also det A = ± 1 and A^-1 = \(\frac { 1 }{det(A)}\) (adj A).
Hence, all entries of A^-1 are integers.

---

Question 18.
The number of values of ‘k’ for which the linear equations:
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
possesses a non-zero solution is
(a) 3
(b) 2
(c) 1
(d) zero.

Answer

Answer: (b) 2
Hint:
The system possesses non-zero solution
If \(\left|\begin{array}{lll}
4 & k & 2 \\
k & 4 & 1 \\
2 & 2 & 1
\end{array}\right|\) = 0
If 4(4 – 2) + k (k – 2) + 2(2k – 8) = 0
if = 8 – k² + 2k + 4k – 16 = 0
if k² – 6k + 8 = 0
if (k – 2)(k – 4) = 0
if k = 2 or 4
k = 2.

---

Question 19.
If A = \(\left[\begin{array}{lll}
1 & \alpha & 3 \\
1 & 3 & 3 \\
2 & 4 & 4
\end{array}\right]\) is the adjoint of a 3 × 3 matrix A and |A| = 4 then α is equal to
(a) 11
(b) 5
(c) 0
(d) 4

Answer

Answer: (a) 11
Hint:
Here |adj A| = |A|^3-1
= |A|² = 4²
= 16
⇒ 1. (12 – 12) -α (4 – 6)+ 3(4 – 6) = 16
⇒ 2α – 6 = 16
⇒ 2α = 22.
Hence, α = 11.

---

Question 20.
If α, ß ≠ 0 and f(x) = α” + ß” and

= k (1 – α)²(1 – ß)²(α – ß)², then ‘4k’ is equal to:
(a) \(\frac { 1 }{αß}\)
(b) 1
(c) -1
(d) αß

Answer

Answer: (b) 1
Hint:

= [(α – 1) (ß² – 1) – (α² – 1) (ß – 1)]²
= (α – 1)²(ß – 1)²(α – ß)2.
Hence, k = 1

---

Question 21.
The system of linear equations:
x + λy – z = 0
λr – y – z = 0
x + y – λz = 0
has a non-trivial solution for
(a) Exactly one value of λ
(b) Exactly two values of λ
(c) Exactly three values of λ
(d) Infinitely many values of λ.

Answer

Answer: (c) Exactly three values of λ
Hint:
The system AX = O has non-trivial solution if det A = 0
i.,e if \(\left|\begin{array}{rrr}
1 & \lambda & -1 \\
\lambda & -1 & -1 \\
1 & 1 & -\lambda
\end{array}\right|\) = 0
⇒ (1)(λ + 1) -λ(-λ² + 1) + (-1)(λ + 1) = 0
⇒ λ + 1 + λ³ – λ – λ – 1 = 0
⇒ λ³ – λ = 0
⇒ λ(λ² – 1) = 0
⇒ λ = 0, 1, -1.
Hence, λ = -1, 0, 1.

---

Question 22.
Let on be a complex number such that 2ω + 1 = z, where z = √-3
If \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -\omega^{2}-1 & \omega^{2} \\
1 & \omega^{2} & \omega
\end{array}\right|\) = 3k the k is equal to
(a) -1
(b) 1
(c) z
(d) -z

Answer

Answer: (c) z
Hint:

[Operating R₁ → R₁ + R₂ + R₃]
= 3[- ω (ω² + 1) – ω⁴]
= 3[- ω³ – ω – ω] = 3[- 1 – 2ω]
= – 3(1 + 2ω) = – 3z.
Thus 3k = – 3z.
Hence, k = -z.

---

Question 23.
If Sis the set of distinct values of ‘h’ for which the following system of linear equations:
x + y + z = 1,
x + ay + z = 1,
ax + by + z = 1
has no solution, then S is
(a) a finite set containing two or more elements
(b) a singleton
(c) an empty set
(d) an infinite set.

Answer

Answer: (b) a singleton
Hint:
D = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & a & 1 \\
a & b & 1
\end{array}\right|\) = 0
⇒ a – 1
⇒ x + y + z = 1
and x + by + z = 0.
The planes are parallel
⇒ b = 1.
Hence, S is a singleton.

---

Question 24.
If A = \(\left[\begin{array}{rr}
2 & -3 \\
-4 & 1
\end{array}\right]\), then adj. (3A² + 12A) is equal to

Answer

Answer:

Hint:

---

Question 25.
If the system of linear equations:
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0
has a non zero solution (x, y, z), then \(\frac {xz}{y^2}\) is equal to:
(a) -10
(b) 10
(c) -30
(d) 30

Answer

Answer: (b) 10
Hint:
The given system has non-zero solution
⇒ \(\left|\begin{array}{ccc}
1 & k & 3 \\
3 & k & -2 \\
2 & 4 & -3
\end{array}\right|\)
⇒ 1(-3k + 8)-k (-9 + 4) + 3 (12 – 2k) = 0
⇒ 44 – 4k = 0
⇒ k = 11
Let z = λ
Thus x + 11 y = -3λ
and 3x + 11 y = 2λ

---

Fill in the blanks

Question 1.
If \(\left|\begin{array}{ll}
x & 2 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{ll}
3 & 2 \\
9 & 6
\end{array}\right|\), then the value of x is ……………..

Answer

Answer: ±4
Hint:
\(\left|\begin{array}{ll}
x & 2 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{ll}
3 & 2 \\
9 & 6
\end{array}\right|\)
⇒ x² – 16 = 18 – 18
⇒ x² = 16
⇒ x = ±4.

---

Question 2.
Let A be a 3 x 3 determinant and |A| = 7. Then the value of |2A| is ……………….

Answer

Answer: 56
Hint:
|2A| = 2³ |A| = 8 x 7 = 56.

---

Question 3.
If A = \(\left[\begin{array}{ll}
1 & 2 \\
4 & 2
\end{array}\right]\) Then the value of k = ……………
if |2A| = k|A|

Answer

Answer: 4
Hint:
if |2A| = 2²|A| = 4|A|
k = 4

---

Question 4.
If A is a skew-symmetric matrix of order 3, then det A = ……………..

Answer

Answer: 0.

---

Question 5.
The value of \(\left[\begin{array}{ccc}
102 & 18 & 36 \\
1 & 3 & 4 \\
17 & 3 & 6
\end{array}\right]\) is ……………..

Answer

Answer: 0
Hint:
Δ = 6\(\left|\begin{array}{ccc}
17 & 3 & 6 \\
1 & 3 & 4 \\
17 & 3 & 6
\end{array}\right|\) = 6(0) = 0

---

Question 6.
If Δ = \(\left|\begin{array}{ll}
1 & a \\
1 & b
\end{array}\right|\), then minor of ‘b’ is ………………

Answer

Answer: 1

---

Question 7.
Minor of ‘d’ is = \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\), is ………………

Answer

Answer: a

---

Question 8.
A square matrix A has inverse if and only if A is ………………

Answer

Answer: Invertible.

---

Question 9.
Co-factor of ‘b’ in \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\) is ……………

Answer

Answer: -c

---

Question 10.
If Δ = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & 0 & 1 \\
5 & 3 & 8
\end{array}\right|\) then minor of a₂₂ is …………….

Answer

Answer: -7

---

We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 4 Determinants with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Determinants MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Continuity and Differentiability Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 5 Continuity and Differentiability Objective Questions.

Continuity and Differentiability Class 12 MCQs Questions with Answers

Students are advised to solve the Continuity and Differentiability Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Continuity and Differentiability Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Continuity and Differentiability Class 12 with answers provided with detailed solutions by looking below.

Question 1.
The function
f(x) =
is continuous at x = 0, then the value of ‘k’ is:
(a) 3
(b) 2
(c) 1
(d) 1.5.

Answer

Answer: (b) 2

---

Question 2.
The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at:
(a) 4
(b)-2
(c) 1
(d) 1.5.

Answer

Answer: (d) 1.5.

---

Question 3.
The value of ‘k’ which makes the function defined by

continuous at x = 0 is
(a) -8
(b) 1
(c) -1
(d) None of these.

Answer

Answer: (d) None of these.

---

Question 4.
Differential coefficient of sec (tan^-1 x) w.r.t. x is
(a) \(\frac { x }{\sqrt{1+x^2}}\)
(b) \(\frac { x}{1+x^2}\)
(c) x\(\sqrt { 1+x^2}\)
(d) \(\frac { 1 }{\sqrt{1+x^2}}\)

Answer

Answer: (a) \(\frac { x }{\sqrt{1+x^2}}\)

---

Question 5.
If y = log (\(\frac { 1-x^2 }{1+x^2}\)) then \(\frac { dy }{dx}\) is equal to:
(a) \(\frac { 4x^3 }{1-x^4}\)
(b) \(\frac { -4x}{1-x^4}\)
(c) \(\frac {1}{ 4-x^4}\)
(d) \(\frac { -4x^3 }{1-x^4}\)

Answer

Answer: (b) \(\frac { -4x}{1-x^4}\)

---

Question 6.
If y = \(\sqrt { sin x+ y}\), then \(\frac { dy }{dx}\) is equal to
(a) \(\frac { cos x }{2y-1}\)
(b) \(\frac { cos x}{1-2y}\)
(c) \(\frac {sin x}{1-2y}\)
(d) \(\frac { sin x }{2y-1}\)

Answer

Answer: (a) \(\frac { cos x }{2y-1}\)

---

Question 7.
If u = sin^-1 (\(\frac { 2x }{1+x^2}\)) and u = tan^-1 (\(\frac { 2x }{1-x^2}\)) then \(\frac { dy }{dx}\) is
(a) \(\frac { 1 }{2}\)
(b) x
(c) \(\frac {1-x^2}{1+x^2}\)
(d) 1

Answer

Answer: (d) 1

---

Question 8.
If x = t², y = t³, then \(\frac { d^2y }{dx^2}\) is
(a) \(\frac { 3 }{2}\)
(b) \(\frac { 3 }{4t}\)
(c) \(\frac {3}{2t}\)
(d) \(\frac { 3t }{2}\)

Answer

Answer: (b) \(\frac { 3 }{4t}\)

---

Question 9.
The value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in the interval [0, √3] is
(a) 1
(b) -1
(c) \(\frac {3}{2}\)
(d) \(\frac {1}{3}\)

Answer

Answer: (a) 1

---

Question 10.
The value of ‘c’ in Mean Value Theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is
(a) \(\frac {3}{2}\)
(b) \(\frac {2}{3}\)
(c) \(\frac {1}{2}\)
(d) \(\frac {3}{4}\)

Answer

Answer: (a) \(\frac {3}{2}\)

---

Question 11.
Let f : (- 1, 1) → R be a differentiable function with f(0) = – 1 and f'(0) = 1.
Let g(x) = [f (2f(x) + 2)]². Then g'(0) =
(a) 4
(b) -4
(c) log 2
(d) -log 2.

Answer

Answer: (b) -4
Hint:
Here g (x)= [f (2 f(x) + 2)]²
g'(x) = 2[f(2f(x) + 2)] \(\frac { d }{dx}\) [2f(x) + 2 ]
= 2f(2f(x) + 2) . [2 f'(x)]
∴ g'(0) = 2f(2f(0) + 2) . [2f'(0)]
= 2f(2 (-1) +2). 2f’/(0)
= 2f(0) . 2f'(0) = 4f(0) f'(0)
= 4 (-1) (1) = -4.

---

Question 12.
\(\frac { d^2x }{dy^2}\) equals

Answer

Answer: d

Hint:

---

Question 13.
If function f(x) is differentiable at x = a, then
\(\lim _{x \rightarrow a}\) \(\frac { x^2 f(a) – a^2 f(x) }{x-a}\) is
(a) a² f(a)
(b) af(a) – a² f'(a)
(c) 2a f(a) – a² f'(a)
(d) 2a f (a) + a² f'(a).

Answer

Answer: (c) 2a f(a) – a² f'(a)
Hint:

---

Question 14.
If f: R → R is a function defined by
f(x) = [x] cos (\(\frac { 2x-1 }{2}\))π, where [x] denotes the greatest integer function, then ‘f’ is
(a) continuous for every real x
(b) discontinuous only at x = 0
(c) discontinuous only at non-zero integral values of x
(d) continuous only at x = 0.

Answer

Answer: (a) continuous for every real x
Hint:
Continuous for every real x.

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Question 15.
If y = sec (tan^-1 x), then \(\frac { dy }{dx}\) at x = 1 is equal to
(a) \(\frac {1}{2}\)
(b) 1
(c) √2
(d) \(\frac {1}{√2}\)

Answer

Answer: (d) \(\frac {1}{√2}\)
Hint:
Here y = sec (tan^-1 x).
∴ \(\frac { dy }{dx}\) = sec (tan^-1 x) tan (tan^-1 x). \(\frac { 1 }{1+x^2}\)
= sec (tan^-1 x). x . (tan^-1 x). \(\frac { 1 }{1+x}\)
\(\left.\frac{d y}{d x}\right]_{x=1}\) = sec (tan^-1 1). \(\frac { 1 }{1+1}\)
= sec (\(\frac { π }{4}\)) \(\frac { 1 }{2}\) = \(\frac { √2 }{2}\) = \(\frac { 1 }{√2}\)

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Question 16.
If g is the inverse of a function f and f'(x) = \(\frac { 1 }{1+x^5}\), then g'(x) is equal to
(a) 5x⁴
(b) \(\frac {1}{1+{g(x)}^5}\)
(c) 1 + {g(x)}⁵
(d) 1 + x⁵

Answer

Answer: (b) \(\frac {1}{1+{g(x)}^5}\)
Hint:
Here f(g(x)) = x. [∵ g is the inverse of f]
f'(g(x)) g'(x) = 1
⇒ g'(x) = \(\frac { 1 }{f'{g(x)}}\) = \(\frac { 1 }{1+{g(x)}^5}\)

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Question 17.
If the function
g(x) = \(\left\{\begin{array}{ll}
k \sqrt{x+1} & ; 0 \leq x \leq 3 \\
m x+2 & ; 3 \end{array}\right.\)
is differentiable, then the value of k + m is
(a) 2
(b) \(\frac {16}{5}\)
(c) \(\frac {10}{3}\)
(d) 4

Answer

Answer: (a) 2
Hint:
We have
g(x) = \(\left\{\begin{array}{ll}
k \sqrt{x+1} & ; 0 \leq x \leq 3 \\
m x+2 & ; 3 \end{array}\right.\)
When this function is differentiable, then it is continuous
⇒ \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\lim _{x \rightarrow 3^{+}}\) g(x) = g(3)
⇒ 2k = 3m + 2 = 2k
⇒ 2k = 3m + 2 ………… (1)
Also, LHD = \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\frac {k}{4}\)
RHD = \(\lim _{x \rightarrow 3^{+}}\) g(x) = m
∴ LHD = RHD k
⇒ \(\frac {k}{4}\) = m
Solving (1) and (2),
k = \(\frac {8}{5}\) and m = \(\frac {2}{5}\)
Hence, k + m = \(\frac {8}{5}\) + \(\frac {2}{5}\) = \(\frac {10}{2}\) = 2.

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Question 18.
For x ∈ R, f(x) = |log 2 – sin x| and g(x) =f(f(x)), then
(a) g is not differentiable at x = 0
(b) g'(0) = cos (log 2)
(c) g'(0) = -cos (log 2)
(d) g is differentiable at x = 0 and g'(0) = – sin (log 2).

Answer

Answer: (b) g'(0) = cos (log 2)
Hint:
We have : f(x) = log 2 – sin x
and g(x) = f(f cos x)
= log 2 – sin (log 2 – sin x).
Since ‘g’ is the sum of two differentiable functions,
∴ g is differentiable.
g'(x) = 0 – cos (log 2 – sin x) (0 – cos x)
= cos (log 2 – sin x) cos x.
Hence, g’ (x) = cos (log 2).

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Fill in the blanks

Question 1.
If f(x) = \(\left\{\begin{array}{c}
\frac{x^{2}-1}{x-1}, \text { when } x \neq 1 \\
k, \text { when } x=1
\end{array}\right.\) is continuous then the value of k = …………………

Answer

Answer: 2.

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Question 2.
If f(x) = x + 7, and g(x) = x – 7, x ∈R, then \(\frac { d }{dx}\) (fog) (x) = ……………….

Answer

Answer: 1.

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Question 3.
If 2x + 3y = sin x, then \(\frac { dy }{dx}\) = …………………..

Answer

Answer: \(\frac { cos x-2 }{3}\)

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Question 4.
\(\frac { d }{dx}\) (cosec^-1 x) = …………………

Answer

Answer: \(\frac { -1 }{|x|\sqrt{x^2-1}}\)

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Question 5.
\(\frac { d }{dx}\) (\(\sqrt { e^{ \sqrt{x}} }\)) = …………………

Answer

Answer: \(\frac { 1 }{4√x}\) \(\sqrt { e ^{\sqrt{x}} }\)

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Question 6.
If x = at², y = 2at, then \(\frac { dy }{dx}\) = ……………….

Answer

Answer: \(\frac { 1 }{t}\)

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Question 7.
The derivative of x^x w.r.t. x is.

Answer

Answer: x^x (1 + log x).

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Question 8.
If y = x² + 3x + 2, then \(\frac { d^2y }{dx^2}\) = ………………

Answer

Answer: 2.

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Question 9.
Value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in [-√3, 0] is ……………..

Answer

Answer: c = -1.

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Question 10.
Value of ‘c’ in LMV Theorem for f(x) = x² in [2, 4] is …………………

Answer

Answer: c = 3.

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