Question Set C

cbse 12th Standard Maths Inverse Trigonometric Functions Model Questions Papers Study Material - QB365

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Questions:

Multiple Choice Question

48 x 1 = 48

The function y = x²e^-x is decreasing in the interval

(a)

(0,2)

(b)

(2, ∞)

(c)

(-∞, 2)

(d)

(-∞, 2) U (2, ∞)

If \(e^x+e^y=e^{x+y}\), then \(\frac{d y}{d x}\) is

(a)

\(e^{y-x}\)

(b)

\(e^{x+y}\)

(c)

\(-e^{y-x}\)

(d)

\(2 e^{x-y}\)

If \(A(\operatorname{adj} A)=\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]\) then the value of \(|A|+|\operatorname{adj} A|\) is equal to

(a)

(b)

(c)

(d)

Value of k, for which \(A=\left[\begin{array}{cc}k & 8 \\ 4 & 2 k\end{array}\right]\) is a singular matrix, is

(a)

(b)

-4

(c)

± 4

(d)

If \(A=\left[\begin{array}{lll}2 & -3 & 4\end{array}\right], B=\left[\begin{array}{l}3 \\ 2 \\ 2\end{array}\right], X=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\) and \(Y=\left[\begin{array}{l}2 \\ 3 \\ 4\end{array}\right]\), then AB+XY is equal to

(a)

[28]

(b)

[24]

(c)

(d)

A and B are square matrices of same order. If (A+B)²= A² + B², then

(a)

AB = BA

(b)

AB =-BA

(c)

AB = 0

(d)

BA = 0

\(\sin \left(\tan ^{-1} x\right)\), where |x| < 1 is equal to

(a)

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

(b)

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

(c)

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

(d)

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

If \(A=\left[\begin{array}{ll} x & 0 \\ 1 & 1 \end{array}\right]\) and \(B=\left[\begin{array}{cc} 4 & 0 \\ -1 & 1 \end{array}\right]\) then the value of x for which A² = B is

(a)

-2

(b)

(c)

2 or -2

(d)

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. Based on the given information f is best defined as

(a)

surjective function

(b)

injective function

(c)

bijective function

(d)

None of the above

10.

If inverse of matrix \(\left[\begin{array}{ccc} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]\)is the matrix \(\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & \lambda & 3 \\ 1 & 3 & 4 \end{array}\right]\), then the value of \(\lambda\) is

(a)

-4

(b)

(c)

(d)

11.

If a, b. c are all distinct, and \(\left|\begin{array}{lll} a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & a+c^{3} \end{array}\right|=0\) then the value of abc is

(a)

(b)

-1

(c)

(d)

-3

12.

Approximate value of \(\sqrt[3]{25} \text { is }\)

(a)

3.074

(b)

2.926

(c)

(d)

none of these

13.

Derivative of \(\frac{x}{x-1}\) with respect to x, is

(a)

(b)

\(\frac{1}{(x-1)^{2}}\)

(c)

\(\frac{2 x-1}{(x-1)^{2}}\)

(d)

\(\frac{-1}{(x-1)^{2}}\)

14.

\(\text { The principal value of } \sin ^{-1}\left(\sin \frac{2 \pi}{3}\right) \text { is }\)

(a)

\(\frac{2 \pi}{3}\)

(b)

\(\frac{ \pi}{3}\)

(c)

\(-\frac{ \pi}{6}\)

(d)

\(\frac{ \pi}{6}\)

15.

A binary operation a o b = a, for a, b \(\in\) N is

(a)

commutative

(b)

not associative

(c)

commutative and associative

(d)

associative but not commutative

16.

If an error of 10 is made. in measuring the angle of a sector of radius 30 cm, then the approximate error in its area is

(a)

450 cm²

(b)

\(25 \pi \mathrm{cm}^{2}\)

(c)

\(2.5 \pi \mathrm{cm}^{2}\)

(d)

None of these

17.

The tangent to the curve \(y=e^{2 x}\) at the point (0,1) meets X-axis at

(a)

(0,1

(b)

\(\left(-\frac{1}{2}, 0\right)\)

(c)

(2,0)

(d)

(0,2)

18.

A kite is moving horizontally at a height of 151.5m. If the speed of kite is 10 m/ s, then the rate at which the string is being let out when the kite is 250 m away from the boy who is flying the kite and the height of the boy 1.5m is

(a)

4 m/s

(b)

6 m/s

(c)

7 m/s

(d)

8 m/s

19.

If \(y=\left(x+\sqrt{1+x^{2}}\right)^{n}, \text { then }\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}\) is equal to

(a)

n²y

(b)

-n^Zy

(c)

-y

(d)

2x²y

20.

If \(f(x)=\left[\begin{array}{c} m x+1, \text { if } x \leq \frac{\pi}{2} \\ \sin x+n, \text { if } x>\frac{\pi}{2} \end{array}\right.\) is continuous at \(x=\frac{\pi}{2}\)

(a)

m = 1,1 = 0

(b)

\(m=\frac{n \pi}{2}+1\)

(c)

\(n=\frac{m \pi}{2}\)

(d)

\(m=n=\frac{\pi}{2}\)

21.

The function f(x) = cot x is discontinuous on the set

(a)

\(\left\{x=n \frac{1}{4}: n \in Z\right\}\)

(b)

\(\{x=2 n \pi: n \in Z\}\)

(c)

\(\left\{x=(2 n+1) \frac{\pi}{2} ; n \in Z\right\}\)

(d)

\(\left\{x=\frac{n \pi}{2} ; n \in Z\right\}\)

22.

if a, b, c are in AP, then determinant
\(\left|\begin{array}{lll} x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right|\) is

(a)

zero

(b)

(c)

(d)

23.

If A is an invertible matrix of order 2, then det (A^-1) is equal to

(a)

det (A)

(b)

\(\frac{1}{\operatorname{det}(A)}\)

(c)

(d)

zero

24.

If \(\Delta=\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|\) then the cofactor A₂₁ is

(a)

-(he + fg)

(b)

fg -hc

(c)

fg + hc

(d)

hc-fg

25.

The area of the triangle formed by 3 collinear points is

(a)

one

(b)

two

(c)

zero

(d)

four

26.

If \(\left|\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right|=\left|\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right|\) then, x is equal to

(a)

(b)

\(\pm 6\)

(c)

-6

(d)

zero

27.

The value of x such that
\(\left[\begin{array}{lll} 1 & 2 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{array}\right]\left[\begin{array}{l} 0 \\ 2 \\ x \end{array}\right]=O, \mathrm{i}\)

(a)

(b)

(c)

-1

(d)

28.

If A is a 3 x 2 matrix, B is a 3 x 3 matrix and Cis a 2 x 3 matrix, then the elements in A, Band C are respectively

(a)

6,9,8

(b)

6,9,6

(c)

9,6,6

(d)

6,6,9

29.

The equation \(\tan ^{-1} x-\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)\) has

(a)

no solution

(b)

unique solution

(c)

infinite number of solutions

(d)

two solutions

30.

The domain in which sine function will be one-one, is

(a)

\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

(b)

\(\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]\)

(c)

\([0, \pi]\)

(d)

Both 'a' and 'b'

31.

The value of \(\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)+\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)\) is

(a)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{6}\)

(d)

\(\frac{2 \pi}{3}\)

32.

\(Let f(x)=\left\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \text { and } g(x)=1+x-[x]\end{array}\right.\)
where [x] denotes the greatest integer less than or equal to x. Then, for all x, f(g(x)) is equal to

(a)

(b)

(c)

g(x)

(d)

f(x)

33.

If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

(a)

reflexive

(b)

transitive

(c)

symmetric

(d)

None of these

34.

Using approximation find the value of \(y=\sqrt{4.01}\)

(a)

2.025

(b)

2.001

(c)

2.01

(d)

2.0025

35.

What is the point of discontinuity for signum function?

(a)

x = 1

(b)

x = -1

(c)

x = 0

(d)

function is continuous on R

36.

\(\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}\) is a matrix of order

(a)

(b)

(c)

(d)

37.

The cosine function can be restricted to any interval of the type_______, for its inverse to exist

(a)

[nπ/2, (n + 1) π/2]

(b)

(nπ, (n + 1) π)

(c)

[nπ, (n + 1) π]

(d)

(nπ/2, (n + 1) π/2)

38.

What is the principal value of \({ sec }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \)

(a)

\(\frac { \pi }{ 3 } \)

(b)

not defined

(c)

\(-\frac { \pi }{ 3 } \)

(d)

\(\frac { 2\pi }{ 3 } \)

39.

If A = {1, 2, 3, 4} and B = {1, 3, 5} and R is a relation from A to B defined by (a, b) ∈ element of R ⇔ a < b. Then, R = ?

(a)

{(2, 3), (4, 5), (1, 3), (2, 5)}

(b)

{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

(c)

{(2, 3), (4, 5), (1, 3), (2, 5), (5, 3)}

(d)

{(5, 3), (3, 5), (5, 4), (4, 5)}

40.

Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then, R is

(a)

Symmetric

(b)

Transitive

(c)

Reflexive

(d)

Equivalence relation

41.

The slope of the tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2,– 1) is

(a)

\(\frac{22}{7}\)

(b)

\(\frac67\)

(c)

\(\frac76\)

(d)

\(\frac{-6}{7}\)

42.

Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer

(a)

Is * both associative and commutative?

(b)

Is * commutative but not associative?

(c)

Is * associative but not commutative?

(d)

Is * neither commutative nor associative?

43.

The point on the curve where tangent to the curve y² = x, makes an angle of 45° clockwise with the x-axis is

(a)

\(\left( -\frac { 1 }{ 2 } ,\frac { 1 }{ 4 } \right) \)

(b)

\(\left( \frac { 1 }{ 4 } ,-\frac { 1 }{ 2 } \right) \)

(c)

(-2, 4)

(d)

(4, 2)

44.

If y = Ae^5x,+ Be^-5x x then \(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \) is equal to

(a)

25y

(b)

(c)

-25y

(d)

10y

45.

A function \(f(x)=\begin{cases} \frac { sinx }{ x } +cosx,x\neq 0 \\ 2k\quad \quad \quad \quad ,x=0 \end{cases}\) is continuous at x = 0 for

(a)

k = 1

(b)

k = 2

(c)

K = \(\frac12\)

(d)

k = \(\frac32\)

46.

If A is a square matrix such that A²=A, then (I + A)² – 3A is

(a)

(b)

(c)

(d)

47.

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

(a)

(b)

(c)

(d)

48.

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) = \(\frac{5}{4x-5}\)

(c)

g(x) = \(\frac{x-4}{5}\)

(d)

g(x) = 5x – 4

Assertion and reason

12 x 1 = 12

49.

Assertion: Let A = {-1, 1, 2, 3} and B = {1, 4, 9}, where f: A \(\rightarrow\)B given by f(x) = x², then f is a many-one function.
Reason: If x₁ \(\neq\)x₂ \(\Rightarrow\) f(x₁) \(\neq\)f(x₂), for every x₁,x₂ \(\in\)domain, then f is one-one or else many-one.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

50.

Assertion: f : R \(\rightarrow\) R defined by f(x) = sin x is a bijection.
Reason: If f is both one-one and onto it is bijection.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

51.

Assertion: \(cosec^{-1}\left ( \frac{3}{2} \right )+cos^{-1}\left ( \frac{2}{3} \right )-2cot^{-1}\left ( \frac{1}{7} \right )-cot^{-1}(7)\) is equal to cot^-1 7.
Reason: sin^-1 x + cos^-1 x = \(\frac{\pi}{2}\), tan^-1 x + cot^-1 x = \(\frac{\pi}{2}\), cosec^-1 x = sin^-1 \(\left ( \frac{1}{x} \right )\), cot^-1 (x) = tan^-1\(\left ( \frac{1}{x} \right )\)
(a) Assertion is correct, reason is correct;reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

52.

Assertion: The value of tan^-1 \(\left ( \frac{3}{4} \right )+tan^{-1}\left ( \frac{1}{7} \right )\)is \(\frac{\pi}{4}\)
Reason: If x > 0, y > 0 then \(tan^{-1}\left ( \frac{x}{y} \right )+tan^{-1}\left ( \frac{y-x}{y+x} \right )=\frac{\pi}{4}\)
(a) Assertion is correct, reason is correct;reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

53.

Assertion: The matrix \(A=\begin{bmatrix} 0& -1& -2\\ 1& 0& -3\\ 2& 3& 0\\ \end{bmatrix}\)is a skew symmetric matrix.
Reason: For the given matrix A we have A' = A.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

54.

Assertion: Addition of matrices is an example of binary operation on the set of matrices of the same order.
Reason: Addition of matrix is commutative.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

55.

Consider the system
2x + 3y + 6z = 8
x + 2y + 3z = 5
x + y + 3z = 4
Assertion: The above system of equation has no solution.
Reason: detA = 0 and (adj A A)B = 0, where
\(A=\begin{bmatrix} 2& 3& 6\\ 1& 2& 3\\ 1& 1& 3\\ \end{bmatrix}\)and \(B=\begin{bmatrix} 8 \\ 5 \\ 4 \\ \end{bmatrix}\)
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

56.

Let A = \(\begin{bmatrix} 1& 0& a\\ 2& 3& b\\ -3& 1& c\\ \end{bmatrix}\), B=\(\begin{bmatrix} 1& 0& x\\ 2& 3& y\\ -3& 1& z\\ \end{bmatrix}\)and C=\(\begin{bmatrix} 1& 0& a+x\\ 2& 3& b+y\\ -3& 1& c+z\\ \end{bmatrix}\)
Assertion: det A + det B = det C.
Reason: A + B = C.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

57.

Assertion: \(\frac{d}{dx}\)e^{cos x} = e^{cos x}(- sin x)
Reason: \(\frac{d}{dx}\)e^x = e^x
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

58.

Assertion: If u = f(tan x), v = g(sec x) and f'(1) = 2, g(\(\sqrt{2}\)) = 4, then \(\left ( \frac{du}{dv} \right )_{s-\pi/4}=\frac{1}{\sqrt{2}}\)
Reason: If u = f(x), v = g(x), then the derivative of f with respect to g is \(\frac{du}{dv}=\frac{du/dx}{dv/dx}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

59.

Consider the function

Assertion: f has a local maximum value at x = 0.
Reason: f'(0) = 0 and f''(0) < 0
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

60.

Assertion: The curves x = y² and xy = k cut at right angle, if 8k² =1.
Reason: Two curves intersect at right angle, if the tangents to the curves at the point of intersection are perpendicular to each other i.e., product of their slope is -1.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

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Answers Key:

Multiple Choice Question

48 x 1 = 48

(d)
(-∞, 2) U (2, ∞)

(a)
12

(a)
[28]

(b)
AB =-BA

(d)
\(\frac{x}{\sqrt{1+x^2}}\)

(b)
injective function

10.

(d)
4

11.

(b)
-1

12.

(b)
2.926

13.

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

14.

(b)
\(\frac{ \pi}{3}\)

15.

(d)
associative but not commutative

16.

Let A be the area and \(\theta\) be the sector angle. Then
\( A=\frac{1}{2} \times 30^{2} \times \theta=450 \theta \\ \frac{d A}{d \theta}=450 \)
Let \(\Delta \theta\) be an error in \(\theta \text { and } \Delta A\) be the corresponding error in A.
Then,\(\Delta A=\frac{d A}{d \theta} \Delta \theta\)
\(\begin{array}{ll} \Rightarrow & \Delta A=450 \times \frac{\pi}{180} \quad\left[\because \Delta \theta=1^{\circ}=\frac{\pi}{180} \text { radians }\right] \\ \Rightarrow & \Delta A=2.5 \pi \mathrm{cm}^{2} \end{array}\)

17.

The equation of curve is y = e^2x
Since, it passes through the point (0, 1).
\(\begin{aligned} &\therefore \quad \cdot \frac{d y}{d x}=e^{2 x} \cdot 2=2 \cdot e^{2 x}\\ &\Rightarrow\left(\frac{d y}{d x}\right)_{(0,1)}=2 \cdot e^{2 \cdot 0}=2=\text { Slope of tangent to the curve } \end{aligned}\)
Equation of tangent is y -1 = 2(x - 0)
\(\Rightarrow \quad y=2 x+1\)
Since, tangent to curve y = e^2X at the point (0, 1) meets X-axis i.e., y = 0
\(\therefore \quad 0=2 x+1 \Rightarrow x=-\frac{1}{2}\)
So, the required point is \(\left(\frac{-1}{2}, 0\right)\)

18.

(a)
4 m/s

19.

(a)
n²y

20.

21.

We know that, f(x) = cot x is continuous in \(R-\{n \pi: n \in Z\}\)
Since \(f(x)=\cot x=\frac{\cos x}{\sin x}[\text { since, } \sin x=0 \text { at } n \pi, n \in Z]\)
Hence, f(x) = cotx is discontinuous on the set \(\{x=n \pi: n \in Z\}\)

22.

(a)
zero

23.

We know,\(A A^{-1}=I\)
\(\therefore \left|A A^{-1}\right|=|| I \mid \)
\(\Rightarrow |A|\left|A^{-1}\right|=1\)
\(\Rightarrow \left|A^{-1}\right|=\frac{1}{|A|} \)

24.

\(A_{21}=(-1)^{2+1} M_{21}=-M_{21}=-\left|\begin{array}{ll} h & g \\ f & c \end{array}\right|\)

25.

By definition of collinearity.

26.

Given \(\left|\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right|=\left|\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right| \Rightarrow x^{2}-36=36-36\)
\(\Rightarrow \quad x^{2}=36 \Rightarrow x=\pm 6\)

27.

28.

The number of elements in m x n matrix is equal to mn.

29.

(b)
unique solution

30.

(d)
Both 'a' and 'b'

31.

\( \tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)=\tan ^{-1} \tan \left(\pi-\frac{\pi}{6}\right) \\ =\tan ^{-1}\left(-\tan \frac{\pi}{6}\right) \\ \text { and } \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right) \\ =\cos ^{-1} \cos \left(2 \pi+\frac{\pi}{6}\right)=\cos ^{-1}\left(\cos \frac{\pi}{6}\right) \)

32.

We know that,
\(0 \leq x-[x]<1, \text { for all } x \in R \)
\(\Rightarrow 1 \leq 1+x-[x]<2 \text { for all } x \in R\)
\(\Rightarrow 15 g(x)<2 \text { for all } x \leq R\)
\(\Rightarrow f(g(x))=1, for \ all \ x \in R\)
\([\because f(x)=1, when x>0]\)

33.

R does not have elements of the type (a,b) and (b, c).

34.

(d)
2.0025

35.

36.

37.

(b)
(nπ, (n + 1) π)

38.

(b)
not defined

39.

(b)
{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

40.

(b)
Transitive

41.

(b)
\(\frac67\)

42.

(b)
Is * commutative but not associative?

43.

As 2y \(\frac{dy}{dx}\) = 1
⇒\(\frac{dy}{dx}=\frac{1}{2y}\) (slope of tangent)
⇒ \(\frac{1}{2y}\) = tan(-45^o)
⇒ y = \(-\frac{1}{2}\) ⇒ x = \(\frac14\)
∴ Point as \(\left( \frac { 1 }{ 4 } ,-\frac { 1 }{ 2 } \right) \)

44.

As y' = 5Ae^5x - 5Be^-5x
and y'' = 25Ae^5x + 25Be^-5x
= 25y

45.

As \(\lim _{ x\rightarrow 0 }{ \left( \frac { sinx }{ x } +cosx \right) } \)
= 1 + 1 = 2 2k
⇒ k = 1

46.

(a)
I

47.

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.

48.

y = f(x)
⇒ 5x + 4
⇒ \(\frac{y-4}{5}\)
∴ f^-1(y) = \(\frac{y-4}{5}\)
or f^-1(x) = \(\frac{x-4}{5}\)

Assertion and reason

12 x 1 = 12

49.

(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.

50.

(d) Assertion is incorrect, reason is correct.

51.

(d) Assertion is incorrect, reason is correct.

52.

(a) Assertion is correct, reason is correct;reason is a correct explanation for assertion.

53.

54.

(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion

55.

56.

57.

(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion

58.

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

59.

60.

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

cbse 12th Standard Maths Inverse Trigonometric Functions Model Questions Papers Study Material - QB365 Set C

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