Which subject in class 12 is the most difficult?

The failure rates in economics and math are the highest. Experts believe it is not unexpected that Math and Economics have significant failure rates in the CBSE Class 12 board exam because they require statistical competence.

In real numbers, what is Z?

In real numbers, Z stands for integers.

In real numbers, what is R?

In real numbers, R stands for real numbers.

In real numbers, what is N?

In real numbers, N stands for natural numbers.

In real numbers, what is Q?

In real numbers, Q stands for rational numbers.

In real numbers, what is P?

In real numbers, P stands for irrational numbers.

CBSE Class 12 Maths MCQ Important Questions With Answers -

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CBSE Class 12 Maths MCQ

Now is the official start of exam season and all we want is to make our parents proud or to score as high as we can with our best efforts. But why not combine the finest efforts with clever work? This will undoubtedly result in our receiving the greatest possible score.
Brush up on some good CBSE Class 12 Maths Important MCQ Questions With Answers from Class 12 Mathematics topics to help you prepare for the Boards test and boost your chances of scoring 90% or above. MCQ Questions for all chapters of Mathematics can be found on this page. Students who will be sitting the CBSE Class 12 Mathematics Board Exams are advised to practise as many questions as possible, and these CBSE Class 12 Maths Important MCQ Questions With Answers for Mathematics class 12 have been produced by the Mathematics experts themselves!

MCQ Class 12 Maths

Q. Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a –
b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:
a) {1, 5, 9}

b) {0, 1, 2, 5}

c) ?

d) A

Answer: a) {1, 5, 9}

Q. The function ?: R⟶R defined as ?(?) = ?
3
is:
a) One-on but not onto

b) Not one-one but onto

c) Neither one-one nor onto

d) One-one and onto

Answer: d) One-one and onto

Explanation: ??? ?(?1
)
= ?(?2
) ???ℎ ?ℎ?? ?1?2 ∈ ?
⇒ ?1
3 = ?2
3
⇒ ?1 = ?2
⇒ ? is one – one
??? ? ∈ ?(????????). ?ℎ?? ??? ??? ?, ?(?) = ?
???
3 = ?
?.?. , ? = ?
1
3 ∈ ?(??????)
i.e., every element ? ∈ ?(????????) has a pre
image ?
1
3 in ?(domain)
⇒ ? is onto
∴ ? is one-one and onto

Q. If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to:
a) A

b) I + A

c) I − A

d) I

Answer: d) I

Explanation: (? − ?)
3 − 7? = ? + ? + 3? + 3? − 7? = I

Q. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let ? = {(1, 4), (2, 5), (3, 6)} be a function
from A to B. Based on the given information, ? is best defined as:
a) Surjective function

b) Injective function

c) Bijective function

d) function

Answer: b) Injective function

Explanation: Since, distinct elements of A have distinct f-images in B. Hence, f is injective
and every element of B does not have its pre-image in A, hence f is not
surjective.
∴ ? ?? ????????? ??? ?? ??? ??????????.

Q. The point(s) on the curve y = x 3 – 11x + 5 at which the tangent is y = x – 11
is/are:
a) (-2,19)

b) (2, – 9)

c) (±2, 19)

d) (-2, 19) and (2, -9)

Answer: b) (2, – 9)

Explanation: ? = ?
3 − 11? + 5 ⇒
??
??
= 3?
2 − 11
The slope of line ? = ? − 11 ?? 1 ⇒ 3?
2 − 11 = 1 ⇒ ? = ±2
∴ point is (2, -9) as (-2, 19) does not satisfy the equation of the given line

Q. For an objective function ? = ?? + ??, where ?, ? > 0; the corner points of
the feasible region determined by a set of constraints (linear inequalities) are
(0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the
maximum Z occurs at both the points (30, 30) and (0, 40) is:
a) ? − 3? = 0

b) ? = 3?

c) ? + 2? = 0

d) 2? − ? = 0

Answer: a) ? − 3? = 0

Explanation: As Z is maximum at (30, 30) and (0, 40)
⇒ 30? + 30? = 40? ⇒ ? − 3? = 0

Q. For which value of m is the line y = mx + 1 a tangent to the curve y 2 = 4x?
a) 1
2

b) 1

c) 2

d) 3

Answer: b) 1

Explanation: ? = ?? + 1 … . . (1) and ?
2 = 4? … … (2)
Substituting (1) in (2) : (?? + 1)
2 = 4?
⇒ ?2?
2 + (2? − 4)? + 1 = 0 …… . . (3)
As line is tangent to the curve
⇒ line touches the curve at only one point
⇒ (2? − 4)
2 − 4?
2 = 0 ⇒ ? = 1

Q. In a linear programming problem, the constraints on the decision variables x
and y are ? − 3? ≥ 0, ? ≥ 0, 0 ≤ ? ≤ 3. The feasible region
a) is not in the first
quadrant
b) is bounded in the first
quadrant
c) is unbounded in the
first quadrant
d) does not exist

Answer: b) is bounded in the first
quadrant

Explanation: Feasible region is bounded in the first quadrant

Q. Given that the fuel cost per hour is ? times the square of the speed the train
generates in km/h, the value of ? is:
a) 16/3

b) 1/3

c) 3

d) 3/1

Answer: d) 3/1

Explanation: Fuel cost per hour = ?(?????)
2
⇒ 48 = ?. 162 ⇒ ? =
3/16

Q. Given that matrices A and B are of order 3×n and m×5 respectively, then the
order of matrix C = 5A +3B is:
a) 3×5

b) 5×3

c) 3×3

d) 5×5

Answer: b) 5×3

Q. If y = 5 cos x – 3 sin x, then ?
2?
??
2
is equal to:

a) – y

b) y

c) 25y

d) 9y

Answer: a) – y

Explanation: ?= 5??? ? − 3??? ? ⇒
??
?? = −5??? ? − 3??? ?
⇒
?
2?
??
2 = −5??? ? + 3??? ? = −?

Q. The points on the curve ?
2
9
+
?
2
16
= 1 at which the tangents are parallel to yaxis are:
a) (0,±4)

b) (±4,0)

c) (±3,0)

d) (0, ±3)

Answer: c) (±3,0)

Explanation: ?
2
9
+
?
2
16
= 1 ⇒
2?
9
+
2?
16
??
??
= 0
⇒ ????? ?? ?????? ?? ??? ????? (?, ?) ?? ?ℎ? ????? =
−??
?? =
9?
16?

As tangent to the curve at the point (x, y) is parallel to y-axis
⇒
9?
16?
= 0 ⇒ ? = 0 and ? = ±3
∴ ?????? = (±3, 0)

Q. Given that A = [???] is a square matrix of order 3×3 and |A| = −7, then the
value of ∑ ??2??2
3
?=1
, where ??? denotes the cofactor of element ??? is:
a) 7

b) -7

c) 0

d) 49

Answer: b) -7

Explanation: |?| = −7
∴ ∑ ??2??2
3
?=1 = ?12?12 + ?22?22 + ?32?32 = |?| = −7

Q. Find the intervals in which the function f given by f (x) = x 2 – 4x + 6 is strictly
increasing:
a) (– ∞, 2) ∪ (2, ∞)

b) (2, ∞)

c) (−∞, 2)

d) (– ∞, 2]∪ (2, ∞)

Answer: b) (2, ∞)

Explanation: ?(?) = ?
2 − 4? + 6
?
′
(?) = 2? − 4
??? ?
′
(?) = 0 ⇒ ? = 2
as ?
′
(?) > 0 ⍱ ? ∈ (2, ∞)
⇒ ?(?) is Strictly increasing in (2, ∞)

Q. The real function f(x) = 2×3 – 3×2 – 36x + 7 is:
a) Strictly increasing in (−∞, −2) and strictly decreasing in ( −2, ∞)
b) Strictly decreasing in ( −2, 3)
c) Strictly decreasing in (−∞, 3) and strictly increasing in (3, ∞)
d) Strictly decreasing in (−∞, −2) ∪ (3, ∞)

Answer: b) Strictly decreasing in ( −2, 3)

Explanation: ?′(?) = 6(?² − ? − 6) = 6(? − 3)(? + 2)
As ?′(?) < 0 ⍱ ? ∈ (−2, 3)
⇒ ?(?) is strictly decreasing in (−2, 3)

Q. Given that A is a non-singular matrix of order 3 such that A2 = 2A, then value
of |2A| is:
a) 4

b) 8

c) 64

d) 16

Answer: c) 64

Explanation:
2 = 2?
⇒ |?
2
| = |2?|
⇒ |?|
2 = 2
3
|?| as |??| = ?
?|?| ??? ? ?????? ?????? ?? ????? ?
⇒ either |?| = 0 ?? |?| = 8
But A is non-singular matrix
∴ |?| = 8
2 = 64

Q. The value of ? for which the function ?(?) = ? + ???? + ? is strictly
decreasing over R is:
a) ? < 1

b) No value of b exists

c) ? ≤ 1

d) ? ≥ 1

Answer: b) No value of b exists

Explanation: ?′(?) = 1 − ??? ? ⇒ ?′
(?) ≥ 0 ⍱? ∈ ?
⇒ ?? ????? ?? ? ?????s

Q. 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: c) (6,8) ∈ R

Explanation: ? = ? − 2 ??? ? > 6
⇒ (6, 8) ∈ R

Q. The point(s), at which the function f given by ?(?) ={
?
|?|
, ? < 0
−1, ? ≥ 0
is continuous, is/are:
a) ??R

b) ? = 0

c) ?? R –{0}

d) ? = −1and 1

Answer: a) ??R

Explanation: ?(?) = {
?
−?
= −1 , ? < 0 − 1 , ? ≥ 0
⇒ ?(?) = −1 ⍱ ? ∈ ?
⇒ ?(?)?? ????????? ⍱ ? ∈ ? ?? ?? ?? ? ???????? ???????

Q. A linear programming problem is as follows:
???????? ? = 30? + 50?
subject to the constraints,
3? + 5? ≥ 15
2? + 3? ≤ 18
? ≥ 0, ? ≥ 0
In the feasible region, the minimum value of Z occurs at
a) a unique point

b) no point

c) infinitely many points

d) two points only

Answer: c) infinitely many points

Explanation: Corner points of feasible region ? = 30? + 50?
(5,0) 150
(9,0) 270
(0,3) 150
(0,6) 300
The minimum value of ? occurs at infinitely many points

Q. The area of a trapezium is defined by function ? and given by ?(?) = (10 +
?)√100 − ?
2, then the area when it is maximised is:
a) 75??2

b) 7√3??2

c) 75√3??2

d) 5??2

Answer: c) 75√3??2

Explanation: ?′(?) =−2?
2−10?+100
√100−?
2
?′(?) = 0 ⇒ ? = −10 ?? 5 , But ? > 0 ⇒ ? = 5
?”(?) =2?
3−300?−1000
(100−?)
3
2
⇒ ?”(5) =
−30
√75 < 0
⇒ Maximum area of trapezium is 75√3 ??2 when x = 5

Q. Given that A is a square matrix of order 3 and | A | = – 4, then | adj A | is
equal to:
a) -4

b) 4

c) -16

d) 16

Answer: d) 16

Explanation: as |??? ?| = |?|
?−1
, where ? is order of the square matrix ?
= (−4)
2 = 16

Q. A relation R in set A = {1,2,3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}.
Which of the following ordered pair in R shall be removed to make it an
equivalence relation in A?
a) (1, 1)

b) (1, 2)

c) (2, 2)

d) (3, 3)

Answer: b) (1, 2)

Q. The point at which the normal to the curve y = ? +
1
?
, x > 0 is perpendicular to
the line 3x – 4y – 7 = 0 is:
a) (2, 5/2)

b) (±2, 5/2)

c) (- 1/2, 5/2)

d) (1/2, 5/2)

Answer: a) (2, 5/2)

Explanation: ?(?) = ? +
1
?
, ? > 0 ⇒ ?
′
(?) = 1 −
1
?
2 =
?
2−1
?
2
, ? > 0
As normal to the curve ? = ?(?) at some point (x, y) is Ʇ to given line
⇒ (
?
2
1−?
2
) ×
3
4
= −1 (?1. ?2 = −1)
⇒ ?
2 = 4 ⇒ ? = ±2
But ? > 0, ∴ ? = 2
Therefore point=(2,
5
2
)

Q. The most economical speed to run the train is:
a) 18km/h

b) 5km/h

c) 80km/h

d) 40km/h

Answer: c) 80km/h

Explanation: ??
??
=
375
4
−
600000
?
2
Let ??
??
= 0 ⇒ ? = 80 ??/ℎ

Q. The fuel cost for the train to travel 500km at the most economical speed is:
a) ₹ 3750

b) ₹ 750

c) ₹ 7500

d) ₹ 75000

Answer: c) ₹ 7500

Explanation: Fuel cost for running 500 km 375
4
? =
375
4
× 80 = ??. 7500

Q. The function f : R → R defined by f(x) = 3 – 4x is

(a) Onto

(b) Not onto

(d) None of these

Answer: (a)

Q. Let * be a binary operation on Q, defined by a * b = 3??/5 is

(a) Commutative

(b) Associative

(d) None of these

Answer: (c)

Q. The length of the longer diagonal of the parallelogram is constructed on 5a + 2b and a – 3b. If it is given that |a| = 2√2, |b| = 3 and angle between a and b is ?4, is

(a) 15

(b) √113

(d) √369

Answer: (c) √593

Q. Let S = {1, 2, 3, 4, 5} and let A = S × S. Define the relation R on A as follows:

(a, b) R (c, d) iff ad = cb. Then, R is

(a) reflexive only

(b) Symmetric only

(d) Equivalence relation

Answer: (d)

Q. Let A = {x : -1 ≤ x ≤ 1} and f : A → A is a function defined by f(x) = x |x| then f is

(a) a bijection

(b) injection but not surjection

(d) neither injection nor surjection

Answer: (a)

Q. Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.

(a) 1

(b) 2

(d) 4

Answer: (b)

Q. If sin-1x + sin-1y + sin-1z = then the value of x + y² + z3 is

(a) 1

(b) 3

(d) 5

Answer: (b) 3

Q. The maximum value of f = 4x + 3y subject to constraints x ≥ 0, y ≥ 0, 2x + 3y ≤ 18; x + y ≥ 10 is

(a) 35

(b) 36

(d) none of these

Answer: (d) None of these

Q. Objective function of a L.P.P.is

(a) a constant

(b) a function to be optimised

(d) none of these

Answer: (b) a function to be optimised

Q. The region represented by x ≥ 0, y ≥ 0 is

(a) first quadrant

(b) second quadrant

(d) fourth quadrant

Answer: (a) first quadrant

Q. Maximize Z = 10×1 + 25×2, subject to 0 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 3, x1 + x2 ≤ 5.

(a) 80 at (3, 2)

(b) 75 at (0, 3)

(d) 95 at (2, 3)

Answer: (d) 95 at (2, 3)

Q. Z = 7x + y, subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0. The minimum value of Z occurs at

(a) (3, 0)

(b) (1/2,5/2)

(d) (0, 5)

Answer: (d) (0, 5)

Q. If a matrix has 6 elements, then number of possible orders of the matrix can b

(a)2

(b)4

(d)6

Answer: (b)4

Q. The value of c in mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5] is

(a) 6 ± √(13/3)

(b) 6 + √(13/3)

(d) None of these

Answer: (c)

Total number of possi

(a) 6

(b) 36

(d) 64

Answer: (d) 64

Q. The diagonal elements of a skew symmetric matrix are

(a) all zeroes

(b) are all equal to some scalar k(≠ 0)

(d) none of these

Answer: (a) all zeroes

Q. The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan ?/2)} at the point ‘t’ is

(a) tan t

(b) cot t

(d) None of these

Answer: (a) tan t

Q. If a matrix A is both symmetric and skew symmetric then matrix A is

(a) a scalar matrix

(b) a diagonal matrix

(d) a rectangular matrix.

Answer: (c) a zero matrix of order n × n

Q. If there is an error of a% in measuring the edge of a cube, then percentage error in its surface area is

(a) 2a%

(b) ?/2 %

(d) None of these

Answer: (b)

Q. A function /is said to be continuous for x ∈ R, if

(a) it is continuous at x = 0

(b) differentiable at x = 0

(d) differentiable for x ∈ R

Answer: (d) differentiable for x ∈ R

Q. The function f(x) = log (1 + x) – 2?/2+? is increasing on

(a) (-1, ∞)

(b) (-∞, 0)

(d) None of these

Answer: (a) (-1, ∞)

Q. The number of commutative binary operations that can be defined on a set of 2 elements is

(a) 8

(b) 6

(d) 2

Answer: (d)

Q. The value of ? for which the function ?(?) = ? + ???? + ? is strictly decreasing over R is:

a) ? < 1

b) No value of b exists

c) ? ≤ 1

d) ? ≥ 1

Answer: b) No value of b exists

Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is

(a) reflexive and symmetric

(b) symmetric and transitive

(d) symmetric

Answer: (d) symmetric

Q. Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then R is

(a) reflexive but not symmetric

(b) reflexive but not transitive

(d) neither symmetric, nor transitive

Answer: (a) reflexive but not symmetric

Q. If AB × AC = 2?̂ −4?̂ + 4?̂ , then the are of ΔABC is

(a) 3 sq. units

(b) 4 sq. units

(d) 9 sq. units

Answer: (a) 3 sq. units

Q. A relation S in the set of real numbers is defined as the number, then relation S is

(a) reflexive

(b) reflexive and symmetric

(d) symmetric and transitive

Answer: (a) reflexive

Q. Find the value of λ so that the vectors 2?−4?̂ +?̂ and 4?−8?̂ +??̂ are parallel.

(a) -1

(b) 3

(d) 2

Answer: (d) 2

Q. Set A has 3 elements and the set B has 4 elements. Then the number of functions that can be defined from set A to set B is

(a) 144

(b) 12

(d) 64 injective

Answer: (c) 24

Q. Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a.b + b.c + c.a is

(a) 47

(b) 25

(d) -25

Answer: (d) -25

Q. If |a| = |b| = 1 and |a + b| = √3, then the value of (3a – 4b).(2a + 5b) is

(a) -21

(b) −21/2

(d) 21/2

Answer: (b) −21/2

Q. The domain of the function^ = sin’ -‘(V) is

(a) [0, 1]

(b) (0, 1)

(d) Φ

Answer: (c) [-1, 1]

Q. In a linear programming problem, the constraints on the decision variables x and y are ? − 3? ≥ 0, ? ≥ 0, 0 ≤ ? ≤ 3. The feasible region

a) is not in the first quadrant

b) is bounded in the first quadrant

c) is unbounded in the first quadrant

d) does not exist

Answer: b) is bounded in the first quadrant

Q. Write the number of points where f(x) = |x + 2| + |x – 3| is not differentiable.

(a) 2

(b) 3

(d) 1

Answer: (a) 2

Q. 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: c) (6,8) ∈ R

Q. The function f(x) = x + 4/? has

(a) a local maxima at x = 2 and local minima at x = -2

(b) local minima at x = 2, and local maxima at x = -2

(d) absolute minima at x = 2 and absolute maxima at x = -2

Answer: (b) local minima at x = 2, and local maxima at x = -2

Q. If a matrix A is both symmetric and skew-symmetric, then

(a) A is a diagonal matrix

(b) A is a zero matrix

(d) A is a square matrix

Answer: (b) A is a zero matrix

Q. The total revenue in ₹ received from the sale of x units of an article is given by R(x) = 3x² + 36x + 5. The marginal revenue when x = 15 is (in ₹ )

(a) 126

(b) 116

(d) 90

Answer: (a) 126

Q. The number of binary operations that can be defined on a set of 2 elements is

(a) 8

(b) 4

(d) 64

Answer: (c) 16

Q. The value of tan²(sec-12) + cot2(cosec-13) is

(a) 5

(b) 11

(d) 15

Answer: (b) 11

Q. The side of an equilateral triangle is increasing at the rate of 2 cm/s. The rate at which area increases when the side is 10 is

(a) 10 cm²/s

(b) √3 cm²/s

(d) 10/3 cm²/s

Answer: (c) 10√3 cm²/s

Q. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is

(a) reflexive but not transitive

(b) transitive but not symmetric

(d) None of these

Answer: (c) equivalence

Q. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let ? = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Based on the given information, ? is best defined as:

a) Surjective function

b) Injective function

c) Bijective function

d) function

Answer: b) Injective function

Q. Let us define a relation R in R as aRb if a ≥ b. Then R is

(a) an equivalence relation

(b) reflexive, transitive but not symmetric

(d) neither transitive nor reflexive but symmetric

Answer: (b) reflexive, transitive but not symmetric

Q. Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is

(a) Reflexive and symmetric

(b) Transitive and symmetric

(d) Reflexive, transitive but not symmetric

Answer: (d) Reflexive, transitive but not symmetric

Q. If * is a binary operation on set of integers I defined by a * b = 3a + 4b – 2, then find the value of 4 * 5.

(a) 35

(b) 30

(d) 29

Answer: (b) 30

Q. Given that matrices A and B are of order 3×n and m×5 respectively, then the order of matrix C = 5A +3B is:

a) 3×5

b) 5×3

c) 3×3

d) 5×5

Answer: (b) 5×3

Q. The function f : A → B defined by f(x) = 4x + 7, x ∈ R is

(a) one-one

(b) Many-one

(d) Even

Answer: (a) one-one

Q. Find the height of the cylinder of maximum volume that can be is cribed in a sphere of radius a.

(a) 2?/3

(b) 2?/√3

(d) 2?/3

Answer: (b) 2?/√3

Q. The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is

(a) scalene

(b) equilateral

(d) None of these

Answer: (c) isosceles

Q. The equation of the normal to the curves y = sin x at (0, 0) is

(a) x = 0

(b) x + y = 0

(d) x – y = 0

Answer: (b) x + y = 0

Q. The radius of a cylinder is increasing at the rate of 3 m/s and its height is decreasing at the rate of 4 m/s. The rate of change of volume when the radius is 4 m and height is 6 m, is

(a) 80π cu m/s

(b) 144π cu m/s

(d) 64 cu m/s

Answer: (a) 80π cu m/s

Q. Derivative of cot x° with respect to x is

(a) cosec x°

(b) cosec x° cot x°

(d) -1° cosec x° cot x°

Answer: (c) -1° cosec2 x°