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Maths MCQs for Class 12 with Answers Chapter 5 Continuity and Differentiability

July 11, 2020 by Kishen Leave a Comment

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 5 Continuity and Differentiability. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Continuity and Differentiability MCQs Pdf with Answers to know their preparation level.

Continuity and Differentiability Class 12 Maths MCQs Pdf

1. Given functions f(x) = $\frac{x^{2}-4}{x-2}$ and g(x) = x + 2, x <= R. Then which of the following is correct?
(a) f is continuous at x = 2,
g is continuous at x = 2
(b) f is continuous at x = 2,
g is not continuous at x = 2
(c) f is not continuous at x = 2,
g is continuous at x = 2
(d) f is not continuous at x = 2,
g is not continuous at x = 2

Answer/Explanation

Answer: c
Explaination: (c), as f(2) is not defined so / is not continuous at x = 2 ‘g’ is a polynomial function, so continuous at x = 2.

Answer/Explanation

Answer: d
Explaination:

3.

for x = 2, then value of k for which f is continuous is
(a) -2
(b) -1
(c) 0
(d) 1

Answer/Explanation

Answer: d
Explaination:

4. A function /is said to be continuous for x ∈ R, if
(a) it is continuous at x = 0
(b) differentiable at x = 0
(c) continuous at two points
(d) differentiable for x ∈ R

Answer/Explanation

Answer: d
Explaination: (d), as differentiable functions is continuous also.

5. Afunction

is continuous at x = 0 for
(a) k = 1
(b) k = 2
(c) k = $\frac{1}{2}$
(d) k = $\frac{3}{2}$

Answer/Explanation

Answer: a
Explaination:

6. Write the number of points where f(x) = |x + 2| + |x – 3| is not differentiable.
(a) 2
(b) 3
(c) 0
(d) 1

Answer/Explanation

Answer: a
Explaination: (a), as f(x) = |x – a| is continuous at x = a but not differentiable thereat.

7. Derivative of cot x° with respect to x is
(a) cosec x°
(b) cosec x° cot x°
(c) -1° cosec2 x°
(d) -1° cosec x° cot x°

Answer/Explanation

Answer: c
Explaination:

Answer/Explanation

Answer: a
Explaination:

9. If f(x) = $\log _{x^{2}}(\log x)$ , then f(e) is
(a) 0
(b) 1
(c) $\frac{1}{e}$
(d) $\frac{1}{2e}$

Answer/Explanation

Answer: d
Explaination:

10. If f(x) = e^x and g(x) = log_e x, then (gof)’ (x) is
(a) 0
(b) 1
(c) e
(d) 1 + e

Answer/Explanation

Answer: b
Explaination:

11.

Answer/Explanation

Answer:
Explaination:

12. If y = x^x-∞, , then x(l -y log x) $\frac{d y}{d x}$ is equal to
(a) x²
(b) y²
(c) xy²
(d) x²y

Answer/Explanation

Answer: b
Explaination:

13. The derivative of sin x with respect to log x is
(a) cos x
(b) x cos x
(c) $\frac{\cos x}{\log x}$
(d) $\frac{1}{x}$ cos x

Answer/Explanation

Answer: b
Explaination:

14. Ify = Ae^5x,+ Be^-5x x then $\frac{d^{2} y}{d x^{2}}$ is equal to
(a) 25y
(b) 5y
(c) -25y
(d) 10y

Answer/Explanation

Answer:
Explaination:

15. Given the function
the function is continuous at x = 0, state true or false.

Answer/Explanation

Answer:
Explaination: False, as ‘f’ is not defined at x = 0. i.e.f (0) does not exist.

46. The function f(x) = $\frac{x+1}{1+\sqrt{1+x}}$ continuous at x = 0 if/(0) is _________ .

Answer/Explanation

Answer:
Explaination:

17. A function f(x) = $\frac{x}{x-5}$ x ∈ R, is a continuous function. State true or false.

Answer/Explanation

Answer:
Explaination: False, as for x = 5, f(5) is not defined.

18. A function f(x) = sin x + cos x is continuous function. State true or false.

Answer/Explanation

Answer:
Explaination: True, as sum of two continuous functions is a continuous function.

19. Discuss the continuity of the function fix)= $\frac{1}{x-5}$ for x ∈ R.

Answer/Explanation

Answer:
Explaination: f(x) = $\frac{1}{x-5}$ , as f(5) is not defined, therefore function is not continuous at x = 5.

20. Discuss the continuity of the function

Answer/Explanation

Answer:
Explaination:
f(x)= $\frac{x^{2}-25}{x-5}$ , x ≠ 5. As x ≠ 5, therefore, value of function exists for all x(≠5) ∈ R.
Also $\lim _{x \rightarrow a}$ f(x) = f(a) = a + 5, (a ≠ 5). Hence, continuous.

21. Check whether the function f(x) = 2x² + 1 is continuous at x = 0.

Answer/Explanation

Answer:
Explaination: f(x) = 2x² + 1, as $\frac{x^{2}-25}{x-5}$ f(x) = f(0) = 1. Hence, continuous

22. Give an example of a function which is continuous but not differentiable at exactly two points.

Answer/Explanation

Answer:
Explaination:
We know function f(x)=|x – a| is continuous at x = a but not differentiable at x = a.
∴ functions |x| and |x – 1| are continuous but not differentiable at x = 0 and 1.
∴ function is h(x) = |x| + |x – 1|.

23. Determine the value of the constant ‘k’

Answer/Explanation

Answer:
Explaination:

24. Determine the value of ‘k’ for which the following function is continuous at x = 3:

Answer/Explanation

Answer:
Explaination:

25. For what value of ‘k’ is the function

continuous at x = 0?

Answer/Explanation

Answer:
Explaination:

26. Find the value of k, so that the function

is continuous at x = 1

Answer/Explanation

Answer:
Explaination:

27. Determine the value of the constant ‘k’ so that the function

is continuous at x = 0. [Delhi]

Answer/Explanation

Answer:
Explaination:

28. For what value of ‘k’ is the function

continuous at x = 0? [Foreign]

Answer/Explanation

Answer:
Explaination:

29. The derivative of

State true or false.

Answer/Explanation

Answer:
Explaination:

30. Find $\frac{d y}{d x}$ , if x² + y² = 5

Answer/Explanation

Answer:
Explaination:

31. Differentiate sin^-1x², with resepct to x.

Answer/Explanation

Answer:
Explaination:

32. Find $\frac{d y}{d x}$ , if sin y + x = log x

Answer/Explanation

Answer:
Explaination:

33. Find $\frac{d y}{d x}$ at x = 1, y = $\frac{\pi}{4}$ if sin²y + cos xy = K. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:

34. Differentiate tan^-1 $\left(\frac{1+\cos x}{\sin x}\right)$ with respect to x. [CBSE 2018]

Answer/Explanation

Answer:
Explaination:

35. If y = 2^√x, then $\frac{d y}{d x}$ is _______ .

Answer/Explanation

Answer:
Explaination:

36. If y = log (tan x) + log (cot x), then $\frac{d y}{d x}$ is _______ .

Answer/Explanation

Answer:
Explaination:

37. If $f(x)=9^{x^{2}+2 x}$ , then f(-1) is _______ .

Answer/Explanation

Answer:
Explaination:

38. Differentiate e^-2x with respect to x.

Answer/Explanation

Answer:
Explaination:

39. Differentiate 5^{sin x}, with respect to x.

Answer/Explanation

Answer:
Explaination:

40. Differentiate log_e(sin x) with respect to x.

Answer/Explanation

Answer:
Explaination:

41. Differentiate log x² w.r.t x.

Answer/Explanation

Answer:
Explaination:

42. If y = e^{-3 log x} then find $\frac{d y}{d x}$ .

Answer/Explanation

Answer:
Explaination:

43.

Answer/Explanation

Answer:
Explaination:

44. Find $\frac{d y}{d x}$ at t = $\frac{2 \pi}{3}$ when x = 10(t – sin t) and y = 12(1 – cos t). [Foreign 2017]

Answer/Explanation

Answer:
Explaination:

45. Find $\frac{d^{2} y}{d x^{2}}$ , if y = log x

Answer/Explanation

Answer:
Explaination:

46. If y = sin 3x, find y₂

Answer/Explanation

Answer:
Explaination:
y = sin 3x
y₁ = 3 cos 3x
y₂ = -9 sin 3x.

47. Find $\frac{d^{2} y}{d x^{2}}$ if y = e^-3x

Answer/Explanation

Answer:
Explaination:

48. Verify the Rolle’s Theorem for the function f(x) = x² in the inverval [-1, 1].

Answer/Explanation

Answer:
Explaination:
Function f(x) = x² is continuous in [-1,1 ], differentiable in ( -1, 1) and f(-1) = f(1). Hence, Rolle’s Theorem verified.
⇒ f'(c) = 0
⇒ 2c = 0
⇒ c = 0 for c ∈ (-1, 1)

49. Verify the Rolle’s Theorem for die functiony(x) = |x| in the inverval [-1, 1]. [HOTS]

Answer/Explanation

Answer:
Explaination: Not verified, as /(x) =|x| is not derivable at x = 0.

50. Verify the Rolle’s Theorem for the function f(x) = sin 2x in [0, π].

Answer/Explanation

Answer:
Explaination:
Function f(x) = sin 2x is continuous in [0, π], differentiable in (0, π) and (0)= f(π)
Hence, Rolle’s theorem verified.
⇒ f'(c) = 2 cos 2c
⇒ 2cos 2c = 0

We hope the given Maths MCQs for Class 12 with Answers Chapter 5 Continuity and Differentiability will help you. If you have any query regarding CBSE Class 12 Maths Continuity and Differentiability MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Maths MCQs for Class 12 with Answers Chapter 4 Determinants

July 11, 2020 by Kishen Leave a Comment

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 4 Determinants. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Determinants MCQs Pdf with Answers to know their preparation level.

Determinants Class 12 Maths MCQs Pdf

Answer/Explanation

Answer: d
Explaination:

Answer/Explanation

Answer: a
Explaination: (a), Δ = 6(-1)- 1(1) = -7.

Answer/Explanation

Answer: d
Explaination:

4. Let A be a square matrix of order 2 × 2, then |KA| is equal to
(a) K|A|
(b) K²|A|
(c) K³|A|
(d) 2K|A|

Answer/Explanation

Answer: b
Explaination:

Answer/Explanation

Answer: a
Explaination:

6. Let x, yeR, then the determinant

Answer/Explanation

Answer: a
Explaination:

Answer/Explanation

Answer: d
Explaination: (d), as value of determinant is sum of the product of elements of any row and column and then- respective cofactor.

8. A and B are invertible matrices of the same order such that |(AB)^-1| = 8, If |A| = 2, then |B| is
(a) 16
(b) 4
(c) 6
(d) $\frac{1}{16}$

Answer/Explanation

Answer: d
Explaination:

9. Determinant is a number associated to a matrix, state true or false.

Answer/Explanation

Answer:
Explaination: False, as determinant is a number associated to a square matrix.

10.

Answer/Explanation

Answer:
Explaination: (a + ib) (a – ib) – (c + id) (c – id) = a² + b² – c² – d².

11.

Answer/Explanation

Answer:
Explaination:
sin 30° cos 60° + cos 30° sin 60°
= sin(30° + 60°) = sin 90° = 1.

12.

Answer/Explanation

Answer:
Explaination:
18x + 45 – 15x -6 = 0
⇒ 3x = -39
⇒ x = -13.

13. Evaluate the determinant

Answer/Explanation

Answer:
Explaination:

14.

Answer/Explanation

Answer:
Explaination:

15. Value of determinant

is 0, state true or false.

Answer/Explanation

Answer:
Explaination: True, as on performing C₃ → C₃ + C₂ we notice elements of C₁ and C₃ are proportional.

16. If A is a matrix of order 3 × 3, then |KA| = _______ .

Answer/Explanation

Answer:
Explaination:

17. The determinant

Answer/Explanation

Answer:
Explaination:

18. Evaluate the determinant

Answer/Explanation

Answer:
Explaination:

19. What is the value of the following determinant?

Answer/Explanation

Answer:
Explaination:

20. If for any 2 × 2 square matrix A, A(adj A)

Answer/Explanation

Answer:
Explaination:

21. If area of a triangle with vertices (3, 2), (-1,4) and (6, k) is 7 sq units, then possible values of k are ______ .

Answer/Explanation

Answer:
Explaination:

22. The points (a + 5, a – 4), (a – 2, a + 3) and (a, a) are non-collinear for any value of a. State true or false.

Answer/Explanation

Answer:
Explaination:

23. Find the area of the triangle with vertices (-1, 2), (4, 0) and (3, 9).

Answer/Explanation

Answer:
Explaination:

24. Find the value of x, such that the points (0, 2), (1, x) and (3, 1) are collinear.

Answer/Explanation

Answer:
Explaination:

25. Show that points A(a, b + c), B(b, c + a) and C(c, a + b) are collinear. [NCERT]

Answer/Explanation

Answer:
Explaination:

26. Find the cofactor of a₁₂ of the determinant

Answer/Explanation

Answer:
Explaination:

27.

cofactor of the element a₂₁ [Foreign 2015]

Answer/Explanation

Answer:
Explaination: Cofactor of a2l = (-1)³ (18 – 21) = 3

28. In the determinant
find
(i) minor of element 3
(ii) cofactor of element 5.

Answer/Explanation

Answer:
Explaination:

29. If A and B are invertible matrices of the same order (AB)^-1 is _____ .

Answer/Explanation

Answer:
Explaination:

30.

Answer/Explanation

Answer:
Explaination:

31. A is invertible matrix of order 3 × 3 and |A| = 9, then value of |A^-1|is

Answer/Explanation

Answer:
Explaination:

32. If the value of a third order determinant is 7, then the value of a determinant formed by replacing each element by its cofactor will be 49. State true or false.

Answer/Explanation

Answer:
Explaination: True, as |adj.A| = |A|n-1, A is matrix of order 3 × n

33. Inverse of matrix

exists, state true or false.

Answer/Explanation

Answer:
Explaination:

34. For what value of k, the matrix $\left[\begin{array}{ll}{k} & {2} \\ {3} & {4}\end{array}\right]$ has no inverse?

Answer/Explanation

Answer:
Explaination:

35. Write A^-1 for A = $\left[\begin{array}{ll}{2} & {5} \\ {1} & {3}\end{array}\right]$ [Delhi 2011]

Answer/Explanation

Answer:
Explaination:

36. Given a square matrix A of order 3 × 3, such that |A| = 12, find the value of |A. adj A|. [HOTS]

Answer/Explanation

Answer:
Explaination:
|A adj A| = |A|³ as matrix A is of order 3 × 3.
∴ |A adj A| = |A|³ = (12)³ = 1728

37. If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^-1) = (det A)^k

Answer/Explanation

Answer:
Explaination:

38.

Answer/Explanation

Answer:
Explaination:

39. If for any 2 × 2 square matrix A, A(adj A) = $\left[\begin{array}{ll}{8} & {0} \\ {0} & {8}\end{array}\right]$ , then write the value of |A|. [Delhi 2017]

Answer/Explanation

Answer:
Explaination:

40.

Answer/Explanation

Answer:
Explaination:

We hope the given Maths MCQs for Class 12 with Answers Chapter 4 Determinants will help you. If you have any query regarding CBSE Class 12 Maths Determinants MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Maths MCQs for Class 12 with Answers Chapter 3 Matrices

July 11, 2020 by Kishen Leave a Comment

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 3 Matrices. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Matrices MCQs Pdf with Answers to know their preparation level.

Matrices Class 12 Maths MCQs Pdf

1. If a matrix has 6 elements, then number of possible orders of the matrix can be
(a) 2
(b) 4
(c) 3
(d) 6

Answer/Explanation

Answer: b
Explaination: (b), as 6 → 1 × 6, 2 × 3, 3 × 2, 6 × 1.

2. If A = [a_ij] is a 2 × 3 matrix, such that a_ij = $\frac{(-i+2 j)^{2}}{5}$ .then a₂₃ is

Answer/Explanation

Answer: d
Explaination: (d), as a₂₃ = $\frac{(-2+6)^{2}}{5}=\frac{16}{5}$

3. If A = diag(3, -1), then matrix A is

Answer/Explanation

Answer: c
Explaination: (c), as diag (3, -1) is a diagonal matrix. Its order is 2 × 2 with diagonal elements 3 and (-1).

4. Total number of possible matrices of order 2 × 3 with each entry 1 or 0 is
(a) 6
(b) 36
(c) 32
(d) 64

Answer/Explanation

Answer: d
Explaination: (d), as total elements are 6 and each entry can be done in 2 ways. Hence, total possibilities = 2⁶ = 64.

5. If A is a square matrix such that A²=A, then (I + A)² – 3A is
(a) I
(b) 2A
(c) 3I
(d) A

Answer/Explanation

Answer: a
Explaination: (a), as (I + A)² -3A = I² + IA + AI + A² – 3A = I + A + A + A – 3A=I

6. If matrices A and B are inverse of each other then
(a) AB = BA
(b) AB = BA = I
(c) AB = BA = 0
(d) AB = 0, BA = I

Answer/Explanation

Answer: b
Explaination: (b), by definition.

Answer/Explanation

Answer: d
Explaination:

8. The diagonal elements of a skew symmetric matrix are
(a) all zeroes
(b) are all equal to some scalar k(≠ 0)
(c) can be any number
(d) none of these

Answer/Explanation

Answer: a
Explaination:
(a), as in skew symmetric matrix, a_ij = -a_ji
⇒ a_ii = – a_ii
⇒ 2a_ii = 0
⇒ a_ii = 0, i.e. diagonal elements are zeroes.

9. If A = $\left[\begin{array}{ll}{5} & {x} \\ {y} & {0}\end{array}\right]$ and A = A’ then
(a) x = 0, y = 5
(b) x = y
(c) x + y = 5
(d) x – y = 5

Answer/Explanation

Answer: b
Explaination:

10. If a matrix A is both symmetric and skew symmetric then matrix A is
(a) a scalar matrix
(b) a diagonal matrix
(c) a zero matrix of order n × n
(d) a rectangular matrix.

Answer/Explanation

Answer: c
Explaination: (c), as it satisfies a_ij = a_ji, = -a_ji and a_ii = 0

11.

Answer/Explanation

Answer:
Explaination: False, as their orders are different.

12. If a matrix has 5 elements, write all possible orders it can have. [AI2011]

Answer/Explanation

Answer:
Explaination: Possible orders are 1 × 5, 5 × 1.

13. If A is a 3 × 3 matrix, whose elements are given by ai_ij = $\frac{1}{3}$ |-3i + j|, then write the value a₂₃. [Foreign 2013]

Answer/Explanation

Answer:
Explaination: Given a_ij = $\frac{1}{3}$ |-3i + j|
∴ a₂₃ = $\frac{1}{3}$ |-6 + 3| = 1.

14.

Answer/Explanation

Answer:
Explaination:

15.

Answer/Explanation

Answer:
Explaination: x + y + z = 9, x + z = 5, y + z = 1, on solving we get
x = 2, y = 4, z = 3

16.

Answer/Explanation

Answer:
Explaination:
cos α = 1, sin α = 0
⇒ α = 0°

17.

Answer/Explanation

Answer:
Explaination: 3a₂₂ – 4a₃₃ = 3 × 5 – 4 × 4 = -1

18.

Answer/Explanation

Answer:
Explaination: These matrices are not equal as their orders are not same.

19.

Answer/Explanation

Answer:
Explaination:

20. If matrix X is such that

then order of matrix X is ______ .

Answer/Explanation

Answer:
Explaination: 2 × 2, as (2 × 2) × (2 × 3) = 2 × 3

21. If A and B are matrices of order 3 × m and 3 × n respectively such that m = n, then order of 2A + 7B is _____ .

Answer/Explanation

Answer:
Explaination: 3 × m or 3 × n, as order of sum of two matrices is same as order of given matrices.

22. Matrix multiplication is commutative, state true or false.

Answer/Explanation

Answer:
Explaination: False, as AB ≠ BA in general.

23. The negative of matrix is obtained by multiplying the matrix by _____ .

Answer/Explanation

Answer:
Explaination: -1, as (-1)A = -A.

24.

Answer/Explanation

Answer:
Explaination: a₂₂ + b₂₁ = 4 – 3 = 1

25.

then write the value of k.

Answer/Explanation

Answer:
Explaination:

26.

Answer/Explanation

Answer:
Explaination:

27. Find x and y, if

Answer/Explanation

Answer:
Explaination:

28.

find the value of x [Delhi 2012]

Answer/Explanation

Answer:
Explaination:

29.

Answer/Explanation

Answer:
Explaination:

30. Find a matrix X such that 2A + B + X =

Answer/Explanation

Answer:
Explaination:

31. If A is a square matrix such that A² = A, then write the value of (I + A)² – 3A. [Foreign 2012]

Answer/Explanation

Answer:
Explaination: (I + A)² – 3A = I² + IA+ AI + A² -3A = I + A+ A+ A – 3A = I

32.

Answer/Explanation

Answer:
Explaination:

33. If matrix A = $\left[\begin{array}{cc}{a} & {b} \\ {c} & {-a}\end{array}\right]$ is the square root of the 2 × 2 identity matrix, then find the relation a between a, b and c.

Answer/Explanation

Answer:
Explaination:

34.

Answer/Explanation

Answer:
Explaination:

35. If A and B are symmetric matrices, then AB -BA is a _______ matrix.

Answer/Explanation

Answer:
Explaination:
skew symmetric, as (AB – BA)’
= (AB)’ – (BA)’
= B’A’ – A’B’ = BA- AB = -(AB – BA) (∵ A’ =A,B’ = B)

36. If A is a skew symmetric matrix then A² is a ______ .

Answer/Explanation

Answer:
Explaination:

37. If A and B are two matrices such that their multiplication is defined, then (AB)’ = _____ .

Answer/Explanation

Answer:
Explaination: B’A’, as result

38. If matrix A = [1 2 3] then find AA’, where A’ is the transpose of matrix A.

Answer/Explanation

Answer:
Explaination:

39.

Answer/Explanation

Answer:
Explaination:

40. For what value of x, is the matrix

a skew symmetric matrix ? [AI 2013]

Answer/Explanation

Answer:
Explaination:

41. If A and B are symmetric matrices, show that AB is symmetric, if AB = BA.

Answer/Explanation

Answer:
Explaination: Given, A’ = A, B’ = B and if AB is symmetric, then
(AB)’ =AB … (i)
Also, (AB)’ = B’A’ = BA … (ii)
From (i) and (ii), we get AB = BA.

42.

skew symmetric, find the values of ‘a’ and ‘b’ [CBSE 2018]

Answer/Explanation

Answer:
Explaination: For skew symmetric matrix, a_ij = – a_ji
⇒ a = – 2, b = 3.

43. Show that all the elements on the main diagonal of a skew symmetric matrix are zero. [Delhi 2017]

Answer/Explanation

Answer:
Explaination: A square matrix A = [a_ij is skew symmetric if
a_ij = –_ji, ∀ i,j
Let i=j
⇒ a_ii = – a_ii
⇒ 2a_ii = 0
=> a_ii = 0
Hence, all the diagonal elements of a skew symmetric matrix are always zero.

44. If A and B are symmetric matrices, show that AB + BA is symmetric and AB – BA is skew symmetric. [Dehradun 2019]

Answer/Explanation

Answer:
Explaination: A’ =A,B’= B;
Consider (AB + BA)’ = (AB)’ + (BA)’
= B’A’ + A’B’
= BA+AB
= AB +BA.
Hence, symmetric.
Consider (AB – BA)’ = (AB)’ – (BA)’
= B’ A’- A’ B’
= BA – AB
= -(AB-BA)
Hence, skew symmetric.

45. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Answer/Explanation

Answer:
Explaination: Let A is symmetric then A’ = A …(i)
Now (B’AB)’ = B’A'(B’)’ = B’A’B = B’AB [using (i)]
Hence, symmetric.
Similarly, let A be skew symmetric then A’= -A
(B’AB)’ = B’A'(B’)’ = B’A’B
= B'(-A)B = -B’AB.
Hence, B’AB is skew symmetric.

We hope the given Maths MCQs for Class 12 with Answers Chapter 3 Matrices will help you. If you have any query regarding CBSE Class 12 Maths Matrices MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Maths MCQs for Class 12 with Answers Chapter 2 Inverse Trigonometric Functions

July 11, 2020 by Kishen Leave a Comment

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 2 Inverse Trigonometric Functions. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Inverse Trigonometric Functions MCQs Pdf with Answers to know their preparation level.

Inverse Trigonometric Functions Class 12 Maths MCQs Pdf

1. Principal value of sin^-1 $\frac{1}{2}$ is

Answer/Explanation

Answer: d
Explaination:

2. sin^-1{sin ( $\frac{2 \pi}{3}$ )} = $\frac{2 \pi}{3}$ , state true or false.

Answer/Explanation

Answer:
Explaination: False, as – $\frac{\pi}{2}$ ≤ sin-1 x ≤ $\frac{\pi}{2}$ .

3. tan^-1{sin (- $\frac{\pi}{2}$ )} is equal to
(a) -1
(b) 1
(c) $\frac{\pi}{2}$
(d) – $\frac{\pi}{4}$

Answer/Explanation

Answer: d
Explaination:
(d), as sin (- $\frac{\pi}{2}$ )= -1, and tan-1(-1) = – $\frac{\pi}{4}$ .

4. sec{tan^-1 (- $\frac{y}{3}$ )} is equal to

Answer/Explanation

Answer: b
Explaination:

5. Principal branch of tan^-1 x is ______ .

Answer/Explanation

Answer:
Explaination:

6. Principal value of the expression cos^-1[cos(-680°)] is

Answer/Explanation

Answer: a
Explaination:
(a), as cos(-680°) = cos 680°
= cos(720° – 40°) = cos 40°
∴ cos-1[cos(-680°)J = cos-1 (cos 40°)
= 40° = $\frac{2 \pi}{9}$ .

7. If tan^-1x = sin^-1( $\frac{1}{\sqrt{2}}$ ),then x is ______ .

Answer/Explanation

Answer:
Explaination: 1, as tan-1 x = $\frac{\pi}{4}$ .

8. cos^-1 $\left(\cos \frac{7 \pi}{6}\right)=\frac{7 \pi}{6}$ , state true or false.

Answer/Explanation

Answer:
Explaination: False, as $\frac{7 \pi}{6}$ ∉ [0, π].

9. Find the value of tan^-1 √3 – cot^-1(-√3). [CBSE 2018]

Answer/Explanation

Answer:
Explaination:

10. Find the principal value of cos^-1 $\frac{1}{2}$ .

Answer/Explanation

Answer:
Explaination:

11. Find the principal value of

Answer/Explanation

Answer:
Explaination:

12. What is the domain of the function sin^-1 x?

Answer/Explanation

Answer:
Explaination: -1 ≤ x ≤ 1 or [-1, 1]

13. Find the value of sin $\left[\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right]$ . [NCERT; Delhi 2011]

Answer/Explanation

Answer:
Explaination:

14. Find the principal value of tan^-1√3 sec^-1(-2). [NCERT; AI 2012]

Answer/Explanation

Answer:
Explaination:

15. If sin^-1 x = $\frac{\pi}{10}$ , for some x ∈ R, then the value of cos^-1 is ______ .

Answer/Explanation

Answer:
Explaination:

16. If tan^-1 x -cot^-1 = $\frac{\pi}{6}$ , then x is ______ .

Answer/Explanation

Answer:
Explaination:

17. The domain of the function^ = sin’ -‘(V) is
(a) [0, 1]
(b) (0, 1)
(c) [-1, 1]
(d) Φ

Answer/Explanation

Answer: c
Explaination:
(c), as -1 ≤ -x² < 1
⇒ 1 ≥ x² ≥ -1
⇒ 0 ≤ x² ≤ 1
⇒ |x| ≤ 1
⇒ -1 ≤ x ≤ 1.

18. If sec^-1 x + sec^-1 y = $\frac{\pi}{2}$ the value of cosec^-1x + cosec^-1y is

Answer/Explanation

Answer: b
Explaination:

19. The value of tan²(sec^-12) + cot2(cosec^-13) is
(a) 5
(b) 11
(c) 13
(d) 15

Answer/Explanation

Answer: b
Explaination:
(b), as tan²(sec-12) + cot²cosec-13)
= sec²(sec-12) – 1 + cosec²(cos 3) – 1
= (2)² – 1 +(3)² – 1
= 4 – 1 +9 – 1 = 11.

20. cot( $\frac{\pi}{4}$ – 2 cot^-13) = 7, state true or false.

Answer/Explanation

Answer:
Explaination:

21. If 3 sin^-1x + cos^-1x = π, then x is equal to

Answer/Explanation

Answer: b
Explaination:

22.

Answer/Explanation

Answer:
Explaination:

23. If cos(tan^-1x + cot^-1√3) = 0, then value of x is _____ .

Answer/Explanation

Answer:
Explaination: √3, as tan^-1x + cot^-1 √3 = $\frac{\pi}{2}$
⇒ x = √3

24. If sin^-1x + sin^-1y + sin^-1z = then the value of x + y² + z³ is
(a) 1
(b) 3
(c) 2
(d) 5

Answer/Explanation

Answer: b
Explaination:

25. The domain of y = cos^-1(x² – 4) is
(a) [3, 5]
(b) [0, π]
(c) [-√5 ,-√3] ∩ [-√5,√3]
(d) [-√5 ,-√3] ∪ [√3, √5]

Answer/Explanation

Answer:
Explaination:

26. Show that sin^-1(2x $\sqrt{1-x^{2}}$ ) = 2 sin^-1 – $\frac{1}{\sqrt{2}}$ ≤ x ≤ $\frac{1}{\sqrt{2}}$

Answer/Explanation

Answer:
Explaination:

27. Find the value of sec(tan^-1 – $\frac{1}{2}$ ).

Answer/Explanation

Answer:
Explaination:

28. If sin(sin^-1 $\frac{1}{5}$ + cos^-1x) = 1 then find the value of x. [NCERT; HOTS; Delhi 2014]

Answer/Explanation

Answer:
Explaination:

29. Find the value of
[NCERT; Delhi 2013 (C), 2011]

Answer/Explanation

Answer:
Explaination:

30.

Answer/Explanation

Answer:
Explaination:

31. Prove that 3 sin^-1 x = sin^-1(3x – 4x³),

Answer/Explanation

Answer:
Explaination:
Let x = sin θ
=> θ = sin^-1x
RHS = sin^-1(3x – 4x³)
= sin-^-1 [3 sin θ – 4 sin³ θ]
= sin^-1 [sin 3θ] = 3θ = 3 sin^-1x = LHS

32. Prove that sin^-1x = tan^-1 $\left(\frac{x}{\sqrt{1-x^{2}}}\right)$

Answer/Explanation

Answer:
Explaination:

33.

Answer/Explanation

Answer:
Explaination:

34. If sin^-1x + sin^-1y = $\frac{2 \pi}{3}$ , then find the value of cos^-1x + cos^-1y.

Answer/Explanation

Answer:
Explaination:

35. Find the value oftan(2 tan^-1 $\frac{1}{5}$ ). [Delhi 2013]

Answer/Explanation

Answer:
Explaination:

We hope the given Maths MCQs for Class 12 with Answers Chapter 2 Inverse Trigonometric Functions will help you. If you have any query regarding CBSE Class 12 Maths Inverse Trigonometric Functions MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Maths MCQs for Class 12 with Answers Chapter 1 Relations and Functions

July 11, 2020 by Kishen Leave a Comment

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 1 Relations and Functions. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Relations and Functions MCQs Pdf with Answers to know their preparation level.

Relations and Functions Class 12 Maths MCQs Pdf

1. 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
(c) equivalence relation
(d) symmetric

Answer/Explanation

Answer: d
Explaination: (d), not reflexive, as l₁ R l₂
⇒ l₁ ⊥ l₁ Not true
Symmetric, true as l₁ R l₂ ⇒ l2R h
Transitive, false as l₁ R l₂, l₂ R l₃
⇒ l₁ || l₃ . l₁ R l₂.

2. Given triangles with sides T₁ : 3, 4, 5; T₂ : 5, 12, 13; T₃ : 6, 8, 10; T₄ : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ₁, Δ₂) : Δ₁ is similar to Δ₂}. Which triangles belong to the same equivalence class?
(a) T₁ and T₂
(b) T₂ and T₃
(c) T₁ and T₃
(d) T₁ and T₄

Answer/Explanation

Answer: c
Explaination: (c), T₁ and T₃ are similar as their sides are proportional.

3. Given set A ={1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be
(a) reflexive if (1, 1) is added
(b) symmetric if (2, 3) is added
(c) transitive if (1, 1) is added
(d) symmetric if (3, 2) is added

Answer/Explanation

Answer: c
Explaination: (c), here (1,2) e R, (2,1) € R, if transitive (1,1) should belong to R.

4. Given set A = {a, b, c). An identity relation in set A is
(a) R = {(a, b), (a, c)}
(b) R = {(a, a), (b, b), (c, c)}
(c) R = {(a, a), (b, b), (c, c), (a, c)}
(d) R= {(c, a), (b, a), (a, a)}

Answer/Explanation

Answer: b
Explaination: (b), A relation R is an identity relation in set A if for all a ∈ A, (a, a) ∈ R.

5. A relation S in the set of real numbers is defined as xSy ⇒ x – y+ √3 is an irrational number, then relation S is
(a) reflexive
(b) reflexive and symmetric
(c) transitive
(d) symmetric and transitive

Answer/Explanation

Answer: a
Explaination:

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

Answer/Explanation

Answer: c
Explaination: (c), total injective mappings/functions
= ⁴ P₃ = 4! = 24.

7. Given a function lf as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
(a) g(x) = 4x + 5
(b) g(x) = $\frac{5}{4 x-5}$
(c) g(x) = $\frac{x-4}{5}$
(d) g(x) = 5x – 4

Answer/Explanation

Answer: c
Explaination:

8. Let Z be the set of integers and R be a relation defined in Z such that aRb if (a – b) is divisible by 5. Then R partitions the set Z into ______ pairwise disjoint subsets.

Answer/Explanation

Answer:
Explaination: Five, as remainder can be 0, 1, 2, 3, 4.

9. Consider set A = {1, 2, 3 } and the relation R= {(1, 2)}, then? is a transitive relation. State true or false.

Answer/Explanation

Answer:
Explaination: True, as there is no situation
(a, b) ∈ R, (b, c) ∈ R Hence, transitive. We can also say, a relation containing only one element is transitive.

10. Every relation which is symmetric and transitive is reflexive also. State true or false.

Answer/Explanation

Answer:
Explaination: False,e.g.if R is arelationinset A = {2,3,4} defined as {(2, 3), (3, 2), (2, 2)} is symmetric and transitive but not reflexive.

11. Let R be a relation in set N, given by R = {(a, b): a = b – 2, b > 6} then (3, 8) ∈ R. State true or false with reason.

Answer/Explanation

Answer:
Explaination: False, as in (3, 8), b = 8
⇒ a = 8 – 2
⇒ a = 6, but here a = 3.

12. Let R be a relation defined as R = {(x, x), (y, y), (z, z), (x, z)} in set A = {x, y, z} then R is (reflexive/symmetric) relation.

Answer/Explanation

Answer:
Explaination: Reflexive, as for all a ∈ A, (a, a) ∈ R.

13. Let R be a relation in the set of natural numbers N defined by R = {(a, b) ∈ N × N: a < b}. Is relation R reflexive? Give a reason.

Answer/Explanation

Answer:
Explaination:
Given R = {(a, b) ∈ N × N: a < b}.
Not reflexive, as for (a, a) × R
⇒ a< a, not true.

14. Let A be any non-empty set and P(A) be the power set of A. A relation R defined on P(A) by X R Y ⇔ X ∩ Y = X, X, Y ∈ P(A). Examine whether ? is symmetric.

Answer/Explanation

Answer:
Explaination: X R Y ⇔ X ∩ Y = X ⇒ Y ∩ X = X ⇒ Y R X.
Hence, symmetric.

15. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive. [NCERT; Delhi 2011]

Answer/Explanation

Answer:
Explaination: (1, 2) ∈ R, (2, 1) ∈ R, but (1, 1) ∉ R.

16. Show that the relation R in the set {1,2,3} given by R = {(1,1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. [NGERT]

Answer/Explanation

Answer:
Explaination:
Given R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} defined on R: {1, 2, 3} → {1, 2, 3}
For reflexive: As (1, 1), (2,2), (3, 3) ∈ R. Hence, reflexive
For symmetric: (1, 2) ∈ R but (2, 1) ∉ R. Hence, not symmetric.
For transitive: (1, 2) ∈ R and (2, 3) ∈R but (1, 3) ∉ R. Hence, not transitive.

17. Let A = {3, 4, 5} and relation R on set A is defined as R = {(a, b) e A x A : a – b – 10). Is relation an empty relation?

Answer/Explanation

Answer:
Explaination: We notice for no value of a, b s A, a-b = 10. Hence, (a, b) £ R for a, b e A. Hence, empty relation.

18. Given set A = {a, b} and relation R on A is defined as R = {(a, a), (b, b)}. Is relation an identity relation?

Answer/Explanation

Answer:
Explaination: Yes, as (a, a) ∈ R, for all a ∈ A..

19. Let set A represents the set of all the girls of a particular class. Relation R on A is defined as R = {(a, b) ∈ A × A : difference between weights of a and b is less than 30 kg}. Show that relation R is a universal relation.

Answer/Explanation

Answer:
Explaination: Let a, b ∈ A then a – b < 30 kg, always true for students of a particular class, i.e. aRb ∀ a, b ∈ A. Hence, universal relation.

20. If A = {1, 2, 3} and relation R = {(2, 3)} in A. Check whether relation R is reflexive, symmetric and transitive.

Answer/Explanation

Answer:
Explaination:
Not reflexive, as (1, 1) ∉ R.
Not symmetric, as (2, 3) ∈ R but (3,2) ∉ R.
Transitive, as relation R in a non empty set containing one element is transitive.

21. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive. [Delhi]

Answer/Explanation

Answer:
Explaination: As (1, 2) ∈ R, (2, 1) ∈ R, but (1, 1) ∉ R.

22. Consider the set A containing n elements, then the total number of injective functions from set A onto itself is _____ .

Answer/Explanation

Answer:
Explaination: Total number of injective functions from set containing n elements to a set containing n elements is ⁿ P_n = n!

23. The domain of the function f : R → R defined by f(x) = $\sqrt{4-x^{2}}$ is ______ .

Answer/Explanation

Answer:
Explaination:
[-2, 2]. For domain 4 – x² ≥ 0
⇒ 4 ≥ x²
⇒ x² ≤ 4
⇒ x² ≤ (2)²
⇒ -2 ≤ x ≤ 2, i.e. [-2, 2].

24. Let A = {a, b }. Then number of one-one functions from A to A possible are
(a) 2
(b) 4
(c) 1
(d) 3

Answer/Explanation

Answer:
Explaination: (a), as if n(A) = m, then possible one-one functions from A to A are m!

25. Let A = {1, 2, 3, 4} and B = {a, b, c}. Then number of one-one functions from A to B are ______.

Answer/Explanation

Answer:
Explaination: 0, as n(A) > n(B)

26. If n(A) = p, then number of bijective functions from set A to A are ______ ..

Answer/Explanation

Answer:
Explaination: p!, as for bijective functions from A to B, n(A) = n(B) and function is one-one onto.

27. The function f : R → R defined as f(x) = [x], where [x] is greatest integer ≤ x, is onto function. State true or false.

Answer/Explanation

Answer:
Explaination: False, as range of f is set of integers, i.e.
Z and range of f ⊆ co-domain R. Hence,not onto e.g. for $\frac{1}{2}$ ∈ R (co-domain) there is no x ∈ R (domain) such that y = f(x) or $\frac{1}{2}$ e∈ R has no pre-image.

28. If $f(x)=\frac{x-1}{|x-1|}, x(\neq 1) \in R$ then range of ‘f’ is _______ .

Answer/Explanation

Answer:
Explaination:

29. If f : R → R be defined by f(x) = (3 – x³)^1/3, then find fof(x). [NCERT]

Answer/Explanation

Answer:
Explaination:

30. If f is an invertible function defined as f(x) = $\frac{3x-4}{5}$ , write f^-1(x).

Answer/Explanation

Answer:
Explaination:

31. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not. [AI 2011] f

Answer/Explanation

Answer:
Explaination: One-one, as for x₁ ≠ x₂
⇒ f(x₁) ≠ f(x₂).

32. Let f : R → R is defined by f (x) = | x |. Is function f onto? Give a reason. [HOTS]

Answer/Explanation

Answer:
Explaination: f is not onto, as for some y ∈ R from co-domain, there is no x ∈ R from domain such that y = f(x), e.g. for -2 ∈ R (co-domain) there is no x ∈ R (domain) such that f(x) = -2, i.e. |x| = -2. Hence, not onto.

33. If f : R → R and g : R → R are given by f (x) = sin x and g(x) = 5x², find gof(x).

Answer/Explanation

Answer:
Explaination: gof(x) = g(f(x)) = g(sin x) = 5 sin² x.

34. Let f : {1, 3,4} → {1,2, 5} and g: {1,2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. [NCERT]

Answer/Explanation

Answer:
Explaination:
gof: {1, 3, 4} → {1, 3}.
gof(1) = g(f(1)) = g(2) = 3.
gof(3) = g(K3)) = g(5) = 1
gof(4) = g(f(4)) = g(1) = 3
gof = {(1, 3), (3, 1), (4, 3)}.

35. Prove that f : R → R given by f(x) = x³ + 1 is one-one function.

Answer/Explanation

Answer:
Explaination:
Given f(x) = x³ + 1
For x₁ ≠ x₂
⇒ x₁³ ≠ x₂³
⇒ x₁³ + 1 ≠ x₂³ + 1
⇒ f(x₁) ≠ f(x₂). Hence, one-one

36. Show that the Signum Function f : R → R,

one-one nor onto.

Answer/Explanation

Answer:
Explaination:

Range of function is {-1, 0, 1} and co-domain is set of real numbers R.
⇒ Range ⊆ co-domain.
There is at least one element in R(codomain) which is not image of any element of the domain, e.g. for 2 e R(co-domain), there is no x in domain such that f(x) = 2, x ∈ R.
Hence, function is not onto.
Also, let x₁ = 2 and x₂ = 3 then f(x₁) = 1 and f(x₂) = 1
i.e., x₁ ≠ x₂ ⇒ f(x₁) = f(x₂).
So, function is not one-one.

37. Given f(x) = sin x check if function f is one-one for (i) (0, π) (ii) (- $\frac{π}{2}$ , $\frac{π}{2}$ ).

Answer/Explanation

Answer:
Explaination:

38. If f : R → R is defined by f(x) = 3x + 2, define f (f(x)). [Foreign]

Answer/Explanation

Answer:
Explaination: f(f(x)) = f(3x + 2) = 3(3x + 2) + 2
= 9x+ 8.

39. Write fog, if f : R → R and g : R → R are given by f(x) = |x| and g(x) = |5x – 2|. [Foreign]

Answer/Explanation

Answer:
Explaination: (fog)(x) =f(g(x))
=f(|5x-2|) = ||5x-2||.

40. Write fog, if f : R → R and g : R → R are given by f(x) = 8x³ and g(x) = x^1/3. [Foreign]

Answer/Explanation

Answer:
Explaination: (fog)(x) = f(g(x)) = f(x^1/3) = 8(x^1/3)³ = 8x.

We hope the given Maths MCQs for Class 12 with Answers Chapter 1 Relations and Functions will help you. If you have any query regarding CBSE Class 12 Maths Relations and Functions MCQs Pdf, drop a comment below and we will get back to you at the earliest.

Maths MCQs for Class 12 Chapter Wise with Answers

July 11, 2020 by Kishen Leave a Comment

Maths Multiple Choice Questions With Answers For Class 12 Pdf Free Download is very important for students who want to score good marks in their CBSE board examination. Students who can Practice CBSE Maths MCQs For Class 12 With Answers Pdf to improve your score in Board Exams.

Maths MCQ Questions for Class 12 with Answers Pdf Download

Practicing NCERT Online Maths Objective Questions For Class 12 Pdf Answers is one of the best ways to prepare for the CBSE Class 12 board exam. There is no substitute for consistent practice whether one wants to understand a concept thoroughly or one wants to score better. By practicing more 2nd Year Maths MCQs with Answers Pdf Download, students can improve their speed and accuracy which can help them during their board exam.

Chapter 1 Relations and Functions MCQs
Chapter 2 Inverse Trigonometric Functions MCQs
Chapter 3 Matrices MCQs
Chapter 4 Determinants MCQs
Chapter 5 Continuity and Differentiability MCQs
Chapter 6 Application of Derivatives MCQs
Chapter 7 Integrals MCQs
Chapter 8 Application of Integrals MCQs
Chapter 9 Differential Equations MCQs
Chapter 10 Vector Algebra MCQs
Chapter 11 Three Dimensional Geometry MCQs
Chapter 12 Linear Programming MCQs
Chapter 13 Probability MCQs

We hope the given MCQ Questions For Class 12 Maths With Answers will help you. If you have any query regarding CBSE Class 12 Maths Multiple Choice Questions with Answers, drop a comment below and we will get back to you at the earliest.