CBSE 12th Maths Applications of Derivatives Important Questions And Answers Study Material - QB365 Set A

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Questions:

Multiple Choice Question

453 x 1 = 453

The maximum value of [x(x -1)+1]^1/3, 0 ≤ x ≤ 1 is

(a)

(b)

1/2

(c)

(d)

$3\sqrt{\frac{1}{3}}$

The least value of the function f(x) = 2cos x + x in the closed interval [0, π/2] is

(a)

(b)

π/5 + √3

(c)

π/2

(d)

the least value does not exist

The area of a trapezium is defined by function f and given by f(x) = (10 + x) $ \sqrt{100 - x ^ 2}$ then the area when it is maximised is

(a)

75cm²

(b)

$7\sqrt3$cm²

(c)

$75\sqrt 3$cm²

(d)

5cm²

A function f : R → R is defined as $f(x) = x ^ 3 + 1$ Then, the function has

(a)

no minimum value

(b)

no maximum value

(c)

both maximum and minimum values

(d)

neither maximum nor minimum

Find the intervals in which the function given by f(x) = x²- 4x + 6 is strictly increasing

(a)

(-∞, 2) U (2, ∞)

(b)

(2, ∞)

(c)

(-∞, 2)

(d)

(-∞, 2) U (3, ∞)

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

(a)

(0,2)

(b)

(2, ∞)

(c)

(-∞, 2)

(d)

(-∞, 2) U (2, ∞)

The value of b for which the function f(x) = x + cosx + b is strictly decreasing over R is

(a)

b < 1

(b)

no value of b exists

(c)

b ≤ 1

(d)

b ≥ 1

The real function f(x) = 2x³ - 3x² - 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) U (3, ∞)

The function f(x) = x³ + 3x is increasing in interval.

(a)

(- ∞, 0)

(b)

(0,∞)

(c)

(d)

(0, 1)

10.

If f(x) = a(x - cosx) is strictly decreasing in R, then a belongs to

(a)

{0}

(b)

(0, ∞)

(c)

(-∞, 0)

(d)

(-∞, ∞)

11.

The interval in which the function f(x) = 2x³ + 9x² + 12x - 1 is decreasing is

(a)

(-1,∞)

(b)

(-2, -1)

(c)

(-∞, -2)

(d)

(-1, 1)

12.

If $y=A e^{5 x}+B e^{-5 x}$, then $\frac{d^2 y}{d x^2}$ is equal to

(a)

25 y

(b)

5 y

(c)

-25 y

(d)

15 y

13.

If $y=\log \left(\cos e^x\right)$, then $\frac{d y}{d x}$ is

(a)

$\cos e^{x-1}$

(b)

$e^{-x} \cos e^x$

(c)

$e^x \sin e^x$

(d)

$-e^x \tan e^x$

14.

If $y=5 \cos x-3 \sin x$, then $\frac{d^2 y}{d x^2}$ is equal to

(a)

-y

(b)

(c)

25 y

(d)

9 y

15.

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)

$2 e^{x-y}$

16.

The derivative of $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$ with respect to $\sin ^{-1} x, \frac{1}{\sqrt{2}}<x<1$ is

(a)

(b)

$\frac{\pi}{2}-2$

(c)

$\frac{\pi}{2}$

(d)

-2

17.

If $x=a \sec \theta$ and $y=b \tan \theta$, then $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{6}$ is

(a)

$\frac{-3 \sqrt{3} b}{a^2}$

(b)

$\frac{-2 \sqrt{3} b}{a}$

(c)

$\frac{-3 \sqrt{3} b}{a}$

(d)

$\frac{-b}{3 \sqrt{3} a^2}$

18.

The set of all points, where the function f(x) = x + |x| is differentiable, is

(a)

$(0, \infty)$

(b)

$(-\infty, 0)$

(c)

$(-\infty, 0) \cup(0, \infty)$

(d)

$(-\infty, \infty)$

19.

If $y=\sin ^{-1} x$, then $\left(1-x^2\right) y_2$ is equal to

(a)

$x y_1$

(b)

$x y$

(c)

$x y_2$

(d)

x²

20.

If $f(x)=|\cos x|$, then $f^{\prime}\left(\frac{3 \pi}{4}\right)$ is

(a)

(b)

-1

(c)

$\frac{-1}{\sqrt{2}}$

(d)

$\frac{1}{\sqrt{2}}$

21.

If $y=\log \left(\sin e^x\right)$, then $\frac{d y}{d x}$ is

(a)

$\cot e^x$

(b)

$\operatorname{cosec} e^x$

(c)

$e^x \cot e^x$

(d)

$e^x \operatorname{cosec} e^x$

22.

If $y=\sin ^2\left(x^3\right)$, then $\frac{d y}{d x}$ is equal to

(a)

$2 \sin x^3 \cos x^3$

(b)

$3 x^3 \sin x^3 \cos x^3$

(c)

$6 x^2 \sin x^3 \cos x^3$

(d)

$2 x^2 \sin ^2\left(x^3\right)$

23.

If $y=\frac{\cos x-\sin x}{\cos x+\sin x}$, then $\frac{d y}{d x}$ is

(a)

$-\sec ^2\left(\frac{\pi}{4}-x\right)$

(b)

$\sec ^2\left(\frac{\pi}{4}-x\right)$

(c)

$\log \left|\sec \left(\frac{\pi}{4}-x\right)\right|$

(d)

$-\log \left|\sec \left(\frac{\pi}{4}-x\right)\right|$

24.

If $f(x)=2|x|+3|\sin x|+6$, then the right hand derivative of f(x) at x = 0 is

(a)

(b)

(c)

(d)

25.

The value of k for which function $f(x)=\left\{\begin{array}{ll}x^2, & x \geq 0 \\ k x, & x<0\end{array}\right.$ is differentiable at x = 0 is

(a)

(b)

(c)

any real number

(d)

26.

The function f(x) = x|x| is

(a)

continuous and differentiable at x = 0.

(b)

continuous but not differentiable at x = 0

(c)

differentiable but not continuous at x = 0

(d)

neither differentiable nor continuous at x = 0

27.

The function f(x)=|x| is

(a)

continuous and differentiable everywhere.

(b)

continuous and differentiable nowhere.

(c)

continuous everywhere, but differentiable everywhere except at x=0

(d)

continuous everywhere, but differentiable nowhere.

28.

Let $f(x)=\left|\begin{array}{cc}x^2 & \sin x \\ p & -1\end{array}\right|$, where p is a constant. Then, the value of p for which $f^{\prime}(0)=1$ is

(a)

(b)

(c)

(d)

-1

29.

If $\sin (x y)=1$, then $\frac{d y}{d x}$ is equal to

(a)

$\frac{x}{y}$

(b)

$-\frac{x}{y}$

(c)

$\frac{y}{x}$

(d)

$-\frac{y}{x}$

30.

The value of k (k < 0) for which the function f defined as $f(x)=\left\{\begin{array}{cl}\frac{1-\cos k x}{x \sin x}, & x \neq 0 \\ \frac{1}{2}, & x=0\end{array}\right.$ is continuous at x = 0, is

(a)

$\pm 1$

(b)

-1

(c)

$\pm \frac{1}{2}$

(d)

$\frac{1}{2}$

31.

The points, at which the function f given by $f(x)=\left\{\begin{array}{ll}\frac{x}{|x|}, & x<0 \\ -1, & x \geq 0\end{array}\right.$ is continuous, is/are

(a)

$x \in R$

(b)

x=0

(c)

$x \in R-\{0\}$

(d)

x = -1 and 1

32.

The value of k for which the function $f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{8 x^2}, & \text { if } x \neq 0 \\ k, & \text { if } x=0\end{array}\right.$ is continuous at x = 0 is

(a)

(b)

-1

(c)

(d)

33.

The value of k for which $f(x)=\left\{\begin{array}{cc} 3 x+5, & x \geq 2 \\ k x^2, & x<2 \end{array}\right. $ is a continuous function, is

(a)

$-\frac{11}{4}$

(b)

$\frac{4}{11}$

(c)

(d)

$\frac{11}{4}$

34.

The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x, is continuous at

(a)

x = 1

(b)

x = 1.5

(c)

x = -2

(d)

x = 4

35.

If $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$, then $\operatorname{det}(\operatorname{adj} A)$ equals

(a)

$a^{27}$

(b)

$a^9$

(c)

$a^6$

(d)

$a^2$

36.

If A is a square matrix of order 3 , such that $A(\operatorname{adj} A)=10 I$, then $|\operatorname{adj} A|$ is equal to

(a)

(b)

(c)

100

(d)

10l

37.

If $A=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right]$, then

(a)

$A^{-1}=B$

(b)

$A^{-1}=6 B$

(c)

$B^{-1}=B$

(d)

$B^{-1}=\frac{1}{6} A$

38.

For $A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$, then $14 A^{-1}$ is given by

(a)

$14\left[\begin{array}{cc}2 & -1 \\ 1 & 3\end{array}\right]$

(b)

$\left[\begin{array}{cc}4 & -2 \\ 2 & 6\end{array}\right]$

(c)

$2\left[\begin{array}{ll}2 & -1 \\ 1 & -3\end{array}\right]$

(d)

$2\left[\begin{array}{cc}3 & -1 \\ 1 & 2\end{array}\right]$

39.

For matrix $A=\left[\begin{array}{cc}2 & 5 \\ -11 & 7\end{array}\right]$ then $(\operatorname{adj} A)^{\prime}$ is equal to

(a)

$\left[\begin{array}{cc}-2 & -5 \\ 11 & -7\end{array}\right]$

(b)

$\left[\begin{array}{cc}7 & 5 \\ 11 & 2\end{array}\right]$

(c)

$\left[\begin{array}{cc}7 & 11 \\ -5 & 2\end{array}\right]$

(d)

$\left[\begin{array}{cc}7 & -5 \\ 11 & 2\end{array}\right]$

40.

Given that A is a square matrix of order 3 and |A|=-4, then $|\operatorname{adj} A|$ is equal to

(a)

-4

(b)

(c)

-16

(d)

41.

If A and B are invertible square matrices of the same order, then which of the following is not correct?

(a)

$adj A=|A| \cdot A^{-1}$

(b)

$\operatorname{det}\left(A^{-1}\right)=[\operatorname{det}(A)]^{-1}$

(c)

$(A B)^{-1}=B^{-1} A^{-1}$

(d)

$(A+B)^{-1}=B^{-1}+A^{-1}$

42.

Given that A is a square matrix of order 3 and |A| = -2, then |adj (24)| is equal to

(a)

-2⁶

(b)

(c)

-2⁸

(d)

2⁸

43.

If for a square matrix A, A² - A + l = 0, then A^-1 equals

(a)

(b)

A + l

(c)

l - A

(d)

A - l

44.

If for a square matrix A, A² - 3A + 1 = 0 and A^-1 = xA + yl, then the value of x + y is

(a)

-2

(b)

(c)

(d)

-3

45.

Let A be a 3 x 3 matrix such that $|\operatorname{adj} A|=64$. Then, |A| is equal to

(a)

8 only

(b)

-8 only

(c)

(d)

8 or -8

46.

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)

47.

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)

48.

For the matrix $A=\left[\begin{array}{ccc}2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3\end{array}\right]$ to be invertible, the value of $\lambda$ is

(a)

(b)

(c)

$R-\{10\}$

(d)

$R-\{-10\}$

49.

If A is a square matrix of onder 3 such that the value of $|\operatorname{adj} A|=8$, then the value of |$\mid A^T|$ is

(a)

$\sqrt{2}$

(b)

$-\sqrt{2}$

(c)

(d)

$2 \sqrt{2}$

50.

Let $A=\left[\begin{array}{cc}200 & 50 \\ 10 & 2\end{array}\right]$and $B=\left[\begin{array}{cc}50 & 40 \\ 2 & 3\end{array}\right]$, then |AB| is equal to

(a)

460

(b)

2000

(c)

3000

(d)

-7000

51.

If $\left|\begin{array}{lll}2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1\end{array}\right|+3=0$, then the value of x is

(a)

(b)

(c)

-1

(d)

52.

Given that $A=\left[a_{i j}\right]$ is a square matrix of order 3 x 3 and |A| = -7, then the value of $\sum_{i=1}^3 a_{i 2} A_{i 2}$, where $A_{i j}$ denotes the cofactor of element $a_{i j}$ is

(a)

(b)

-7

(c)

(d)

53.

Let $A=\left[\begin{array}{ccc}1 & \sin \alpha & 1 \\ -\sin \alpha & 1 & \sin \alpha \\ -1 & -\sin \alpha & 1\end{array}\right]$, where $0 \leq \alpha \leq 2 \pi$, then

(a)

|A| = 0

(b)

$|A| \in(2, \infty)$

(c)

$|A| \in(2,4)$

(d)

$|A| \in[2,4]$

54.

Given that A is a non-singular matrix of order 3 such that $A^2=2 A$, then the value of |2A| is

(a)

(b)

(c)

(d)

55.

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)

56.

If $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$, then the possible value(s) of x is/are

(a)

(b)

$\sqrt{3}$

(c)

$-\sqrt{3}$

(d)

$\sqrt{3},-\sqrt{3}$

57.

The value of |A|, if $A=\left[\begin{array}{ccc}0 & 2 x-1 & \sqrt{x} \\ 1-2 x & 0 & 2 \sqrt{x} \\ -\sqrt{x} & -2 \sqrt{x} & 0\end{array}\right]$, where $x \in R^{+}$, is

(a)

$(2 x+1)^2$

(b)

(c)

$(2 x+1)^3$

(d)

None of these

58.

If the area of the tríangle with vertices (-3,0), (3, 0) and (0, k) is 9 sq units, then the value's of k will be

(a)

(b)

土3

(c)

-9

(d)

59.

Let A be the area of a triangle having vertices $\left(x_1, y_1\right),\left(x_2, y_2\right)$ and $\left(x_3, y_3\right)$. Which of the following is correct?

(a)

$\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|= \pm A$

(b)

$\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|= \pm 2 A$

(c)

$\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|= \pm A/2$

(d)

$\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|= \pm A^2$

60.

If $\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1\end{array}\right]$ is non-singular matrix and $a \in A$, then the set A is

(a)

(b)

{0}

(c)

{4}

(d)

R - {4}

61.

Let A be a skew- symmetric matrix of order 3 . If |A| = x, then (2023)^x is equal to

(a)

2023

(b)

$\frac{1}{2023}$

(c)

$(2023)^2$

(d)

62.

If $\left|\begin{array}{lll}\alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1\end{array}\right|=0$, then the value of $\alpha$ is

(a)

(b)

(c)

(d)

63.

The value of the determinant $\left|\begin{array}{ccc}2 & 7 & 1 \\ 1 & 1 & 1 \\ 10 & 8 & 1\end{array}\right|$ is

(a)

(b)

-79

(c)

(d)

-51

64.

If A is a square matrix of order 2 and |4| = -2, then value of |5A' | is

(a)

- 50

(b)

- 10

(c)

(d)

65.

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)

66.

Given that matrices A and B are of order $3 \times n$ and $m \times 5$ respectively, then the order of matrix C = 5A + 3B is

(a)

3 x 5 and m = n

(b)

3 x 5

(c)

3 x 3

(d)

5 x 5

67.

Given that $A=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha\end{array}\right]$ and $A^2=3 I$, then

(a)

$1+\alpha^2+\beta \gamma=0 $

(b)

$1-\alpha^2-\beta \gamma=0 $

(c)

$3-\alpha^2-\beta \gamma=0 $

(d)

$3+\alpha^2+\beta \gamma=0$

68.

If A is a square matrix such that A²=A, then (I + A)³ – 7A is

(a)

(b)

I + A

(c)

I - A

(d)

69.

If $A=\left[\begin{array}{cc}0 & 2 \\ 3 & -4\end{array}\right]$ and $k A=\left[\begin{array}{cc}0 & 3 a \\ 2 b & 24\end{array}\right]$, then the value of k, a and b respectively, are

(a)

-6,-12,-18

(b)

-6,-4,-9

(c)

-6,4,9

(d)

-6,12,18

70.

A matrix $A=\left[a_{i j}\right]_{3 \times 3}$ is defined by $a_{i j}=\left\{\begin{array}{cl}2 i+3 j, & i<j \\ 5, & i=j . \\ 3 i-2 j, & i>j\end{array}\right.$ The number of elements in A which are more than 5 , is

(a)

(b)

(c)

(d)

71.

If $\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]$, then the value of a+b-c+2d is

(a)

(b)

(c)

(d)

-8

72.

If $A=\left[a_{i j}\right]$ is a square matrix of order 2 such that $a_{i j}=\left\{\begin{array}{ll}1, & \text { when } i \neq j \\ 0, & \text { when } i=j\end{array}\right.$, then $A^2$ is

(a)

$\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right]$

(b)

$\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$

(c)

$\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$

(d)

$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $

73.

If $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]$ and $A=B^2$, then x equals

(a)

$\pm 1$

(b)

-1

(c)

(d)

74.

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

75.

If $A=\left[\begin{array}{ll}3 & 4 \\ 5 & 2\end{array}\right]$ and 2 A+B is a null matrix, then B is equal to

(a)

$\left[\begin{array}{cc}6 & 8 \\ 10 & 4\end{array}\right]$

(b)

$\left[\begin{array}{cc}-6 & -8 \\ -10 & -4\end{array}\right]$

(c)

$\left[\begin{array}{cc}5 & 8 \\ 10 & 3\end{array}\right]$

(d)

$\left[\begin{array}{cc}-5 & -8 \\ -10 & -3\end{array}\right]$

76.

If $x\left[\begin{array}{l}1 \\ 2\end{array}\right]+y\left[\begin{array}{l}2 \\ 5\end{array}\right]=\left[\begin{array}{l}4 \\ 9\end{array}\right]$, then

(a)

x = 1, y = 2

(b)

x = 2, y = 1

(c)

x = 1, y = -1

(d)

x = 3, y = 2

77.

Simplest form of $\tan ^{-1}\left(\frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos x}-\sqrt{1-\cos x}}\right), \pi<x<\frac{3 \pi}{2}$ is

(a)

$\frac{\pi}{4}-\frac{x}{2}$

(b)

$\frac{3 \pi}{2}-\frac{x}{2}$

(c)

$-\frac{x}{2}$

(d)

$\pi-\frac{x}{2}$

78.

$\sin \left(\tan ^{-1} x\right)$, 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)

$\frac{x}{\sqrt{1+x^2}}$

79.

If $\tan ^{-1} x=y$, then

(a)

$-1<y<1$

(b)

$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

(c)

$-\frac{\pi}{2}<y<\frac{\pi}{2}$

(d)

$y \in\left\{-\frac{\pi}{2}, \frac{\pi}{2}\right\}$

80.

$\left[\sin ^{-1} \frac{\pi}{3}+\sin ^{-1}\left(\frac{1}{2}\right)\right]$ is equal to

(a)

(b)

$\frac{1}{2}$

(c)

$\frac{1}{3}$

(d)

$\frac{1}{4}$

81.

Let f(x) = $\left|\begin{array}{cc} x^2 & \sin x \\ p & -1 \end{array}\right|$, where p is a constant. Then, the value of p for which f'(0) = 1 is

(a)

(b)

(c)

(d)

-1

82.

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)

83.

For the matrix A = $\left[\begin{array}{ccc} 2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3 \end{array}\right]$ to be invertible, the value of $\lambda$ is

(a)

(b)

(c)

R - {10}

(d)

R - {-10}

84.

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

85.

If value of $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot \theta \operatorname{cosec}^2 \theta d \theta$ is

(a)

$\frac{1}{2}$

(b)

- $\frac{1}{2}$

(c)

(d)

-$\frac{\pi}{8}$

86.

If sin(xy) = 1 then $\frac{dy}{dx}$ is equal to

(a)

$\frac{x}{y}$

(b)

- $\frac{x}{y}$

(c)

$\frac{y}{x}$

(d)

- $\frac{y}{x}$

87.

The product of matrix P and Q is equal to a diagonal matrix. If the order of matrix Q is 3 $\times$ 2, then the order of matrix P is

(a)

2 $\times$ 2

(b)

3 $\times$ 3

(c)

2 $\times$ 3

(d)

3 $\times$ 2

88.

lf A is a square matrix of order 2 and |A| = -2, then value of |5A'| is

(a)

-50

(b)

-10

(c)

(d)

89.

The function f : R→ R defined as f(x) = x³ is

(a)

one-one but not onto

(b)

not one-one but onto

(c)

neither one-one nor onto

(d)

both one-one and onto

90.

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

(a)

{1, 5, 9}

(b)

{0,1,2,5}

(c)

(d)

91.

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)

92.

Let A (3,5). Then, number of reflexive relations on A is

(a)

(b)

(c)

(d)

93.

Select the correct option out of the four given options
Let R be a relation in the set N given by
R = {(a, b):ab - 2,b > 6}
Then,

(a)

(8, 7) ∈ R

(b)

(6,8) ∈ R

(c)

(3,8) ∈ R

(d)

(2,4) ∈ R

94.

Given a curve y = 7x - x³ and x increases at the rate of 2 units per sec. The rate at which the slope of the curve is changing when x = 5 is

(a)

- 60 units/sec

(b)

60 units/sec

(c)

-70 units/sec

(d)

-140 units/sec

95.

The function f(x) = $\frac{x}{2}+\frac{2}{x}$ has a local minima at x equal to

(a)

(b)

(c)

(d)

-2

96.

Derivative of $e^{\sin ^2 x}$ with respect to cos x is

(a)

sin x$e^{\sin ^2 x}$

(b)

cos x$e^{\sin ^2 x}$

(c)

- 2 cos x$e^{\sin ^2 x}$

(d)

-2 sin² x cos x$e^{\sin ^2 x}$

97.

If Xe^y = 1, then the value of $\frac{dy}{dx}$ at x = 1 is

(a)

-1

(b)

(c)

-e

(d)

- $\frac{1}{e}$

98.

Find the matrix A², where A = [a_ij] is a 2 $\times$ 2 matrix whose elements are given by
a_ij = maximum (i, j) - minimum (i, j)

(a)

$\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]$

(b)

$\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$

(c)

$\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$

(d)

$\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]$

99.

If [x 2 0] $\left[\begin{array}{c} 5 \\ -1 \\ x \end{array}\right]$=[3 1] $\left[\begin{array}{c} -2 \\ x \end{array}\right]$, then value of x is

(a)

-1

(b)

(c)

(d)

100.

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)

101.

If A is a square matrix of order 3 such that the value of | adj A | = 8, then the value of |4^T | is

(a)

$\sqrt{2}$

(b)

-$\sqrt{2}$

(c)

(d)

2$\sqrt{2}$

102.

If A = $\left[\begin{array}{ccc} a & c & -1 \\ b & 0 & 5 \\ 1 & -5 & 0 \end{array}\right]$ is a skew-symmetric matrix, then the value of 2a - (b + c) is

(a)

(b)

(c)

-10

(d)

103.

A function f : R $\rightarrow$ R defined as f(x) = x² - 4x + 5 is

(a)

injective but not surjective

(b)

surjective but not injective.

(c)

both injective and surjective.

(d)

neither injective nor surjective.

104.

Find The Number Of Value 11+12+13+............1000 value

(a)

1000

(b)

100

(c)

200

105.

Name the Capital of India ?

(a)

Delhi

(b)

Mumbai

(c)

Kolkallta

(d)

Chennai

106.

If a, b, c are in AP, then the value of
$\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{array}\right| \text { is }$

(a)

(b)

-3

(c)

(d)

abc

107.

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

108.

Let A be a square matrix of order 3 x 3 and k a scalar, then |kA| is equal to

(a)

k|A|

(b)

|k||A|

(c)

k³|A|

(d)

none of these

109.

Let A be a non-angular square matrix of order 3 x 3, then |A . adj A| is equal to

(a)

|A|³

(b)

|A|²

(c)

|A|

(d)

3|A|

110.

If A and B are invertible matrices then which of the following is not correct

(a)

$A d j A=|A| \cdot A^{-1}$

(b)

$\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$

(c)

$(A B)^{-1}=B^{-1} A^{-1}$

(d)

$(A+B)^{-1}=A^{-1}+B^{-1}$

111.

If $A=\left[\begin{array}{ll} 3 & -2 \\ 4 & -2 \end{array}\right]$ then the value of kif, A² = kA - 2I is

(a)

(b)

(c)

-7

(d)

112.

The matrix $\left[\begin{array}{rrr} 2 & -1 & 4 \\ 1 & 0 & -5 \\ -4 & 5 & 7 \end{array}\right]$ is

(a)

a symmetric matrix

(b)

a skew-symmetric matrix

(c)

a diagonal matrix

(d)

none of these

113.

If matrix A is of order m x n, and for matrix B, AB and BA both are defined, then order of matrix B is

(a)

m x n

(b)

n x n

(c)

n x n

(d)

n x m

114.

A matrix has 18 elements, then possible number of orders of a matrix are

(a)

(b)

(c)

(d)

115.

If $A=\left[\begin{array}{rr} 3 & 1 \\ -1 & 2 \end{array}\right]$ then A² - 5A - 7I iS

(a)

a zero matrix

(b)

an identity matrix

(c)

diagonal matrix

(d)

none of these

116.

If $A=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$ then A⁶ is equal to

(a)

zero matrix

(b)

(c)

(d)

none of these

117.

The matrix A satisfies the equation $\left[\begin{array}{rr} 0 & 2 \\ -1 & 1 \end{array}\right] A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$ then matrix A is

(a)

$\left[\begin{array}{rr} 2 & 0 \\ 1 & -1 \end{array}\right]$

(b)

$\left[\begin{array}{rr} 1 & -2 \\ 1 & 0 \end{array}\right]$

(c)

$\left[\begin{array}{cc} \frac{1}{2} & -1 \\ \frac{1}{2} & 0 \end{array}\right]$

(d)

$\left[\begin{array}{rr} 1 & 2 \\ -1 & 0 \end{array}\right]$

118.

If $F(x)=\left[\begin{array}{rr} \cos x & \sin x \\ -\sin x & \cos x \end{array}\right] \text { , }$ then F(x) F(y) is equal to

(a)

F(x)

(b)

F(xy)

(c)

F(x + y)

(d)

F(x - y)

119.

The function f(x) cos x - 2px is monotonically decreasing for

(a)

$p<\frac{1}{2}$

(b)

$p>\frac{1}{2}$

(c)

$p<2$

(d)

$p>2$

120.

The height of the cylinder of greatest volume which can be inscribed in a right circular cone, of height 17, is

(a)

$\frac{h}{2}$

(b)

$\frac{h}{4}$

(c)

$\frac{h}{3}$

(d)

none of these

121.

The maximum value of $\left(\frac{1}{x}\right)^{x} \text { is }$

(a)

(b)

e^e

(c)

$e^{\frac{1}{e}}$

(d)

$\left(\frac{1}{e}\right)^{\frac{1}{e}}$

122.

Which of the following function is decreasing on $\left(0, \frac{\pi}{2}\right)?$

(a)

cos x

(b)

-cos 2x

(c)

cos 3x

(d)

tan x

123.

Given function $f(x)=x^{2} e^{-x}$ then 'f' increases in the interval

(a)

$(-\infty, \infty)$

(b)

$(-2, 0)$

(c)

$(2, \infty)$

(d)

$(0,2)$

124.

Equation of tangent to the curve $y=1-e^{\frac{x}{2}}$ at the point of intersection with the y -axis is

(a)

$x+2 y=1$

(b)

$2 x+y=0$

(c)

$x-y=2$

(d)

none of these

125.

For the function y x³ + 21, the value of x, when y increases 75 times as fast as X, is

(a)

$\pm$3

(b)

$\pm 5 \sqrt{3}$

(c)

$\pm$5

(d)

none of these

126.

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

(a)

3.074

(b)

2.926

(c)

(d)

none of these

127.

It isgiven thatatx= I, the function f(x) = x4 - 62x² + ax + 9 attains its maximum value, on the interval [0, 2]. The value of a is

(a)

(b)

-120

(c)

120

(d)

128.

If the curves 4x = y² and 4xy = k cut at right angles, then

(a)

k² = 64

(b)

k² = 512

(c)

k² = 4

(d)

k² = 256

129.

A spherical balloon has a variable diameter $\frac{3}{2}(2 x+1)$ The rate of change of its volume with respect to x is

(a)

$\frac{27 \pi}{8}(2 x+1)^{2}$

(b)

$\frac{9}{4} \pi(2 x+1)^{3}$

(c)

$\frac{9 \pi}{16}(2 x+1)^{3}$

(d)

$\pi(2 x+1)^{2}$

130.

The maximum value of xy, "subject to x + y = 8 is

(a)

(b)

(c)

(d)

131.

Discuss the applicability of LMV Theorem for the function f(x) = |x|, in [1.1]

(a)

applicable, c = 0

(b)

applicable, c = -1

(c)

applicable, c = 1

(d)

not applicable

132.

Derivative of $\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a} \text { , }$ With respect to x, is

(a)

$\sin ^{-1} \frac{x}{a}$

(b)

$\frac{x}{2} \sqrt{a^{2}-x^{2}}$

(c)

$\sqrt{a^{2}-x^{2}}$

(d)

$\frac{1}{\sqrt{a^{2}-x^{2}}}$

133.

$\text { If } x=a t^{2}, y=2 a t, \text { then } \frac{d^{2} y}{d x^{2}} \text { is }$

(a)

$\frac{1}{t}$

(b)

$-\frac{1}{t^{2}}$

(c)

at²

(d)

$\frac{-1}{2 a t^{3}}$

134.

The function 'f 'defined by $f(x)=\left\{\begin{array}{cc} \frac{x^{3}-8}{x-2}, & x \neq 2 \\ 12, & x=2 \end{array}\right. \text { is }$

(a)

not continuous at x = 2

(b)

continuous at x = 2

(c)

not continuous at x = 3

(d)

not continuous at x = - 2

135.

State the function which is continuous for all x $\in$ R,

(a)

sin x

(b)

$\frac{x^{2}-25}{x-5}$

(c)

[x]

(d)

sgn (x)

136.

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)

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

137.

State which of the following is continuous as well as differentiable for x $\in$ R

(a)

|x|

(b)

[x]

(c)

polynomial function

(d)

sgn (x)

138.

A function 'f is said to be continuous at x = a, if

(a)

$\lim _{x \rightarrow a} f(x)$ exists

(b)

$\lim _{x \rightarrow a} f(x)$ does not exist

(c)

f(a) exists

(d)

none of these

139.

$\text { The value of } \cos ^{-1}\left(\frac{1}{2}\right)+3 \sin ^{-1}\left(\frac{1}{2}\right) \text { is equal to }$

(a)

$\frac{\pi}{4}$

(b)

$\frac{\pi}{6}$

(c)

$\frac{2\pi}{3}$

(d)

$\frac{5\pi}{6}$

140.

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

(a)

$\frac{2 \pi}{3}$

(b)

$\frac{ \pi}{3}$

(c)

$-\frac{ \pi}{6}$

(d)

$\frac{ \pi}{6}$

141.

Let Z be the set of integers. Define a binary operation * in Z x Z as (a, b) * (c, d) = (a + c, b +d), then binary operation * is

(a)

not commutative

(b)

not associative

(c)

commutative and associative

(d)

does not have identity element

142.

Let the function 'f' be defined by $f(x)=5 x^{2}+2, \forall x \in R$ Then f' is

(a)

onto function

(b)

one-one, onto function

(c)

one-one, into function

(d)

many-one, into function

143.

The relation R in the set of real numbers defined as R = {(a, b) $\in$ R x R : 1 + ab > 0} is

(a)

reflexive and transitive

(b)

symmetric and transitive

(c)

reflexive and symmetric

(d)

equivalence 'relation

144.

$\text { Let ' } f^{\prime}: R-\{2\} \rightarrow R-\{1\} \text { be a function defined by }$ $f(x)=\frac{x-1}{x-2}, \text { then } f^{\prime} \text { is }$

(a)

into function

(b)

many one function

(c)

bijective function

(d)

many one, into function

145.

Let R be the relation in the set of natural numbers N defined as $R=\{(x, y) \in N \times N: 3 x+y=11\}$ Then R^-1 is given by

(a)

{0, 11), (1, 8), (2, 5), (3, 2)}

(b)

{(1, 8), (2, 5), (3, 2)}

(c)

{(11, 0), (8, 1), (5, 2), (2, 3)}

(d)

{(8, 1), (5, 2), (2, 3)}

146.

Let function f: R $\rightarrow$ R is defined as f(x) = 2x³ -1 If f^-1 exists, then f^-1 is

(a)

$2 x^{3}+1$

(b)

$(2 x)^{3}+1$

(c)

$(1-2 x)^{3}$

(d)

$\left(\frac{1+x}{2}\right)^{\frac{1}{3}}$

147.

If f(x) = x³ and g(x) = cos 3x , then fog is

(a)

x³.cos 3x

(b)

cos 3x³

(c)

cos³ 3x

(d)

3cos x³

148.

A relation defined in a non-empty set A, having n elements, has

(a)

n relations

(b)

2 relations

(c)

n² relations

(d)

2n² relations

149.

Let the function 'f' : N $\rightarrow$ N be defined by $f(x)= {2} x+3, \forall x \in N \text { . Then } f^{\prime \prime} \text { is }$

(a)

not onto

(b)

bijective function

(c)

many-one, into function

(d)

none of these

150.

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

151.

Let A = {a, b}. Then number of one-one functions from A to A possible are

(a)

(b)

(c)

(d)

152.

At $x=\frac{5 \pi}{6}, f(x)=2 \sin 3 x+3 \cos 3 x$ is

(a)

maximum

(b)

minimum

(c)

zero

(d)

neither maximum nor minimum

153.

The smallest value of polynomial $x^{3}-18 x^{2}+96 x \text { in }[0,9]$ is

(a)

126

(b)

(c)

135

(d)

160

154.

The area of greatest rectangle that can be inscribed in an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is

(a)

ab sq units

(b)

$\frac{a b}{2} \text { sq units }$

(c)

2ab sq units

(d)

3ab sq units

155.

The area of a triangle is computed using the formula $S=\frac{1}{2} b c \sin A$.If the relative errors made in measuring b,e and calcuting S are respectively 0.02, 0.01 and 013 the approximate error in A when $A=\pi / 6$ .

(a)

0.05 radians

(b)

0.01 radians

(c)

0.05 degree

(d)

0.01 degree

156.

The equation s of the tangent and normal to the parabola y²= 4ax at the point (at², 2at) are respectively

(a)

$t y=x+a t^{2}=\text { ind } y=-t x+2 a t+a t^{3}$

(b)

$t y=x-a t^{2} \text { a nd } y=t x-2 a t+a t^{3}$

(c)

$y=t x+2 a t+a t^{3} \text { and } t y=x+a t^{2}$

(d)

$y=-t x+2 a t \cdot a t^{3} \text { and } t y=x+a t^{2}$

157.

If the tange Jltat any point on the curve $x^{4}+y^{4}=c^{4}$ cut off inter cepts a and b on the coordinate axes, then the val ue of a^-4/3 +b^-4/3 is

(a)

$c^{-4 / 3}$

(b)

$c^{-1 / 2}$

(c)

$c^{1 / 2}$

(d)

None of these

158.

Let $g(x)=2 f\left(\frac{x}{2}\right)+f(2-x) \text { and } f^{\prime \prime}(x)<0$ for all $x \in(0,2) \text { then, } g(x) \text { is }$

(a)

increasing on (4/3,2) and increasing on (0,4/3)

(b)

decreas ling on (0,4/3) and increasing on (4 / 3, 2)

(c)

increasing on (0, 4 / 3) and decreasing on (4 / 3, 2)

(d)

None of the above

159.

If $f: R \rightarrow R \text { be defined by } f(x)=2 x+\cos x \text { , then } f$

(a)

has a minimum at $x=\pi$

(b)

has at maximum at x = 0

(c)

is a decreasing function

(d)

is an i ucreasing function

160.

For $O t<\theta<\frac{\pi}{2}$ the value of θ, if it increases twice as fast and its sine is

(a)

$\frac{\pi}{2}$

(b)

$\frac{\pi}{3}$

(c)

$\frac{\pi}{6}$

(d)

None of these

161.

The distance covered by a particle in time t is given by $x=3+8 t-4 t^{2}$ After 1s, its velocity will be

(a)

0 units/s

(b)

3 units/s

(c)

4 units/s

(d)

7 units/s

162.

Aright circular cylinder which is open at the top and has a given surface area, will have the greatest volume, if its height h and radius rare related by

(a)

2h =r

(b)

h = 4r

(c)

h =2r

(d)

h=r

163.

The functin f(x) = x^x has a stationary point at

(a)

x =e

(b)

$x=\frac{1}{e}$

(c)

x = 1

(d)

$x=\sqrt{e}$

164.

The maximum slope of curve $y=-x^{3}+3 x^{2}+9 x-27$ is

(a)

(b)

(c)

(d)

165.

The function $f(x)=2 x^{3}-3 x^{2}-12 x+4$ has

(a)

two points of local maximum

(b)

two points of local minimum

(c)

one maxima and one minima

(d)

no maxima or minima

166.

If x is real, then the minimum value of $x^{2}-8 x+17$ is

(a)

-1

(b)

(c)

(d)

167.

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

168.

The circumference of a circle is measured as 56 cm with an error of 0.02 cm. The percentage error in area is

(a)

$\frac{1}{2}$

(b)

$\frac{1}{28}$

(c)

$\frac{1}{14}$

(d)

$\frac{1}{56}$

169.

If the radius of a sphere is measured as 7m with an error of 0.02 m, then the approximate error in calculating its volume is

(a)

$1.92 \pi \mathrm{m}^{3}$

(b)

$3.92 \pi \mathrm{m}^{3}$

(c)

$0.285 \pi \mathrm{m}^{3}$

(d)

$2.98 \pi \mathrm{m}^{3}$

170.

Using differentials, the approximate value of $\sqrt{0.082}$ is

(a)

2.867

(b)

0.2867

(c)

0.356

(d)

None of these

171.

If $y=x^{4}-10$ and x changes from 2 to 1.99,then what is the change in y?

(a)

0.32

(b)

0.032

(c)

5.68

(d)

5.98

172.

The points at which the tangents to the curve $y=x^{3}-12 x+18$ are parallel to X-axis are

(a)

(2, - 2), (- 2, - 34)

(b)

(2, 34), (- 2, 0)

(c)

(0, 34), (- 2, 0)

(d)

(2,2), (- 2,34)

173.

If the curve $a y+x^{2}=7 \text { and } x^{3}=y$ cut orthogonally at (1, 1), then the value of a is

(a)

(b)

(c)

-6

(d)

174.

If x + y = K is normal to $y^{2}=12 x$ then K is

(a)

(b)

(c)

-9

(d)

-3

175.

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)

176.

The curve $y=x^{1 / 5}$ has at (0, 0)

(a)

a vertical tangent (parallel to Y-axis)

(b)

a horizontal tangent (parallel to X-axis)

(c)

an oblique tangent

(d)

no tangent

177.

Which of the following functions is decreasing on $\left(0, \frac{\pi}{2}\right)$?

(a)

sin2x

(b)

tanx

(c)

cosx

(d)

cos3x

178.

The function $f(x)=4 \sin ^{3} x-6 \sin ^{2} x+12 \sin x$ + 100 is strictly

(a)

increasing in $\left(\pi, \frac{3 \pi}{2}\right)$

(b)

decreasing in $\left(\frac{\pi}{2}, \pi\right)$

(c)

decreasing in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$

(d)

decreasing in $\left[0, \frac{\pi}{2}\right]$

179.

The function $f(x)=\tan x-x$

(a)

always increases

(b)

always decreases

(c)

never increases

(d)

sometimes increases and sometimes decreases

180.

If $y=x(x-3)^{2}$ decreases for the values of x given by

(a)

$1<x<3$

(b)

$x<0$

(c)

$x>0$

(d)

$0<x<\frac{3}{2}$

181.

The total cost C(x) (in Rs) associated with the production of x units of an item is given by C(x) = 0.007x3 - 0.003x² + 15x + 4000. The marginal cost when 17units are produced, is

(a)

Rs. 20.967

(b)

Rs. 21.96

(c)

Rs. 81.968

(d)

Rs. 11.967

182.

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

183.

The radius of the base of a cone is increasing at the rate of 3 cm/min and the altitude is decreasing at the rate of 4 cm/min. The rate of change of lateral surface when the , radius = 7 cm and altitude 24 cm, is

(a)

$54 \pi \mathrm{cm}^{2} / \mathrm{min}$

(b)

$7 \pi \mathrm{cm}^{2} / \mathrm{min}$

(c)

$27 \mathrm{~cm}^{2} / \mathrm{min}$

(d)

None of the above

184.

A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

(a)

$\frac{1}{10} \mathrm{rad} / \mathrm{s}$

(b)

$\frac{1}{20} \mathrm{rad} / \mathrm{s}$

(c)

$20 \mathrm{rad} / \mathrm{s}$

(d)

10 rad/s

185.

If the sides of an equilateral triangle are increasing at the rate of 4 cm/s, then the rate at which the area increases, when side is 5 cm, is

(a)

$10 \mathrm{~cm}^{2} / \mathrm{s}$

(b)

$\sqrt{3} \mathrm{~cm}^{2} / \mathrm{s}$

(c)

$10 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{s}$

(d)

$\frac{10}{3} \mathrm{~cm}^{2} / \mathrm{s}$

186.

If $y=\left(\tan ^{-1} x\right)^{2}$ ,then the value of $\left(x^{2}+1\right)^{2} y_{2}+2 x\left(x^{2}+1\right) y_{1}$ is

(a)

(b)

(c)

(d)

None of these

187.

If $y=3 \cos (\log x)+4 \sin (\log x)$,then

(a)

$x y_{2}+y_{1}+y=0$

(b)

$x y_{2}+y_{1}-y=0$

(c)

$x^{2} y_{2}+x y_{1}+y=0$

(d)

None of these

188.

If $y=\cos ^{-1} x$ then the value of $\frac{d^{2} y}{d x^{2}}$ in terms of y alone is

(a)

$-\cot y \operatorname{cosec}^{2} y$

(b)

cosec y cot² y

(c)

-cot y cosec y

(d)

None of these

189.

For the function $f(x)=x+\frac{1}{x}, x \in[1,3]$ the value of C for mean value theorem is

(a)

(b)

$\sqrt{3}$

(c)

(d)

None of these

190.

For the function $f(x)=x^{3}-5 x^{2}-3 x, x \in[1,3]$ the value of C for mean value theorem is

(a)

$\frac{7}{4}$

(b)

$\frac{7}{3}$

(c)

$\frac{3}{7}$

(d)

None of these

191.

The value of c in Rolle's theorem for the function $f(x)=x^{3}-3 x$ in the interval $[0, \sqrt{3}]$

(a)

(b)

-1

(c)

$\frac{3}{2}$

(d)

$\frac{1}{3}$

192.

The value of c in Rolle's theorem for the function $f(x)=x^{2}+2 x-8, x \in[-4,2]$ is

(a)

(b)

-1

(c)

(d)

-2

193.

The Rolle's theorem is applicable in the interval $-1 \leq x \leq 1$ for the function

(a)

f(x) = x

(b)

f(x) = x²

(c)

f(x) = 2x³ + 3

(d)

(x) = lxl

194.

If $x=\sin t \text { and } y=\sin p t, \text { then }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}$ is equal to

(a)

-y

(b)

(c)

(d)

-p²y

195.

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

196.

If $y=\left(x+\sqrt{1+x^{2}}\right)^{n}$ is

(a)

-xcosx -2sinx

(b)

xcosx + 2sinx

(c)

xsinx + cosx

(d)

None of these

197.

If $y=x \cos x, \text { then } \frac{d^{2} y}{d x^{2}}$ is

(a)

$-x \cos x-2 \sin x$

(b)

xcosx + 2sinx

(c)

xsinx + cosx

(d)

None of these

198.

If (cosx)y = (cosy)^x, then $\frac{d y}{d x}$ is equal to

(a)

$\frac{\log (\cos y)+y(\tan x)}{\log (\cos x)+x \tan y}$

(b)

$\frac{\log (\cos y)-y(\tan x)}{\log (\cos x)-x(\tan y)}$

(c)

$\frac{\log (\tan x)+y(\cos x)}{\log (\cos x)+x(\tan y)}$

(d)

None of these

199.

If $y=a^{t+\frac{1}{t}} \text { and } x=\left(t+\frac{1}{t}\right)^{a}, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{a^{\left(1+\frac{1}{t}\right)} \log a}{\left(t+\frac{1}{t}\right)^{a-1}}$

(b)

$\frac{a^{\left(1+\frac{1}{t}\right)}}{\left(t+\frac{1}{b}\right)^{a-1}}$

(c)

$\frac{a^{\left(1+\frac{1}{t}\right)} \log a}{a\left(t+\frac{1}{t}\right)^{a-1}}$

(d)

None of these

200.

If $y=\log _{7}(\log x), \text { then } \frac{a y}{d x}$ is equal to

(a)

$\frac{1}{x \log x \log 7}$

(b)

$\frac{-1}{x \log x \log 7}$

(c)

$\frac{1}{x \log x}$

(d)

None of these

201.

If $y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)$,then $\frac{d y}{d x}$ is equal to

(a)

$\frac{4 x^{3}}{1-x^{4}}$

(b)

$\frac{-4 x}{1-x^{4}}$

(c)

$\frac{1}{4-x^{4}}$

(d)

$\frac{-4 x^{3}}{1-x^{4}}$

202.

If $f(x)=|\cos x|$,then

(a)

f is everywhere differentiable

(b)

f is everywhere coqtinuous but not differentiable at $x=n \pi, n \in Z$

(c)

is everywhere continuous but not differentiable at $x=(2 n+1) \frac{\pi}{2}, n \in Z$

(d)

None ofthe above

203.

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)

$m=n=\frac{\pi}{2}$

204.

If $y=(\cos x)^{(\cos x)^{(\cos x) \ldots \infty}}$,then $\frac{d y}{d x}$ is equal to

(a)

$\frac{y \tan x}{y \log \cos x-1}$

(b)

$\frac{y^{2} \tan x}{y \log \cos x-1}$

(c)

$\frac{y \tan x}{1+y \log \cos x}$

(d)

None ofthese

205.

If $y=x^{x^{x^{x^{x}}}}$ then $\frac{d y}{d x}$ is equal to

(a)

yx^y-1

(b)

$\frac{y^{2}}{x(1-y \log x)}$

(c)

$\frac{y}{x(1+y \log x)}$

(d)

None of these

206.

The derivative of b tans with respect to asec θ is

(a)

$\frac{b}{a} \operatorname{cosec} \theta$

(b)

$\frac{a}{b} \operatorname{cosec} \theta$

(c)

$\frac{b}{a} \cot \theta$

(d)

$\frac{a}{b} \cot \theta$

207.

The derivative ofsin2 x with respect to e^cosx is

(a)

$\frac{2 \cos x}{e^{\cos x}}$

(b)

$-\frac{2 \cos x}{e^{\cos x}}$

(c)

$\frac{2}{e^{\cos x}}$

(d)

None of these

208.

The derivative of $\cos ^{-1}\left(2 x^{2}-1\right) \text { w.r.t. } \cos ^{-1} x$

(a)

(b)

$\frac{-1}{2 \sqrt{1-x^{2}}}$

(c)

$\frac{2}{x}$

(d)

1- x²

209.

If $x^{y}=y^{x}, \text { then } x(x-y \log x) \frac{d y}{d x}$ is equal to

(a)

y(y-xlogy)

(b)

y(y + xlogy)

(c)

x(x + ylogx)

(d)

x(y -xlogy)

210.

If $x=e^{x / y}, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{x-y}{x \log x}$

(b)

$\frac{y-x}{\log x}$

(c)

$\frac{y-x}{x \log x}$

(d)

$\frac{x-y}{\log x}$

211.

If $y^{x}=e^{y-x}, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{1+\log y}{y \log y}$

(b)

$\frac{(1+\log y)^{2}}{y \log y}$

(c)

$\frac{1+\log y}{(\log y)^{2}}$

(d)

$\frac{(1+\log y)^{2}}{\log y}$

212.

If $y=\log _{a} x+\log _{x} a+\log _{x} x+\log _{a} a, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{1}{x}+x \log a$

(b)

$\frac{\log a}{x}+\frac{x}{\log a}$

(c)

$\frac{1}{x \log a}+x \log a$

(d)

None of these

213.

If y = log x^x, then the value of $\frac{d y}{d x}$ is

(a)

$x^{x}(1+\log x)$

(b)

log (ex)

(c)

$\log \frac{e}{x}$

(d)

$\log \left(\frac{x}{e}\right)$

214.

Derivate of $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$ $0<x<\frac{\pi}{2}$ is

(a)

$\frac{1}{2}$

(b)

(c)

(d)

None of these

215.

If $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2},-1 \leq x<1,}$ then $\frac{d y}{d x}$ is equal to

(a)

(b)

(c)

(d)

216.

If $y=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right),-\frac{1}{\sqrt{3}}, then \(\frac{d y}{d x}$ is

(a)

$\frac{3}{1+x^{2}}$

(b)

$\frac{1}{1+x^{2}}$

(c)

$\frac{-3}{1+x^{2}}$

(d)

$\frac{3}{1-x^{2}}$

217.

If cos y = xcos(a + y) with $\cos a \neq 1, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{\sin ^{2}(a+y)}{\sin \phi}$

(b)

$\frac{\cos ^{2}(a+y)}{\sin a}$

(c)

$\sin ^{2}(a+y) \sin a$

(d)

None of these

218.

If $y=\sqrt{\sin x+y}, \text { then } \frac{d y}{d x}$ is equal to

(a)

$\frac{\cos x}{2 y-1}$

(b)

$\frac{\cos x}{1-2 y}$

(c)

$\frac{\sin x}{1-2 y}$

(d)

$\frac{\sin \psi}{2 y-1}$

219.

If 2x + 3y = sinx, then $\frac{d y}{d x}$ is equal to

(a)

$\frac{\cos x+2}{\ 3}$

(b)

$\frac{\cos x-2}{3}$

(c)

cosx + 2

(d)

None of the above

220.

If y + siny = cosx, then $\frac{d y}{d x}$ is equal to

(a)

$-\frac{\sin x}{1+\cos y}, y=(2 n+1) \pi$

(b)

$\frac{\sin x}{1+\cos y}, y \neq(2 n+1) \pi$

(c)

$\frac{\sin x}{1+\cos y}, y \neq(2 n+1) \pi$

(d)

None of the above

221.

Let $f(x)=\left\{\begin{array}{cl} (x-1) \sin \frac{1}{(x-1)} & \text { ,if } x \neq 1 \\ 0 & , \text { if } x=1 \end{array}\right.$ Then, which of the following is true?

(a)

tis differentiable at x = 1 but not at x = 0

(b)

t is neither differentiable at x = 0 nor at x = 1

(c)

f is differentiable at x = 0 and at x = 1

(d)

f is differentiable at x = 0 but not at x = 1

222.

If $y=\sqrt{3 x+2}+\frac{1}{\sqrt{2 x^{2}+4}}$ ,then $\frac{d y}{d x}$ is equal to

(a)

$\frac{3}{2 \sqrt{3 x+2}}-\frac{2 x}{\left(2 x^{2}+4\right)^{3 / 2}}$

(b)

$\frac{3}{2 \sqrt{3 x+2}}+\frac{2 x}{\left(2 x^{2}+4\right)^{3 / 2}}$

(c)

$\frac{3}{2 \sqrt{3 x+2}}+\frac{2}{\left(2 x^{2}+4\right)^{3 / 2}}$

(d)

None of the above

223.

The differential coefficient of sin (cos(x²) with respect to x is.

(a)

-2xsinx²cos(~os x²)

(b)

2xsin(x²)cos(x²)

(c)

2xsin(x²) cos(x²) cosx

(d)

None of the above

224.

If $f(x)=|\sin x|$ then

(a)

f is everywhere differentiable

(b)

f is everywhere continuous but not differentiable at $x=n \pi, n \in Z$

(c)

f is everywhere continuous but not differentiable at $x=(2 n+1) \frac{\pi}{2}, n \in Z$

(d)

None of he above

225.

The set of points, where the function f given by $f(x)=|2 x-1| \sin x$ is differentiable, is

(a)

(b)

$R-\left\{\frac{1}{2}\right\}$

(c)

$(0, \infty)$

(d)

None of these

226.

If $f(x)=2 x \text { and } g(x)=\frac{x^{2}}{2}+1$ then which of the following can be a discontinuous function?

(a)

f(x) + g (x)

(b)

f(x) - g(x)

(c)

f(x). g (x)

(d)

$\frac{g(x)}{f(x)}$

227.

If $f(x)=\left\{\begin{array}{cl} \frac{\sqrt{1+k x}-\sqrt{1-k x}}{x}, & \text { for }-1 \leq x<0 \\ 2 x^{2}+3 x-2, & \text { for } 0 \leq x \leq 1 \end{array}\right.$ is continuous at x = 0, then k is equal to

(a)

-4

(b)

-3

(c)

-2

(d)

-1

228.

The number of points at which the function $f(x)=\frac{1}{x-[x]}[\cdot]$ denotes the greatest integer function is not continuous is

(a)

(b)

(c)

(d)

None of these

229.

The function $f(x)=\left\{\begin{array}{cl} \frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \text { is } \\ 3, & \text { if } x=\frac{\pi}{2} \end{array}\right.$ continuous at $x=\frac{\pi}{2}$

(a)

-6

(b)

(c)

(d)

-5

230.

The function defined by g(x) = x - [x] is discontinuous at

(a)

all rational points

(b)

all irrational points

(c)

all integral points

(d)

None of the above

231.

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)

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

232.

The function $f(x)=\frac{4-x^{2}}{4 x-x^{3}}$ is

(a)

discontinuous at only one point

(b)

discontinuous at exactly two points

(c)

discontinuous at exactly three points

(d)

None of the above

233.

If $f(x)=\left\{\begin{array}{ll} \lambda\left(x^{2}-2 x\right), & \text { if } x \leq 0 \\ 4 x+1, & \text { if } x>0 \end{array}\right.$ then which one of the following is correct.

(a)

f(x) is continuous at x = 0 for any value of λ

(b)

f(x) is discontinuous at x = 0 for any value of λ

(c)

f(x) is discontinuous at x = 1for any value of

(d)

None of the above

234.

The point of discontinuity of the function
$f(x)=\left\{\begin{array}{ll} 2 x+3, & \text { if } x \leq 2 \\ 2 x-3, & \text { if } x>2 \end{array}\right.$ is

(a)

x = 0

(b)

x = 1

(c)

x = 2

(d)

None of these

235.

The function $f(x)=\left\{\begin{array}{ll} 1, & \text { if } x \neq 0 \\ 2, & \text { if } x=0 \end{array}\right.$ is not continuous at

(a)

x = 0

(b)

x = 1

(c)

x = -1

(d)

None of these

236.

The adjoint of the matrix $A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$ is

(a)

$\left[\begin{array}{ll} 4 & 2 \\ 3 & 1 \end{array}\right]$

(b)

$\left[\begin{array}{rr} -4 & 2 \\ 3 & -1 \end{array}\right]$

(c)

$\left[\begin{array}{rr} 4 & -2 \\ -3 & 1 \end{array}\right]$

(d)

$\left[\begin{array}{rr} 1 & -2 \\ -3 & 4 \end{array}\right]$

237.

If A and B are square matrices of same order,then

(a)

$|A B|=|A| \cdot|B|$

(b)

$|A B| \neq|A| \cdot|B|$

(c)

$|A B|=\frac{|A|}{|B|},|B| \neq 0$

(d)

$|A B|=\frac{|B|}{|A|},|A| \neq 0$

238.

Asquare matrix A is said to be non-singular, if

(a)

$|A|=0$

(b)

$|A| \neq 0$

(c)

$|A|=-1$

(d)

$|A|=1$

239.

If $f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}$ is equal to

(a)

(b)

-1

(c)

(d)

240.

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)

241.

For what value of k, the following system of linear equations will have infinite solutions?
$ x-y+z=3 $
$2 x+y-z=2 $
$-3 x-2 k y+6 z=3$

(a)

k ≠ 2

(b)

k = 0

(c)

k = 3

(d)

k = -1

242.

Given,2x - y + 2z = 2, x - 2y + Z = - 4 and x + y + λz= 4, then the value of Asuch that the given system of equation has no solution is

(a)

(b)

(c)

(d)

-3

243.

The simultaneous equations kx + 2y -z = 1, (k -1)y - 2z = 2, (k + 2)z = 3 have only one solution when

(a)

k = -2

(b)

k = -1

(c)

k = 0

(d)

k = 1

244.

For the system of equations 5x + 2y = 4; 7x +3y = 5 the values of x and yare respectively.

(a)

x = 2, y = -3

(b)

x = 2, y = 3

(c)

x = -2, y = -3

(d)

x = -2, y = 3

245.

If A is singular matrix and $(\operatorname{adj} A) B \neq O$ then

(a)

there is unique solution

(b)

solution does not exist

(c)

there are infinitely many solutions

(d)

None of the above

246.

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

247.

If $A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|$ then A^-I exists, if

(a)

$\lambda=2$

(b)

$\lambda \neq 2$

(c)

$\lambda \neq-2$

(d)

None of these

248.

If $A=\left[\begin{array}{cc} 2 & 3 \\ -4 & -6 \end{array}\right]$ then which of the following is true?

(a)

$A(\operatorname{adj} A) \neq|A| I$

(b)

$A(\operatorname{adj} A) \neq(\operatorname{adj} A) A$

(c)

$A(\operatorname{adj} A)=(\operatorname{adj} A) A=|A| I=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]$

(d)

None of the above

249.

Let A be the non-singular square matrix of order 3 x 3, then [adj A Iis equal to

(a)

|A|

(b)

IAI²

(c)

IA|³

(d)

3|A|

250.

If $\Delta=\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$ and A_ij is cofactor of a_ij, then value of Δ is given by

(a)

a₁₁A₃₁+a₁₂+A₃₂+a₁₃A₃₃

(b)

a₁₁A₁₁ + a₁₂A₂₁+ a₁₃A₃₁

(c)

a₂₁A₁₁ + a₂₂A₁₂ + a₂A₁₃

(d)

a₁₁A₁₁ + a₂₁A₂₁+ a₃₁A₃₁

251.

If M_u = - 40, M₁₂ = - 10 and M₁₃ = 35 of the determinant $\Delta=\left|\begin{array}{rrr} 1 & 3 & -2 \\ 4 & -5 & 6 \\ 3 & 5 & 2 \end{array}\right|$ then the value of $\Delta$ is

(a)

-80

(b)

(c)

(d)

100

252.

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

253.

If $\Delta=\left|\begin{array}{lll} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{array}\right|$ then the minor M₃₁ is

(a)

-c(a² - b²)

(b)

c(b²-a²)

(c)

c(a² + b²)

(d)

c(a²-b²)

254.

Minor of an element of a determinant of order $n(n \geq 2)$ is a determinant of order.

(a)

(b)

n-1

(c)

n-2

(d)

n+1

255.

The area of the triangle formed by 3 collinear points is

(a)

one

(b)

two

(c)

zero

(d)

four

256.

If area of a triangle is 35 sq. units with vertices (2, - 6), (5, 4) and (k, 4),then k is

(a)

(b)

-2

(c)

-12, -2

(d)

12, -2

257.

Area of the triangle whose vertices are (a, b + c), (b, c + a) and (c, a + b), is

(a)

2 sq units

(b)

3 sq unit

(c)

0 sq unit

(d)

None of the above

258.

The area of triangle with vertices (x₁ y_l), (x₂'y₂) and (x₃'y₃) is

(a)

$\Delta=\frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|$

(b)

$\Delta=\frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ y_{1} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|$

(c)

$\Delta=\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|$

(d)

None of these

259.

The determinant $\left|\begin{array}{rrr} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{array}\right|$ is

(a)

independent of θ only

(b)

independent of x only

(c)

independent of both ө and x

(d)

None of the above

260.

Iff $f(x)=\left|\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right|$ then

(a)

f(a) = 0

(b)

f(b) = 0

(c)

f(0) = 0

(d)

f(1) = 0

261.

Let $\Delta=\left|\begin{array}{lll} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right| \text { and } \Delta_{1}=\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right|$ then

(a)

$\Delta_{1}=-\Delta$

(b)

$\Delta \neq \Delta_{1}$

(c)

$\Delta^{2}-\Delta_{1}=0$

(d)

None of these

262.

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

263.

If A and B are symmetric matrices of same order, then (AB' - BA') is a

(a)

skew-symmetric matrix

(b)

null matrix

(c)

symmetric matrix

(d)

unit matrix

264.

For any two matrices A and B, we have

(a)

AB=BA

(b)

AB ≠BA

(c)

AB = 0

(d)

None of these

265.

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)

266.

If matrix $A=\left[a_{i j}\right]_{2 \times 2},\ where \ a_{i j}=\left\{\begin{array}{l}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.$ Then $A^{2}$ is equal to

(a)

(b)

(c)

(d)

None ofthese

267.

If $\left[\begin{array}{cc}2 x+y & 4 x \\ 5 x-7 & 4 x\end{array}\right]=\left[\begin{array}{cc}7 & 7 y-13 \\ y & x+6\end{array}\right]$, then

(a)

x = 3, y = 1

(b)

x = 2, y = 3

(c)

x = 2, y = 4

(d)

x = 3, y = 3

268.

If A and B are square matrices of the same order and AB = 3I,then A^-1 is equal to

(a)

(b)

$\frac{1}{3}B$

(c)

3B^-1

(d)

$\frac{1}{3}B^{-1}$

269.

Matrices A and B will be inverse of each other only if

(a)

AB = BA

(b)

AB = BA = 0

(c)

AB = 0,

(d)

AB = BA = I

270.

If X, A and B are matrices of the same order such that X = AB, then we apply elementary row transformations simultaneously on X and on the matrix

(a)

(b)

(c)

(d)

Both A and B

271.

On using elementary row operation $R_{1} \rightarrow R_{1}-3 R_{2}$ in the following matrix equation $\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right],$ we have

(a)

$\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1-7 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$

(b)

$\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{cc}-1 & -3 \\ 1 & 1\end{array}\right]$

(c)

$\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ 1 & -7\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$

(d)

$\left[\begin{array}{rr}4 & 2 \\ -5 & -7\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ -3 & -3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$

272.

On using elementary column operations $C_{2} \rightarrow C_{2}-2 C_{1}$ in the following matrix equation $\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]$, we have

(a)

$\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]$

(b)

$\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -0 & 2\end{array}\right]$

(c)

$\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -3 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -2 & 4\end{array}\right]$

(d)

$\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]$

273.

On usiig elementary column operations C₂➝ C₂ - 2C₁ in the following matrix equation$\left[\begin{array}{cc} 1 & -3 \\ 2 & 4 \end{array}\right]=\left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 3 & 1 \\ 2 & 4 \end{array}\right]$ ,we have

(a)

$\left[\begin{array}{cc} 1 & -5 \\ 0 & 4 \end{array}\right]=\left[\begin{array}{cc} 1 & -1 \\ -2 & 2 \end{array}\right]\left[\begin{array}{cc} 3 & -5 \\ 2 & 0 \end{array}\right]$

(b)

$\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -0 & 2\end{array}\right]$

(c)

$\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -3 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -2 & 4\end{array}\right]$

(d)

$\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]$

274.

The matrix $\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]$ is a

(a)

diagonal matrix

(b)

symmetric matrix

(c)

skew-symmetric matrix

(d)

scalar matrix

275.

$A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$, then if the value of $\alpha$ is

(a)

$\frac{\pi}{6}$

(b)

$\frac{\pi}{3}$

(c)

$\frac{3 \pi}{2}$

(d)

$\pi$

276.

The set of all 2x 2 matrices which is commutative with the matrix.$\left[\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right]$ with respect to matrix multiplication is

(a)

$\left[\begin{array}{ll} p & q \\ r & r \end{array}\right]$

(b)

$\left[\begin{array}{ll} p & q \\ q & r \end{array}\right]$

(c)

$\left[\begin{array}{cc} p-q & p \\ q & r \end{array}\right]$

(d)

$\left[\begin{array}{cc} p & q \\ q & p-q \end{array}\right]$

277.

If A and B are square matrices of the sameorder, then (A + B) (A - B) is equal to

(a)

A²-B²

(b)

A² - BA - AB - B²

(c)

A² - B² + BA - AB

(d)

A² - BA + B² + AB

278.

If $A=\left[\begin{array}{ccc} 2 & -1 & 3 \\ -4 & 5 & 1 \end{array}\right] \text { and } B=\left[\begin{array}{cc} 2 & 3 \\ 4 & -2 \\ 1 & 5 \end{array}\right] \text { , then }$

(a)

only AB is defined

(b)

only BA is defined

(c)

AB and BA both are defined

(d)

AB and BA both are not defined

279.

If the product of two matrices is a zero matrix, then

(a)

atleast one of the matrix is a zero matrix

(b)

both the matrices are zero matrices

(c)

it is not necessary that one of the matrices is a zero matrix

(d)

None of the above

280.

The product $\left[\begin{array}{rr} a & b \\ -b & a \end{array}\right]\left[\begin{array}{rr} a & -b \\ b & a \end{array}\right]$ is equal to

(a)

$\left[\begin{array}{cc}a^{2}+b^{2} & 0 \\ 0 & a^{2}+b^{2}\end{array}\right]$

(b)

$\left[\begin{array}{ll}(a+b)^{2} & 0 \\ (a+b)^{2} & 0\end{array}\right]$

(c)

$\left[\begin{array}{ll}a^{2}+b^{2} & 0 \\ a^{2}+b^{2} & 0\end{array}\right]$

(d)

$\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]$

281.

If A and B are two matrices of the order $3 \times m$ and $3 \times n$ respectively and m=n, then the order of the matrix $(5 A-2 B)$ is

(a)

m x 3

(b)

3 x 3

(c)

m x n

(d)

3 x n

282.

If $\left[\begin{array}{rr}1 & 2 \\ -2 & -b\end{array}\right]+\left[\begin{array}{ll}a & 4 \\ 3 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 0\end{array}\right]$, then $a^{2}+b^{2}$ is equal to

(a)

(b)

(c)

(d)

283.

1f If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{lll}1 & 3 & 2 \\ 4 & 3 & 1\end{array}\right], C=\left[\begin{array}{l}1 \\ 2\end{array}\right]$ and $D=\left[\begin{array}{lll}4 & 6 & 8 \\ 5 & 7 & 9\end{array}\right]$, then which of the following is defined?

(a)

A + B

(b)

B + C

(c)

C + D

(d)

B + D

284.

The matrix $P=\left[\begin{array}{lll} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{array}\right]$ is not A.

(a)

square matrix

(b)

diagonal matrix

(c)

unit matrix

(d)

None of these

285.

Total number of possible matrices of order 3 x 3 with each entry 2 or 0 is

(a)

(b)

(c)

(d)

512

286.

If a matrix has 8 elements, then which of the following will not be a possible order of the matrix?

(a)

1x 8

(b)

2 x 4

(c)

4x2

(d)

4 x 4

287.

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

288.

If $\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{2}{3}=\sin ^{-1} x,$ then x is equal to

(a)

(b)

$\frac{\sqrt{5}-4 \sqrt{2}}{9}$

(c)

$\frac{\sqrt{5}+4 \sqrt{2}}{9}$

(d)

$\frac{\pi}{2}$

289.

The greatest and least value of $$\left(\sin ^{-1} x\right)^{2} +\left(\cos ^{-1} x\right)^{2}$$ are respectively

(a)

$\frac{5 \pi^{2}}{4} and \frac{\pi^{2}}{8}$

(b)

$\frac{\pi}{2} and -\frac{\pi}{2}$

(c)

$\frac{\pi^{2}}{4} and -\frac{\pi^{2}}{4}$

(d)

$\frac{\pi^{2}}{4}$ and 0

290.

The value of $\tan ^{2}\left(\sec ^{-1} 2\right)+\cot ^{2}\left(\operatorname{cosec}^{-1} 3\right)$ is

(a)

(b)

(c)

(d)

291.

If $\alpha \leq 2 \sin ^{-1} x+\cos ^{-1} x \leq \beta$, then

(a)

$\alpha=-\frac{\pi}{2}, \beta=\frac{\pi}{2}$

(b)

$\alpha=0, \beta=\pi$

(c)

$\alpha=-\frac{\pi}{2}, \beta=\frac{3 \pi}{2}$

(d)

$\alpha=0, \beta=2 \pi$

292.

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

293.

The value of $\cos \left[\tan ^{-1}\left\{\sin \left(\cot ^{-1} x\right)\right\}\right]$ is

(a)

$\frac{1}{\sqrt{x^{2}+2}}$

(b)

$\sqrt{\frac{x^{2}+2}{x^{2}+1}}$

(c)

$\sqrt{\frac{x^{2}+1}{x^{2}+2}}$

(d)

$\frac{1}{\sqrt{x^{2}+1}}$

294.

If $0 \leq x<1$, then $\sin \left\{\tan ^{-1}\left(\frac{1-x^{2}}{2 x}\right)+\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right\}$ is equal to

(a)

(b)

-1

(c)

(d)

295.

The value of $\sin ^{-1}\left\{\cot \left(\sin ^{-1} \frac{\sqrt{2-\sqrt{3}}}{2}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sec ^{-1} \sqrt{2}\right)\right\}$ is

(a)

(b)

$\frac{\pi}{2}$

(c)

$\frac{\pi}{3}$

(d)

$\frac{\pi}{4}$

296.

If $\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2}$, then the value of $x^{100}+y^{100}+z^{100}-\frac{9}{x^{101}+y^{101}+z^{101}}$ is

(a)

(b)

(c)

(d)

297.

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'

298.

If $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4},$ then x is equal to

(a)

(b)

$-1, \frac{1}{10}$

(c)

$\frac{1}{6}$

(d)

None of these

299.

The value of $\tan \left(\cos ^{-1} \frac{3}{5}+\tan ^{-1} \frac{1}{4}\right)$is

(a)

$\frac{19}{8}$

(b)

$\frac{8}{19}$

(c)

$\frac{19}{12}$

(d)

$\frac{3}{4}$

300.

The value of $\sin \left(2 \tan ^{-1} \frac{2}{3}\right)-\cos \left(2 \tan ^{-1} \sqrt{3}\right)$ is

(a)

$\frac{26}{37}$

(b)

$\frac{37}{26}$

(c)

$\frac{12}{13}$

(d)

$\frac{13}{15}$

301.

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)

$\frac{2 \pi}{3}$

302.

The value of $\tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right]$ is

(a)

$\frac{\pi}{3}$

(b)

$\frac{2 \pi}{3}$

(c)

$\frac{-\pi}{3}$

(d)

$\frac{\pi}{6}$

303.

$\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)$ is equal to

(a)

1/2

(b)

1/3

(c)

1/4

(d)

304.

If $\sin ^{-1} x=y$, then

(a)

$0 \leq y \leq x$

(b)

$\frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

(c)

\(0

(d)

\(\frac{-\pi}{2}

305.

The inverse of cosine function is defined in th intervals

(a)

$\left[-\pi, 0]\right.$

(b)

$\left[\frac{-\pi}{2}, 0\right]$

(c)

$\left[0, \frac{\pi}{2}\right]$

(d)

$\left[\frac{\pi}{2}, \pi]\right.$

306.

Let $f: R \rightarrow R$ be the functions defined by $f(x)=x^{3}+5$. Then, $f^{-1}(x)$ is

(a)

$(x+5)^{1 / 3}$

(b)

$(x-5)^{1 / 3}$

(c)

$(5-x)^{1 / 3}$

(d)

5-x

307.

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be the bijective functions. Then, $(g o f)^{-1}$ is

(a)

$f^{-1} o g^{-1}$

(b)

fog

(c)

$g^{-1} o f^{-1}$

(d)

gof

308.

If $f: X \rightarrow Y$ is a function such that there exists a function $g: Y \rightarrow X$ such that gof $=I_{X}$ and $f o g=I_{\gamma},$ then f must be

(a)

one-one

(b)

onto

(c)

one-one and onto

(d)

None of these

309.

$Let f(x)=\left\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \text { and } g(x)=1+x-[x]\end{array}\right.$
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)

310.

Let $f(y)=\frac{a y}{y+1}, y \neq-1 .$Then, for what value of a is $f(f(y))=y ?$

(a)

(b)

-1

(c)

(d)

311.

Which of the following options is correct?

(a)

gof is one one $\Rightarrow$ g is one-one

(b)

gof is one-one $\Rightarrow$f is one-one

(c)

gof is onto $\Rightarrow$ is not onto

(d)

gof is onto $\Rightarrow$ is onto

312.

Let f, g and h be functions from R to R. Then,

(a)

$(f+g) oh =\operatorname{fog}+\mathrm{goh}$

(b)

$ (f+g) o h=f o h+g o h$

(c)

$(f \cdot g) oh =( foh )+( goh )$

(d)

$(f \cdot g) oh =(f o g) \cdot( goh )$

313.

If f: R $\rightarrow$ R and g: R $\rightarrow$ R are given by $f(x)=\cos x $ and $ g(x)=3 x^{2}$, then

(a)

$ \operatorname{gof}(x)=\cos ^{2} x$

(b)

$\operatorname{fog}(x)=\cos x^{2}$

(c)

$gof \neq fog$

(d)

$ f o g=g o f$

314.

The greatest integer function : R ➝R, given by [(x) = [x] is

(a)

one-one

(b)

onto

(c)

both one-one and onto

(d)

neither one-one nor onto

315.

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

(a)

720

(b)

120

(c)

(d)

None of these

316.

Let A = {1, 2, 3, ... , n} and B = {a, b}. Then the number of surjections from A into B is

(a)

ⁿP₂

(b)

2ⁿ-2

(c)

2ⁿ-1

(d)

None of these

317.

The number of all one-one functions from set A = {1, 2, 3} to itself is

(a)

(b)

(c)

(d)

318.

f : X⟶Y is onto, if and only if

(a)

range of f = Y

(b)

range of f ≠ Y

(c)

range of f < Y

(d)

range of f ≥Y

319.

If A = {x ∈ Z : 0 ≤ x ≤ 12}and R is the relation in A given by R = {(a, b): a = b}. Then, the set of all elements related to 1 is

(a)

{1, 2}

(b)

{2, 3}

(c)

{1}

(d)

{2}

320.

For the set A = {1, 2, 3}, define a relation R in the set A as follows
R = {(1, 1), (2,2), (3, 3), (1, 3)}
Then, the ordered pair to be added to R to make it the smallest equivalence relation is

(a)

(1, 3)

(b)

(3, 1)

(c)

(2, 1)

(d)

(1, 2)

321.

The relation R in the set of natural numbers N defined as R = {(x, y) : y = x + 5 and x < 4} is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

None of these

322.

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

323.

A relation f from C to R is defined by $x f y \Leftrightarrow|x|=y$.Then, the correct option is

(a)

(2+i)f3

(b)

3f(-3)

(c)

i f 1

(d)

(2 +3i) f 13

324.

Given a function y = f(x). Let Δx be the very small change in the value of x , then the corresponding change in the value of y that is Δy is approximately given by

(a)

$\frac{f(x)}{f'(x)}∆x$

(b)

f'(x).∆x

(c)

$\frac{dy}{dx}$.x

(d)

F'(x)

325.

Find the approximate value of f(10.01) where f(x) = 5x² + 6x + 3

(a)

564.06

(b)

564.01

(c)

563.00

(d)

563.01

326.

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

(a)

2.025

(b)

2.001

(c)

2.01

(d)

2.0025

327.

A stone is dropped into a quiet lake and waves move in circles at a speed of 2cm per second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area changing ?

(a)

Decreasing at the rate of 48π cm² / sec

(b)

Increasing at the rate of 24π cm² / sec

(c)

Increasing at the rate of 48π cm² / sec

(d)

Decreasing at the rate of 24π cm² / sec

328.

The total cost associated with the production of x units of a product is given by c(x) = 5x² + 14x + 6. Find marginal cost when 5 units are produced

(a)

Rs. 64

(b)

Rs. 70

(c)

Rs. 50

(d)

Rs. (10x + 14)

329.

The volume of cube is increasing at the constant rate of 3cm³/s. Find the rate of change of edge of the cube when its edge is 5 cm.

(a)

25 cm³/sec

(b)

25 cm/s

(c)

1/25 cm/s

(d)

1/25 cm³/s

330.

Find the approximate change in total surface area of a cube of side x metre caused by increase in side by 1%

(a)

12 m²

(b)

0.12x² m²

(c)

1.2x m²

(d)

12x m²

331.

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 5x²+ 22x + 35. Find the marginal revenue, when x = 7, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.

(a)

(b)

Rs 127

(c)

Rs 92

(d)

Rs 48

332.

The radius of air bubble is increasing at the rate of 0. 25 cm/s. At what rate the volume of the bubble is increasing when the radius is 1 cm.

(a)

4π cm³/s

(b)

22π cm³/s

(c)

2π cm³/s

(d)

π cm³/s

333.

A real function f is said to be continuous if it is continuous at every point in …… .

(a)

[-∞,∞]

(b)

The range of f

(c)

The domain of f

(d)

Any interval of real numbers

334.

The function f(x) = $\begin{cases} \frac { { e }^{ 1/x }-1 }{ { e }^{ 1/x }+1 } ,x\neq 0 \\ 0\quad \quad x=0 \end{cases}$

(a)

is continuous at x = 0

(b)

Continuous everywhere

(c)

Not continuous at x = 0 but can be made continuous

(d)

Not continuous at x = 0

335.

Function f(x) = log x + $\sqrt { 1-{ x }^{ 2 } } $ is continuous at

(a)

(0, 1)

(b)

(-1, 1)

(c)

(0, ∞)

(d)

(0, 1)

336.

What is the point of discontinuity for signum function?

(a)

x = 1

(b)

x = -1

(c)

x = 0

(d)

function is continuous on R

337.

Examine the continuity of function f(x) = (x - 1) (x - 2)

(a)

Discontinuous at x = 1, 2

(b)

Discontinuous at x = 1

(c)

Continuous everywhere

(d)

Discontinuous at x = 2

338.

Discuss the continuity of function f(x) = |x - 1| + |x + 1|

(a)

discontinuous at x = 1

(b)

discontinuous at x = ±1

(c)

continuous everywhere

(d)

discontinuous at x = -1

339.

Let $\begin{cases} 4x \ >2 \\ ax \ 0\le x\le 2 \\ b \ x<0 \end{cases}$ For what values of a and b, f is a continuous function.

(a)

a = 2, b = 0

(b)

a = 1, b = 0

(c)

a = 0, b = 2

(d)

a = 0, b = 0

340.

Examine the continuity of the function $f(x)=\frac { { x }^{ 2 }-4 }{ x-2 } $

(a)

Discontinuous at x = -2

(b)

Discontinuous at x = 2

(c)

Continuous everywhere

(d)

Discontinuous at x = 4

341.

The Function $f(x)=\frac { 4-{ x }^{ 2 } }{ 4x-{ x }^{ 3 } } $

(a)

Discontinuous exactly at two points

(b)

Discontinuous at every points

(c)

Discontinuous at only one point

(d)

Discontinuous exactly at three points

342.

Which of the following functions are not continuous.

(a)

[x]

(b)

|x|

(c)

e^x

(d)

$\frac1x$, x ≠ 0

343.

To construct a 2 x 3 matrix [a_ij], such that aij = – $\frac { i-3j }{ 4 } $ The values that i and j can take are …….

(a)

i = 1, 2, 3 ; j = 1, 2, 3

(b)

i = 1, 2 ; j = 1, 2, 3

(c)

i = 1, 2 ; j = 1, 2

(d)

i = 1, 2, 3 ; j = 1, 2

344.

If, ${ a }_{ ij }=\frac { 1 }{ 2 } |i-3j|$ the value of a₂₂ is

(a)

(b)

-2

(c)

(d)

345.

What is the element in the 2^nd row and 1^st column of a 2 x 2 Matrix A= [ a_ij], such that a = (i + 3) (j – 1)

(a)

(b)

(c)

-5

(d)

346.

$\left[ \begin{matrix} 2 & 3 & 1 \\ 1 & 2 & 4 \end{matrix}\begin{matrix} 5 & 1 \\ 2 & 2 \end{matrix} \right] $ is a matrix of order

(a)

2 x 5

(b)

2 x 2

(c)

5 x 2

(d)

5 x 5

347.

$\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$ is example of

(a)

an identity matrix

(b)

a zero matrix.

(c)

a Scalar m

(d)

diagonal matrix.

348.

Consider the following information regarding the number of men and women workers in three BPOs I, II and III

	Men	Women
I	35	20
II	20	23
III	25	25

What does the entry in the second row and first column represent if the information is represented as a 3 x 2 matrix?

(a)

The number of Men in BPO II

(b)

The number of Women in BPO II

(c)

The number of Women in BPO I

(d)

The number of Men in BPO I

349.

[5] is a scalar matrix of order

(a)

(b)

(c)

(d)

350.

For what real value of y will matrix A be equal to matrix B, where
$A=\begin{bmatrix} 3x-4 & 5y \\ 8 & { y }^{ 2 }-4y \end{bmatrix};B=\begin{bmatrix} x+1 & 6{ y }^{ 2 }+1 \\ 8 & -3 \end{bmatrix}$

(a)

1, 3

(b)

No real value

(c)

1/3, 1/2

(d)

2 and 3

351.

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

(a)

(b)

(c)

(d)

352.

$\begin{bmatrix} x+10 & { y }^{ 2 }+2y \\ 0 & -4 \end{bmatrix} =\begin{bmatrix} 3x+4 & 3 \\ 0 & { y }^{ 2 }-5y \end{bmatrix}$ Then the value of x is ________

(a)

(b)

(c)

(d)

353.

Value of $sin\left( 2{ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \right) $

(a)

$\sqrt3$/2

(b)

-1

(c)

-$\sqrt3$/2

(d)

-1/2

354.

Domain of function ${ cos }^{ -1 }\left( \frac { 2x+1 }{ 3 } \right) $ is

(a)

(-2,0)

(b)

[-2,0]

(c)

[-2,1]

(d)

(-2,1)

355.

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)

356.

Value of ${ sin }^{ -1 }\left( sin\frac { 7\pi }{ 4 } \right) $ in the range of sin ^-1x is

(a)

3π/4

(b)

π/4

(c)

7π/4

(d)

-π/4

357.

The principal value of tan^-1 1 is given by

(a)

π/2

(b)

π/3

(c)

π/6

(d)

π/4

358.

The value of $cos\left\{ \frac { \pi }{ 3 } \left( { sin }^{ -1 }\left( \frac { -1 }{ 2 } \right) \right) \right\} $ is given by

(a)

-1

(b)

(c)

(d)

1/3

359.

Value of ${ cot }^{ -1 }\left( sin\left( -\frac { \pi }{ 2 } \right) \right) $

(a)

$\frac { 3\pi }{ 4 } $

(b)

$-\frac { \pi }{ 4 } $

(c)

-1

(d)

$\frac { \pi }{ 4 } $

360.

Evaluate $cosec\left( { cosec }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } \right) +\frac { \pi }{ 6 } \right) $

(a)

π/3

(b)

(c)

(d)

not defined

361.

Identify the graph above

(a)

y = sin^-1x

(b)

y = cos^-1x

(c)

y = sin x

(d)

y = cos x

362.

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 } $

363.

Let R be an equivalence relation on Z, the set of integers. R = { (a,b): a,b ∈ Z and a – b is a multiple of 3 } The Equivalence class of [1] is

(a)

{..-7,-4,2,5,8,.}

(b)

{.-4,-1,2,5,8,.}

(c)

{.-4,-1,2,5,8,.}

(d)

{…..-5,-2,1,4,7,.}

364.

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. Then, R is

(a)

Symmetric only

(b)

An equivalence relation

(c)

Reflexive and symmetric only

(d)

Reflexive and transitive only

365.

Let R be a relation on N (set of natural numbers) such that (m, n) R (p, q)mq(n + p) = np(m + q). Then, R is

(a)

An Equivalence Relation

(b)

Only Reflexive

(c)

Symmetric and reflexive

(d)

Only Transitive

366.

Let R be a relation on a finite set A having n elements. Then, the number of relations on A is

(a)

n x n

(b)

2ⁿ

(c)

n²

(d)

2^nxn

367.

If R be a relation “less than” from set A = {1, 2, 3, 4} to B = {1, 3, 5}, i.e. (a, b) ∈ R if a < b, if (b,a) ∈ R^-1elements in R^-1 are

(a)

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

(b)

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

(c)

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

(d)

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

368.

Let R be a relation on N, set of natural numbers such that m R n ⇔ m divides n. Then R is

(a)

Reflexive and symmetric

(b)

Neither reflexive nor transitive

(c)

Reflexive and transitive

(d)

Symmetric and transitive

369.

Let A = {1,2,3,4} and B = {x,y,z}. Then R = {(1,x) , ( 2,z), (1,y), (3,x)} is

(a)

relation from B to A

(b)

Is not a relation

(c)

relation from A to B

(d)

relation from B to B

370.

Let C = {(a, b): a² + b² = 1; a, b ∈ R} a relation on R, set of real numbers. Then C is

(a)

Equivalence relation

(b)

Reflexive

(c)

Transitive

(d)

Symmetric

371.

Let R be a relation on set A of triangles in a plane. R = { (T₁ , T₂) : T₁, T₂element of A and T₁ is congruent to T₂} Then the relation R is ______

(a)

Equivalence relation

(b)

Transitive

(c)

Symmetric

(d)

Reflexive

372.

Let A = {1,2,3,4,5,6,7}. P={1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation

(a)

{4,5,6}

(b)

{0}

(c)

{1,2,3,4,5,6,7}

(d)

{ }

373.

Let R be the relation on the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3,3), (3,2)}. then R is

(a)

R is reflexive and symmetric but not transitive.

(b)

R is symmetric and transitive but not reflexive.

(c)

R is an equivalence relation.

(d)

R is reflexive and transitive but not symmetric

374.

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)}

375.

In the set N x N the relation R is defined by (a, b) R (c, d) ⇔ ad = bc. Then R is

(a)

symmetric and transitive but not reflexive

(b)

reflexive and transitive but not symmetric

(c)

Equivalence relation

(d)

Partial order relation

376.

If A = {1,3,5,7} and define a relation, such that R = { (a,b) a,b ∈ A : |a+b| = 8}. Then how many elements are there in the relation R

(a)

(b)

(c)

(d)

377.

If A = {1,3,5,7} and we define a relation R = {(a,b), a,b ∈ A:|a - b| = 8} Then the number of elements in the relation R is

(a)

(b)

(c)

(d)

378.

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

379.

For real number x and y, we write xRy ⇔ x - y + $\sqrt2$ irrational number. Then the relation R is

(a)

Reflexive

(b)

Symmetric

(c)

Transitive

(d)

Equivalence

380.

Let a relation T on the set R of real numbers be T = { (a,b) : 1 + ab < 0, a,∈R}. Then from among the ordered pairs (1,1) (1,2)(1,-2)(2,2), the only pair that belongs to T is________.

(a)

(2,2)

(b)

(1,1)

(c)

(1,-2)

(d)

(1,2)

381.

Let R = { (P,Q) : OP = OQ , O being the origin} be an equivalence relation on A . The equivalence class [( 1,2)] is

(a)

{(x, y): x² + y²= 5}

(b)

{(x, y): x² = y²}

(c)

{(x, y): x² + y²= 1}

(d)

{(x, y): x² + y²= 4}

382.

The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

(a)

$\left( 4,\pm \frac { 8 }{ 3 } \right) $

(b)

$4,\frac { -8 }{ 3 } $

(c)

$\left( 4,\pm \frac { 3 }{ 8 } \right) $

(d)

$\left( 4,\pm \frac { 3 }{ 8 } \right) $

383.

The normal to the curve x² = 4y passing (1,2) is

(a)

x + y = 3

(b)

x – y = 3

(c)

x + y = 1

(d)

x – y = 1

384.

The normal at the point (1,1) on the curve 2y + x² = 3 is

(a)

x + y = 0

(b)

x – y = 0

(c)

x + y +1 = 0

(d)

x – y = 1

385.

The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is

(a)

(b)

(c)

(d)

$\frac12$

386.

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)

$\frac{-6}{7}$

387.

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

(a)

0.06 x³ m³

(b)

0.6 x³ m³

(c)

0.09 x³m³

(d)

0.9 x³ m³

388.

If f(x) = 3x² + 15x + 5, then the approximate value of f (3.02) is

(a)

47.66

(b)

57.66

(c)

67.66

(d)

77.66

389.

The line y = x + 1 is a tangent to the curve y² = 4x at the point

(a)

(1, 2)

(b)

(2, 1)

(c)

(1, – 2)

(d)

(– 1, 2)

390.

The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

(a)

(b)

$\frac13$

(c)

-3

(d)

-$\frac13$

391.

The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is

(a)

116

(b)

(c)

(d)

126

392.

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

393.

${ tan }^{ -1 }\left( \frac { x }{ y } \right)- { tan }^{ -1 }\frac { x-y }{ x+y } $ is equal to

(a)

$\frac { \pi }{ 2 } $

(b)

$\frac { \pi }{ 3 } $

(c)

$\frac { \pi }{ 4} $

(d)

$\frac { 3\pi }{ 4 } $

394.

Number of binary operations on the set {a, b} are

(a)

(b)

(c)

(d)

395.

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?

396.

Let f : R - $\left\{ \frac { 4 }{ 3 } \right\} $ ⟶ R be a function defined as $f(x)=\frac { 4x }{ 3x+4 } $. The inverse of f is the map g : Range f ⟶ R - $\left\{ \frac { 4 }{ 3 } \right\} $ 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) = $\frac{3y}{4-3y}$

397.

If f : R ⟶ R be given by f (x) = (3 − x³ )^$\frac13$ , then fof (x) is

(a)

x^$\frac13$

(b)

x³

(c)

(d)

(3 – x³).

398.

The absolute maximum value of y = x³ – 3x + 2 in 0 ≤ x ≤ 2 is

(a)

(b)

(c)

(d)

399.

The angle between the curve y² = x and x² = y at (1, 1) is

(a)

60°

(b)

tan^-1$\frac43$

(c)

cot^-1$\frac43$

(d)

90°

400.

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

(a)

(0, 1)

(b)

(2, 0)

(c)

($-\frac12$, 0)

(d)

(-2, 0)

401.

If the curves ay + x² = 7 and x³ = y cut orthogonally at (1,1), then the value of a is

(a)

(b)

(c)

-6

(d)

402.

The curves y = ae^-x and y = be^x are orthogonal if

(a)

a = b

(b)

a = -b

(c)

ab = -1

(d)

ab = 1

403.

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)

404.

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

(a)

x = 0

(b)

y = 0

(c)

x + y = 0

(d)

x – y = 0

405.

The point(s) on the curve y = x², at which y-coordinate is changing six times as fast as x-coordinate is/are

(a)

(2, 4)

(b)

(3, 9)

(c)

(3, 9), (9, 3)

(d)

(6, 2)

406.

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)

$\sqrt3$ cm²/s

(c)

10 $\sqrt3$ cm²/s

(d)

$\frac{10}{3}$cm²/s

407.

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

(a)

25y

(b)

(c)

-25y

(d)

10y

408.

The derivative of sin x with respect to log x is

(a)

cos x

(b)

x cos x

(c)

$\frac{cosx \ x}{log \ x}$

(d)

$\frac{1}{x} cos \ x$

409.

If y = x^x-∞, then x(l -y log x)$\frac { dy }{ dx } $ is equal to

(a)

x²

(b)

y²

(c)

xy²

(d)

x²y

410.

If y = tan^-1 $\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) $, then $\frac { dy }{ dx } $ is equal to

(a)

$\frac { 1 }{ 1+{ x }^{ 4 } } $

(b)

$\frac { -2x }{ 1+{ x }^{ 4 } } $

(c)

$\frac { -1 }{ 1+{ x }^{ 4 } } $

(d)

$\frac { { x }^{ 2 } }{ 1+{ x }^{ 4 } } $

411.

If f(x) = e^x and g(x) = log_e x, then (gof)’ (x) is

(a)

(b)

(c)

(d)

1 + e

412.

If f(x) = log^_x2 (log x), then f(e) is

(a)

(b)

(c)

$\frac1e$

(d)

$\frac{1}{2e}$

413.

If y = sin^-1 $\left( \frac { 3x }{ 2 } -\frac { { x }^{ 3 } }{ 2 } \right) $, then $\frac { dy }{ dx } $ is

(a)

$\frac { 3 }{ \sqrt { 4-{ x }^{ 2 } } } $

(b)

$\frac { -3 }{ \sqrt { 4-{ x }^{ 2 } } } $

(c)

$\frac { 1 }{ \sqrt { 4-{ x }^{ 2 } } } $

(d)

$\frac {- 1 }{ \sqrt { 4-{ x }^{ 2 } } } $

414.

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°

415.

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

(a)

(b)

(c)

(d)

416.

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 = $\frac32$

417.

A function f 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

418.

If $f(x)=\frac { sin({ e }^{ x-2 }-1) }{ log(x-1) } $, x ≠ 2 and f(x) = k for x = 2, then value of k for which f is continuous is

(a)

-2

(b)

-1

(c)

(d)

419.

$\lim _{ x\rightarrow 0 }{ \frac { \sqrt { \frac { 1 }{ 2 } (1-cosx) } }{ x } } $ is equal to

(a)

(b)

-1

(c)

(d)

none of this

420.

Given functions f(x) = $\frac { { x }^{ 2 }-4 }{ x-2 } $ and g(x) = x + 2, x <= R. Then which of the following is

(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

421.

A and B are invertible matrices of the same order such that |(AB)^-1| = 8, If |A| = 2, then |B| is

(a)

(b)

(c)

(d)

$\frac{1}{16}$

422.

Let x, yeR, then the determinant $\triangle =$ $\left| \begin{matrix} cosx & -sinx & 1 \\ sinx & cosx & 1 \\ cos(x+y) & -sin(x+y) & 0 \end{matrix} \right| $, lies in the interval

(a)

$[-\sqrt { 2 } ,\sqrt { 2 } ]$

(b)

[-1, 1]

(c)

$[-\sqrt { 2 } ,1]$

(d)

$[-1,\sqrt { 2 } ]$

423.

Let f(x) = $\left| \begin{matrix} cos \ x & 2 \ sin \ x & sin \ x \\ x & x & x \\ 1 & 2x & x \end{matrix} \right| $, then $\lim _{ x\rightarrow 0 }{ \frac { f(x) }{ { x }^{ 2 } } } $ is equal to

(a)

(b)

-1

(c)

(d)

424.

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|

425.

Let Δ = $\left| \begin{matrix} { Ax }^{ 2 } & x^{ 3 } & 1 \\ { By }^{ 2 } & { y }^{ 3 } & 1 \\ { Cz }^{ 2 } & { z }^{ 3 } & 1 \end{matrix} \right| $ and ${ \triangle }_{ 1 }=\left| \begin{matrix} Ax & By & Cz \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ yz & zx & xy \end{matrix} \right| $, then

(a)

Δ + Δ₁ = 0

(b)

Δ ≠ Δ₁

(c)

Δ = xΔ₁

(d)

Δ - Δ₁ = 0

426.

The value $\left| \begin{matrix} 6 & 0 & -1 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{matrix} \right| $ is

(a)

-7

(b)

(c)

(d)

427.

If $\begin{vmatrix} 2x & -1 \\ 4 & 2 \end{vmatrix}=\begin{vmatrix} 3 & 0 \\ 2 & 1 \end{vmatrix}$ then x is

(a)

(b)

$\frac { 2 }{ 3 } $

(c)

$\frac { 3 }{ 2 } $

(d)

$-\frac { 1 }{ 4 } $

428.

If A = $\begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}$ and A = A’ then

(a)

x = 0, y = 5

(b)

x = y

(c)

x + y = 5

(d)

x – y = 5

429.

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

430.

If A = $\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$

(a)

$\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$

(b)

$\begin{bmatrix} 0 & 4 \\ 4 & 0 \end{bmatrix}$

(c)

$\begin{bmatrix} 4 & 0 \\ 4 & 0 \end{bmatrix}$

(d)

$\begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}$

431.

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

432.

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

(a)

(b)

(c)

(d)

433.

Total number of possible matrices of order 2 × 3 with each entry 1 or 0 is

(a)

(b)

(c)

(d)

434.

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

(a)

$\begin{bmatrix} 0 & 3 \\ 0 & -1 \end{bmatrix}$

(b)

$\begin{bmatrix} -1 & 0 \\ 3 & 0 \end{bmatrix}$

(c)

$\begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix}$

(d)

$\begin{bmatrix} 3 & -1 \\ 0 & 0 \end{bmatrix}$

435.

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

(a)

$\frac15$

(b)

$\frac25$

(c)

$\frac95$

(d)

$\frac{16}{5}$

436.

If a matrix has 6 elements, then number of possible orders of the matrix can be

(a)

(b)

(c)

(d)

437.

The domain of y = cos^-1(x² – 4) is

(a)

[3, 5]

(b)

[0, π]

(c)

[-$\sqrt5$ ,-$\sqrt3$] ∩ [-$\sqrt5$,$\sqrt3$]

(d)

[-$\sqrt5$ ,-$\sqrt3$] ∪ [-$\sqrt5$,$\sqrt3$]

438.

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

(a)

(b)

(c)

(d)

439.

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

(a)

(b)

$\frac { 1 }{ \sqrt { 2 } } $

(c)

(d)

$\frac { 1 }{ 2 } $

440.

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

(a)

(b)

(c)

(d)

441.

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

(a)

$\pi$

(b)

$\frac{\pi}{2}$

(c)

$\frac{3\pi}{2}$

(d)

≥-ㅠ

442.

The domain of the function y = sin^-1(x²) is

(a)

[0, 1]

(b)

(0, 1)

(c)

[-1, 1]

(d)

443.

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

(a)

$\frac{2\pi}{9}$

(b)

-$\frac{2\pi}{9}$

(c)

$\frac{34\pi}{9}$

(d)

$\frac{\pi}{9}$

444.

sec{tan^-1 (-$\frac y3$)} is equal to

(a)

$\frac { \sqrt { 9+{ y }^{ 2 } } }{ 9 } $

(b)

$\frac { \sqrt { 9+{ y }^{ 2 } } }{ 3 } $

(c)

$\frac { 3 }{ \sqrt { 9+{ y }^{ 2 } } } $

(d)

$\frac { 9 }{ \sqrt { 9+{ y }^{ 2 } } } $

445.

tan^-1{sin (-$\frac{\pi}{2}$)} is equal to

(a)

-1

(b)

(c)

$\frac{\pi}{2}$

(d)

$-\frac{\pi}{4}$

446.

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

(a)

$\frac{\pi}{3}$

(b)

-$\frac{\pi}{3}$

(c)

$\frac{5\pi}{3}$

(d)

$-\frac{\pi}{6}$

447.

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

448.

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)

(c)

(d)

449.

A relation S in the set of real numbers is defined as xSy ⇒ x – y+ $\sqrt3$ is an irrational number, then relation S is

(a)

reflexive

(b)

reflexive and symmetric

(c)

transitive

(d)

symmetric and transitive

450.

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)}

451.

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

452.

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

453.

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

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Answers Key:

Multiple Choice Question

453 x 1 = 453

(d)
neither maximum nor minimum

(b)
(2, ∞)

(d)
(-∞, 2) U (2, ∞)

(b)
no value of b exists

(b)
strictly decreasing in (-2,3)

10.

11.

(b)
(-2, -1)

12.

(a)
25 y

13.

(d)
$-e^x \tan e^x$

14.

(a)
-y

15.

16.

(a)
2

17.

(a)
$\frac{-3 \sqrt{3} b}{a^2}$

18.

19.

(a)
$x y_1$

20.

(a)
1

21.

22.

23.

(a)
$-\sec ^2\left(\frac{\pi}{4}-x\right)$

24.

(b)
5

25.

(d)
0

26.

(a)
continuous and differentiable at x = 0.

27.

28.

(d)
-1

29.

(d)
$-\frac{y}{x}$

30.

(b)
-1

31.

(a)
$x \in R$

32.

33.

(d)
$\frac{11}{4}$

34.

(b)
x = 1.5

35.

36.

37.

(d)
$B^{-1}=\frac{1}{6} A$

38.

(b)
$\left[\begin{array}{cc}4 & -2 \\ 2 & 6\end{array}\right]$

39.

40.

(d)
16

41.

(d)
$(A+B)^{-1}=B^{-1}+A^{-1}$

42.

(d)
2⁸

43.

44.

(b)
2

45.

(d)
8 or -8

46.

(a)
12

47.

(d)
4

48.

(d)
$R-\{-10\}$

49.

(d)
$2 \sqrt{2}$

50.

(d)
-7000

51.

52.

(b)
-7

53.

(d)
$|A| \in[2,4]$

54.

55.

56.

(d)
$\sqrt{3},-\sqrt{3}$

57.

(b)
0

58.

(b)
土3

59.

(b)
$\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|= \pm 2 A$

60.

(d)
R - {4}

61.

(d)
1

62.

(d)
4

63.

(a)
47

64.

(a)
- 50

65.

(a)
[28]

66.

(b)
3 x 5

67.

68.

(d)
I

69.

(b)
-6,-4,-9

70.

(b)
4

71.

(a)
8

72.

(d)
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] $

73.

74.

(b)
AB =-BA

75.

(b)
$\left[\begin{array}{cc}-6 & -8 \\ -10 & -4\end{array}\right]$

76.

(b)
x = 2, y = 1

77.

(a)
$\frac{\pi}{4}-\frac{x}{2}$

78.

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

79.

80.

(a)
1

81.

(d)
-1

82.

83.

(d)
R - {-10}

84.

(b)
injective function

85.

(a)
$\frac{1}{2}$

86.

(d)
- $\frac{y}{x}$

87.

88.

(a)
-50

89.

(d)
both one-one and onto

90.

(a)
{1, 5, 9}

91.

(b)
(1, 2)

92.

(b)
4

93.

(b)
(6,8) ∈ R

94.

(a)
- 60 units/sec

95.

(a)
2

96.

97.

(a)
-1

98.

(b)
$\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$

99.

(a)
-1

100.

(d)
4

101.

(d)
2$\sqrt{2}$

102.

(a)
0

103.

(d)
neither injective nor surjective.

104.

(a)
1000

105.

Delhi

106.

107.

(b)
-1

108.

109.

(a)
|A|³

110.

(d)
$(A+B)^{-1}=A^{-1}+B^{-1}$

111.

(d)
1

112.

(d)
none of these

113.

(d)
n x m

114.

115.

116.

117.

118.

119.

(b)
$p>\frac{1}{2}$

120.

121.

122.

(a)
cos x

123.

(d)
$(0,2)$

124.

(a)
$x+2 y=1$

125.

126.

(b)
2.926

127.

128.

(b)
k² = 512

129.

(a)
$\frac{27 \pi}{8}(2 x+1)^{2}$

130.

(b)
16

131.

(d)
not applicable

132.

133.

(d)
$\frac{-1}{2 a t^{3}}$

134.

(b)
continuous at x = 2

135.

(a)
sin x

136.

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

137.

138.

(d)
none of these

139.

(d)
$\frac{5\pi}{6}$

140.

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

141.

142.

(d)
many-one, into function

143.

144.

145.

(d)
{(8, 1), (5, 2), (2, 3)}

146.

(d)
$\left(\frac{1+x}{2}\right)^{\frac{1}{3}}$

147.

148.

(d)
2n² relations

149.

(a)
not onto

150.

(d)
associative but not commutative

151.

(a)
2

152.

(b)
minimum

153.

(b)
0

154.

155.

(a)
0.05 radians

156.

(a)
$t y=x+a t^{2}=\text { ind } y=-t x+2 a t+a t^{3}$

157.

(a)
$c^{-4 / 3}$

158.

159.

(d)
is an i ucreasing function

160.

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

161.

(a)
0 units/s

162.

(d)
h=r

163.

(b)
$x=\frac{1}{e}$

164.

(b)
12

165.

166.

167.

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,$\Delta A=\frac{d A}{d \theta} \Delta \theta$
$\begin{array}{ll} \Rightarrow & \Delta A=450 \times \frac{\pi}{180} \quad\left[\because \Delta \theta=1^{\circ}=\frac{\pi}{180} \text { radians }\right] \\ \Rightarrow & \Delta A=2.5 \pi \mathrm{cm}^{2} \end{array}$

168.

Given, circumference of a circle $S=2 \pi r=56$
$\Rightarrow r=\frac{28}{\pi}$
$\therefore \text { Error } \delta S=2 \pi \delta r=0.02$
$\Rightarrow \delta r=\frac{0.02}{2 \pi}$
Let area of circle, $A=\pi r^{2}$
$ \therefore \text { Percentage error in } A=\frac{\delta A}{A} \times 100 $
$=2 \times \frac{\delta r}{r} \times 100 $
$=2 \times \frac{0.02 \times \pi}{2 \pi \times 28} \times 100=\frac{1}{14} $

169.

Let r be the radius of the sphere and $\Delta r$ be the error in measuring radius
Then, $r=7 \mathrm{~m} \text { and } \Delta r=0.02 \mathrm{~m}$
Now, volume of a sphere is given by $V=\frac{4}{3} \pi r^{3}$
On differentiate W.r.t. r, we get $\frac{d V}{d r}=\left(\frac{4}{3} \pi\right)\left(3 r^{2}\right)=4 \pi r^{2}$
$ \therefore \Delta V =\left(\frac{d V}{d r}\right) \Delta r=\left(4 \pi r^{2}\right) \Delta r $
$=4 \pi \times 7^{2} \times 0.02=3.92 \pi \mathrm{m}^{3} $
Hence, the approximate error in calculating the volume is $3.92 \pi \mathrm{m}^{3}$.

170.

Let $f(x)=\sqrt{x}$
Using $f(x+\Delta x) \approx f(x)+\Delta x \cdot f^{\prime}(x)$ ,taking x = 0.09 and $\Delta x=-0.008$
We get,$f(0.09-0.008)=f(0.09)+(-0.008) f^{\prime}(0.09)$
$ \Rightarrow \sqrt{0.082} =\sqrt{0.09}-0.008 \cdot\left(\frac{1}{2 \sqrt{0.09}}\right) $
$=0.3-\frac{0.008}{0.6}=0.3-0.0133=0.2867 $

171.

We have, $y=x^{4}-10 \Rightarrow d y / d x=4 x^{3}$
and $\Delta x=2.00-1.99=0.01$
$ \therefore \quad \Delta y =\frac{d y}{d x} \times \Delta x=4 x^{3} \times \Delta x \\ =4 \times 2^{3} \times 0.01=32 \times 0.01=0.32 $
So, the approximate change in y is 0.32.

172.

The given equation of curve is
$\begin{aligned} &y=x^{3}-12 x+18\\ &\therefore \quad \frac{d y}{d x}=3 x^{2}-12\\ &\text { [on differentiating w.r.t. } x] \end{aligned}$
So, the slope of line parallel to the X-axis.
$ therefore \quad\left(\frac{d y}{d x}\right)=0\\\Rightarrow \quad 3 x^{2}-12=0\\ \Rightarrow \quad x^{2}=\frac{12}{3}=4\\ \therefore\\ x=\pm 2\\ \text { For } x=2, y=2^{3}-12 \times 2+18=2\\ \text { and for } x=-2, y=(-2)^{3}-12(-2)+18=34\\ \text { So, the points are }(2,2) \text { and }(-2,34) $

173.

We have, $a y+x^{2}=7 \text { and } x^{3}=y$
On differentiating w.r.t. X in both equations, we get
$ \quad a \cdot \frac{d y}{d x}+2 x=0 \quad \text { and } 3 x^{2}=\frac{d y}{d x} \\ \Rightarrow \quad \frac{d y}{d x}=-\frac{2 x}{a} \text { and } \frac{d y}{d x}=3 x^{2} \\ \Rightarrow \quad\left(\frac{d y}{d x}\right)_{(1,1)}=\frac{-2}{a}=m_{1} \\ \text { and }\left(\frac{d y}{d x}\right)_{(1,1)}=3 \cdot 1=3=m_{2} $
Since, the curves cut orthogonally at (1, 1).
$\begin{array}{ll} \therefore & m_{1} \cdot m_{2}=-1 \\ \Rightarrow & \left(\frac{-2}{a}\right)+3=-1 \\ \therefore & a=6 \end{array}$

174.

We have,$y^{2}=12 x \Rightarrow 2 y \frac{d y}{d x}=12$
$\Rightarrow \quad \frac{d y}{d x}=\frac{6}{y}$
Let x + y = K be normal to y² = 12x at point P(x₁, y₁) then
$\begin{array}{l} \left(\frac{-1}{d y / d x}\right)_{\text {at } p}=(\text { Slope of the line } x+y=K) \\ \Rightarrow \quad-\frac{y_{1}}{6}=-1 \Rightarrow y_{1}=6 \end{array}$
Since, (x₁ y_I) lies on l = 12x, therefore
$ y_{1}^{2}=12 x_{1} \Rightarrow 12 x_{1}=36 \\ \Rightarrow x_{1}=3 $
Also P( x₁ y₁) lies on x + y = K, therefore
$x_{1}+y_{1}=K \Rightarrow K=9$

175.

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

176.

We have,$y=x^{1 / 5}$
$\begin{array}{l} \Rightarrow \quad \frac{d y}{d x}=\frac{1}{5} x^{\frac{1}{5}-1}=\frac{1}{5} x^{-4 / 5} \\ \therefore \quad\left(\frac{d y}{d x}\right)_{(0,0)}=\frac{1}{5} \times(0)^{-4 / 5}=\infty \end{array}$
S6, the curve Y = x^1/5 has a vertical tangent at (0, 0), which is parallel to Y-axis.

177.

In the interval $\left(0, \frac{\pi}{2}\right), f(x)=\cos x$
$\Rightarrow \quad f^{\prime}(x)=-\sin x$
which gives $f^{\prime}(x)<0 \text { in }\left(0, \frac{\pi}{2}\right)$
Hence, f(x) = cos x is decreasing in $\left(0, \frac{\pi}{2}\right)$

178.

We have,
$ f(x)=4 \sin ^{3} x-6 \sin ^{2} x+12 \sin x+100 \\ \therefore f^{\prime}(x) =12 \sin ^{2} x \cdot \cos x-12 \sin x \cdot \cos x+12 \cos x \\ =12\left[\sin ^{2} x \cdot \cos x-\sin x \cdot \cos x+\cos x\right] \\ =12 \cos x\left[\sin ^{2} x-\sin x+1\right] \\ \Rightarrow f^{\prime}(x) =12 \cos x\left[\sin ^{2} x+(1-\sin x)\right] \\ \because \quad 1-\sin x \geq 0 \text { and } \sin ^{2} x \geq 0 \\ \therefore \sin ^{2} x +1+\sin x \geq 0 $
Hence,$f^{\prime}(x)>0$ when $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \text { and } f^{\prime}(x)<0$
when $\cos x<0 \text { i.e., } x \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Hence, f(x) is decreasing when $x \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Since,$\left(\frac{\pi}{2}, \pi\right) \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Hence, f(x) is decreasing in $\left(\frac{\pi}{2}, \pi\right)$

179.

We have, f(x) = tan x - x
$\therefore \quad f^{\prime}(x)=\sec ^{2} x-1 \Rightarrow f^{\prime}(x) \geq 0, \forall x \in R$
So, f(x) always increases

180.

We have,$y=x(x-3)^{2}$
$ \therefore \quad \frac{d y}{d x} =x \cdot 2(x-3) \cdot 1+(x-3)^{2} \cdot 1 \\ =2 x^{2}-6 x+x^{2}+9-6 x=3 x^{2}-12 x+9 \\ =3\left(x^{2}-3 x-x+3\right)=3(x-3)(x-1) $
So, y = x(x - 3)² decreases for (1, 3).
[since, y' < 0 for all x E (1,3), hence y is decreasing on (1, 3)]

181.

(a)
Rs. 20.967

182.

(a)
4 m/s

183.

Let r, l and h denote respectively the radius, slant height and height of the cone at any time t. Then,
$\begin{aligned} l^{2} &=r^{2}+h^{2} \\ \Rightarrow & 2 l \frac{d l}{d t}=2 r \frac{d r}{d t}+2 h \frac{d h}{d t} \\ \Rightarrow & l \frac{d l}{d t}=r \frac{d r}{d t}+h \frac{d h}{d t} \end{aligned}$
$\begin{array}{ll} \Rightarrow & l \frac{d l}{d t}=7 \times 3+24 \times(-4) \quad\left[\because \frac{d h}{d t}=-4 \text { and } \frac{d r}{d t}=3\right] \\ \Rightarrow & l \frac{d l}{d t}=-75 \end{array}$
When r = 7 and h = 24, then we have
$ l^{2} =7^{2}+24^{2} \\ \Rightarrow \quad l =25 \\ \therefore \quad l \frac{d l}{d t} =-75 \Rightarrow \frac{d l}{d t}=-3 $
Let S denote the lateral surface area, then
$ S=\pi r l \\ \Rightarrow \frac{d S}{d t}=\pi\left(\frac{d r}{d t} l+r \frac{d l}{d t}\right)=\pi(3 \times 25+7 \times(-3)) \\ \Rightarrow 54 \pi \mathrm{cm}^{2} / \mathrm{min} $

184.

(b)
$\frac{1}{20} \mathrm{rad} / \mathrm{s}$

185.

Let the side of an equilateral triangle be x cm.
$\therefore$ Area of equilateral triangle $A=\frac{\sqrt{3}}{4} x^{2}$
Also,$\frac{d x}{d t}=4 \mathrm{~cm} / \mathrm{s}$
On differentiating Eq. (i) w.r.t. t, we get
$\frac{d A}{d t}=\frac{\sqrt{3}}{4} \cdot 2 x \cdot \frac{d x}{d t}$
$ =\frac{\sqrt{3}}{4} \cdot 2 \cdot 5 \cdot 4 {\left[\because x=5 \text { and } \frac{d x}{d t}=4\right]} \\ = 10 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{s} $

186.

(b)
3

187.

We have,$f(x)=x^{3}-3 x, x \in[0, \sqrt{3}]$
For f(x), Rolle's theorem is satisfied
$ f^{\prime}(c) =0 \left[\because f^{\prime}(x)=3 x^{2}-3\right] $
$\Rightarrow 3 c^{2}-3 =0 $
$ \Rightarrow c^{2}=\frac{3}{3}=1 $
$ \Rightarrow c=\pm 1, \text { where } 1 \in(0, \sqrt{3}) $
$\therefore c=1 $

188.

We have,
$f(x)=x^{2}+2 x-8, x \in[-4,2]$
Rolle'stheoremis satisfied
$ \therefore f^{\prime}(c)=0 $
$\Rightarrow f^{\prime}(c)=2 c+2=0 $
$c+1=0 \Rightarrow c=-1 $

189.

We,have $f(x)=x+\frac{1}{r}, x \in[1,3]$
For f(x), mean value theorem is satisfied
$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$
$\Rightarrow 1-\frac{1}{c^{2}}=\frac{\left[3+\frac{1}{3}\right]-\left[1+\frac{1}{1}\right]}{3-1}$ $\left[\begin{array}{l} \because f^{\prime}(x)=1-\frac{1}{x^{2}} \\ \text { and } b=3, a=1 \end{array}\right]$
$ \Rightarrow \frac{c^{2}-1}{c^{2}}=\frac{\frac{10}{3}-2}{2} $
$\Rightarrow \frac{c^{2}-1}{c^{2}}=\frac{4}{3 \times 2}=\frac{2}{3} $
$\Rightarrow 3\left(c^{2}-1\right)=2 c^{2} \Rightarrow 3 c^{2}-2 c^{2}=3 $
$\Rightarrow c^{2}=3 \Rightarrow c=\pm \sqrt{3} $
$\because c=\sqrt{3} \in(1,3) $

190.

(b)
$\frac{7}{3}$

191.

We have,$f(x)=x^{3}-3 x, x \in[0, \sqrt{3}]$
For (x), Rolle's theorem is satisfied
$f^{\prime}(c)=0$ $\left[\because f^{\prime}(x)=3 x^{2}-3\right]$
$ \Rightarrow 3 c^{2}-3=0$
$\Rightarrow c^{2}=\frac{3}{3}=1$
$\Rightarrow c=\pm 1, \text { where } 1 \in(0, \sqrt{3})$
$\therefore c=1 $

192.

We have
$f(x)=x^{2}+2 x-8, x \in[-4,2]$
Rolle'stheorem is satisfied
$\therefore f^{\prime}(c)=0 $
$\Rightarrow f^{\prime}(c)=2 c+2=0 $
$c+1=0 \Rightarrow c=-1 $

193.

(b)
f(x) = x²

194.

(d)
-p²y

195.

(a)
n²y

196.

(a)
-xcosx -2sinx

197.

(a)
$-x \cos x-2 \sin x$

198.

(a)
$\frac{\log (\cos y)+y(\tan x)}{\log (\cos x)+x \tan y}$

199.

200.

(a)
$\frac{1}{x \log x \log 7}$

201.

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

202.

203.

204.

Given that,$y=(\cos x)^{(\cos x)^{(\cos x)} \cdots^{\infty}}$
$\Rightarrow \quad y=(\cos x)^{y}$
Taking log on both sides, we get
log y = y log(cos x)
Now, differentiating w.r.t. x, we get
$\frac{1}{y} \cdot \frac{d y}{d x}=y \cdot \frac{1}{\cos x}(-\sin x)+\log (\cos x) \cdot \frac{d y}{d x}$
$\Rightarrow \frac{d y}{d x}=y\left\{-y \tan x+\log \cos x \cdot \frac{d y}{d x}\right\} $
$\Rightarrow(1-y \log \cos x) \frac{d y}{d x}=-y^{2} \tan x $
$\Rightarrow \frac{d y}{d x}=\frac{y^{2} \tan x}{(y \log \cos x-1)} $

205.

We have, $y=x^{x^{x^{x^{x}}}}$
$ \text { Then } y=x^{y} \Rightarrow y=e^{y \log x} \\ \Rightarrow \frac{d y}{d x}=e^{y \log x} \frac{d}{d x}(y \log x) \\ \Rightarrow \frac{d y}{d x}=x^{y}\left(\frac{d y}{d x} \log x+\frac{y}{x}\right) \\ \Rightarrow \frac{d y}{d x}=y\left(\frac{d y}{d x} \log x+\frac{y}{x}\right) \quad\left[\because y=x^{y}\right] \\ \Rightarrow \frac{d y}{d x}=\frac{y^{2}}{x(1-y \log x)} $

206.

Let $x=b \tan \theta \text { and } y=a \sec \theta$
On differentiating w.r.t. θ, we get
$ \frac{d x}{d \theta} =b \sec ^{2} \theta \text { and } \frac{d y}{d \theta}=a \sec \theta \tan \theta \\ \therefore \quad \frac{d x}{d y} =\frac{d x / d \theta}{d y / d \theta}=\frac{b \sec ^{2} \theta}{a \sec \theta \tan \theta} \\ =\frac{b}{a}\left(\frac{\sec \theta}{\tan \theta}\right)=\frac{b}{a} \operatorname{cosec} \theta $

207.

Let $u(x)=\sin ^{2} x \text { and } v(x)=e^{\cos x}$ We want to find
$\frac{d u}{d v}=\frac{d u / d x}{d v / d x}$ Clearly $\frac{d u}{d x}=2 \sin x \cos x$ and
$\frac{d v}{d x}=e^{\cos x}(-\sin x)=-(\sin x) e^{\cos x}$
$\frac{d u}{d v}=\frac{2 \sin x \cos x}{-\sin x e^{\cos x}}=-\frac{2 \cos x}{e^{\cos x}}$

208.

Let $u=\cos ^{-1}\left(2 x^{2}-1\right) \text { and } v=\cos ^{-1} x$
$ \therefore \frac{d u}{d x} =-\frac{1}{\sqrt{1-\left(2 x^{2}-1\right)^{2}}} \cdot 4 x=\frac{-4 x}{\sqrt{1-\left(4 x^{4}+1-4 x^{2}\right.}} $
$=\frac{-4 x}{\sqrt{-4 x^{4}+4 x^{2}}}=\frac{-4 x}{\sqrt{4 x^{2}\left(1-x^{2}\right)}}=\frac{-2}{\sqrt{1-x^{2}}} $
and $\frac{d v}{d x}=\frac{-1}{\sqrt{1-x^{2}}}$
$\therefore \frac{d u}{d v}=\frac{d u / d x}{d v / d x}=\frac{-2 / \sqrt{1-x^{2}}}{-1 / \sqrt{1-x^{2}}}=2$

209.

Since $x^{y}=y^{x} \Rightarrow y \log x=x \log y$
On differentiating w.r.t. x, we get
$ \Rightarrow y \cdot \frac{1}{x}+\log x \frac{d y}{d x}=\frac{x}{y} \frac{d y}{d x}+\log y $
$\Rightarrow \frac{d y}{d x}\left(\frac{x}{y}-\log x\right)=\frac{y}{x}-\log y $
$\Rightarrow x(x-y \log x) \frac{d y}{d x}=y(-x \log y+y) $

210.

Given that $x=e^{x / y}$
Taking log on both sides, we get
$ \log x =\frac{x}{y} \cdot \log e=\frac{x}{y} $
$\Rightarrow=y \log x $
Now, differentiating w.r.t. x, we get
$\Rightarrow 1=y \cdot \frac{1}{x}+\log x \cdot \frac{d y}{d x}$
$ \Rightarrow \frac{d y}{d x}=\frac{x-y}{x \log x} $

211.

We have $y^{x}=e^{y-x}$
Taking log both sides, we get
$ x \log y =y-x $
$ \Rightarrow x(\log y+1) =y $
$\Rightarrow \frac{y}{1+\log y} =x $
$\left(\frac{(1+\log y)-\left(\frac{1}{y}\right) y}{(1+\log y)^{2}}\right) \frac{d y}{d x}=1$
$ \frac{\log y}{(1+\log y)^{2}} \frac{d y}{d x}=1 $
$\Rightarrow \frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y} $

212.

GIven,
$y=\frac{\log x}{\log a}+\frac{\log a}{\log x}+1+1$
$\Rightarrow \frac{d y}{d x}=\frac{1}{x \log a}-\frac{\log a}{x(\log x)^{2}}$

213.

Given y = x log x
$\frac{d y}{d x}=\frac{x}{x}+\log x$
$\begin{array}{ll} \Rightarrow & \frac{d y}{d x}=\log e+\log x \\ \Rightarrow & \frac{d y}{d x}=\log (e x) \end{array}$

214.

$ \sqrt{1+\sin x}=\cos \frac{x}{2}+\sin \frac{x}{2} \text { and }\\ \sqrt{1-\sin x}=\cos \frac{x}{2}-\sin \frac{x}{2} $

215.

$\sin ^{-1} \sqrt{1-x^{2}}=\cos ^{-1} x$

216.

$\text { Put } x=\tan \theta$

217.

Given, cos y = xcos(a + y)
$ \Rightarrow x =\frac{y}{\cos (a+y)} $
$\frac{d x}{d y} =\frac{d}{d y}\left\{\frac{\cos y}{\cos (a+y)}\right\} $
$=\frac{\cos (a+y)(-\sin y)-\cos y(-\sin (a+y) 1)}{\cos ^{2}(a+y)} $
$[\because \sin (A-B)=\sin A \cos B-\cos A \sin B]$
$\therefore \frac{d y}{d x}=\frac{1}{\frac{d x}{d y}}=\frac{\cos ^{2}(a+y)}{\sin a}$

218.

$\because \ y=(\sin x+y)^{1 / 2}$
$\therefore \ \frac{d y}{d x}=\frac{1}{2}(\sin x+y)^{-1 / 2} \cdot \frac{d}{d x}(\sin x+y)$
[by chain rule of derivative]
$\Rightarrow \frac{d y}{d x}=\frac{1}{2} \cdot \frac{1}{(\sin x+y)^{1 / 2}} \cdot\left(\cos x+\frac{d y}{d x}\right) $
$\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}\left(\cos x+\frac{d y}{d x}\right) \quad\left[\because(\sin x+y)^{1 / 2}=y\right] $
$\Rightarrow \frac{d y}{d x}\left(1-\frac{1}{2 y}\right)=\frac{\cos x}{2 y} $
$\therefore \frac{d y}{d x}=\frac{\cos x}{2 y} \cdot \frac{2 y}{2 y-1}=\frac{\cos x}{2 y-1} $

219.

Given, 2x + 3y = sin x
On differentiating both sides w.r.t. x, we get
$\frac{d}{d x}(2 x+3 y) =\frac{d}{d x}(\sin x) $
$2+3 \frac{d y}{d x} =\cos x $
$ \Rightarrow 3 \frac{a y}{d x}=\cos x-2$
$ \Rightarrow \frac{d y}{d x}=\frac{\cos x-2}{3} $

220.

We differentiate the relationship directly with respect to x,we get
$\frac{d y}{d x}+\frac{d}{d x}(\sin y)=\frac{d}{d x}(\cos x)$
[by chain rule of derivative]
$\frac{d y}{d x}+\cos y \cdot \frac{d y}{d x}=-\sin x$
This gives $\frac{d y}{d x}=-\frac{\sin x}{1+\cos y}$
where $y \neq(2 n+1) \pi$

221.

(d)
f is differentiable at x = 0 but not at x = 1

222.

Let $y=\sqrt{3 x+2}+\frac{1}{\sqrt{2 x^{2}+4}}$
$=\mid(3 x+2)^{\frac{1}{2}}+\left(2 x^{2}+4\right)^{-\frac{1}{2}}$
Therefore,
$\frac{d y}{d x}=\frac{1}{2}(3 x+2)^{\frac{1}{2}-1} \cdot \frac{d}{d x}(3 x+2)$ $+\left(-\frac{1}{2}\right)\left(2 x^{2}+4\right)^{-\frac{1}{2}-1} \cdot \frac{d}{d x}\left(2 x^{2}+4\right)$
$=\frac{1}{2}(3 x+2)^{-\frac{1}{2}} \cdot(3)-\left(\frac{1}{2}\right)\left(2 x^{2}+4\right)^{-\frac{3}{2}} \cdot 4 x $
$=\frac{3}{2 \sqrt{3 x+2}}-\frac{2 x}{\left(2 x^{2}+4\right)^{\frac{3}{2}}} $

223.

$y=\sin \left(\cos x^{2}\right) $
Therefore,$\frac{d y}{d x}=\frac{d}{d x} \sin \left(\cos x^{2}\right)$
$ =\cos \left(\cos x^{2}\right) \frac{d}{d x}\left(\cos x^{2}\right) \\ =\cos \left(\cos x^{2}\right)\left(-\sin x^{2}\right) \frac{d}{d x}\left(x^{2}\right) \\ =-\sin x^{2} \cos \left(\cos x^{2}\right)(2 x) \\ =-2 x \sin x^{2} \cos \left(\cos x^{2}\right) $

224.

(b)
f is everywhere continuous but not differentiable at $x=n \pi, n \in Z$

225.

$\because \text { At } x=\frac{1}{2}$ curve have two tangents at that point therefore from the graph it is clear that $y=|2 x-1|$
$\therefore \ f(x) \text { is differentiable in } R-\left\{\frac{1}{2}\right\}$

226.

We know that, if and g are continuous functions, then
(a) f + g is continuous
(b) f - g is continuous.
(c) fg is continuous
(d) $\frac{f}{g}$ is continuous at these points, where $g(x) \neq 0$
Here,$\frac{g(x)}{f(x)}=\frac{\frac{x^{2}}{2}+1}{2 x}=\frac{x^{2}+2}{4 x}$
which is discontinuous at x = o

227.

$\begin{array}{l} \text { LHL }=\lim _{x \rightarrow 0^{-}} \frac{\sqrt{1+k x}-\sqrt{1-k x}}{x} \\ =\lim _{x \rightarrow 0^{-}} \frac{2 k x}{x(\sqrt{1+k x}+\sqrt{1-k x})}=k \end{array}$
$\begin{aligned} \mathrm{RHL} &=\lim _{x \rightarrow 0^{+}}\left(2 x^{2}+3 x-2\right)=-2 \\ f(0) &=-2 \end{aligned}$
$\therefore$ It is given that f(x) is continuous at x = 0.
$\therefore$ LHL= RHL = f(0) ⇒ k = - 2

228.

x - [x] = 0 when x is an integer, so that f(x) is discontinuous for all x ∈ I i.e. f(x) is discontinuous at infinite number of points.

229.

We have $f(x)=\left\{\begin{array}{cl} \frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\ 3, & \text { if } x=\frac{\pi}{2} \end{array}\right.$
f(x) is continuous at $x=\frac{\pi}{2}$
$\therefore \ \lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{k \cos x}{\pi-2 x}=3 $
$\Rightarrow \lim _{h \rightarrow 0} \frac{k \cos \left(\frac{\pi}{2}-h\right)}{\pi-2\left(\frac{\pi}{2}-h\right)}=3 $
$\Rightarrow \lim _{h \rightarrow 0} \frac{k \sin h}{2 h}=3 $
$\therefore \frac{k}{2}=3 \Rightarrow k=6 $

230.

f(x) = [x] is discontinuous at every integer

231.

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 $\{x=n \pi: n \in Z\}$

232.

$\text { We have, } f(x)=\frac{4-x^{2}}{4 x-x^{3}}=\frac{\left(4-x^{2}\right)}{x\left(4-x^{2}\right)}$
$\begin{array}{l} =\frac{\left(4-x^{2}\right)}{x\left(2^{2}-x^{2}\right)} \\ =\frac{4-x^{2}}{x(2+x)(2-x)} \end{array}$
Clearly, f(x) is discontinuous at exactly three points
x = 0, x = - 2 and x = 2.

233.

$ \lim _{x \rightarrow 0^{-}} f(x)=0 \text { and } \lim _{x \rightarrow 0^{+}} f(x)=1$

234.

$\lim _{x \rightarrow 2^{-}} f(x)=7 \text { and } \lim _{x \rightarrow 2^{+}} f(x)=1$

235.

$\lim _{x \rightarrow 0^{-}} f(x)=1=\lim _{x \rightarrow 0^{+}} f(x) \neq f(0)=2$

236.

237.

(a)
$|A B|=|A| \cdot|B|$

238.

(b)
$|A| \neq 0$

239.

(a)
0

240.

(a)
zero

241.

The given system will have infinite solution, if
$\begin{aligned} \Rightarrow &\begin{array}{l} \left|\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ -3 & -2 k & 6 \end{array}\right|=0 \\ 6 k-18=0 \Rightarrow k=3 \end{array} \end{aligned}$
Note There is no need to verify (adj A) B = O. For k = 3.

242.

The given system of equations will have no solution, if $|A|=0$
$\Rightarrow \left|\begin{array}{ccc} 2 & -1 & 2 \\ 1 & -2 & 1 \\ 1 & 1 & \lambda \end{array}\right|=0$
$\Rightarrow 2(-2 \lambda-1)+(\lambda-1)+2(1+2)=0$
$\Rightarrow-3 \lambda+3=0 \Rightarrow \lambda=1 $

243.

Given system of equations has unique solution
$\text { if }\left|\begin{array}{ccc} k & 2 & -1 \\ 0 & k-1 & -2 \\ 0 & 0 & k+2 \end{array}\right| \neq 0$
$\Rightarrow k \neq-2,0,1$
$\therefore$k = - 1 is the required value.

244.

From the option, we can see only option (a) satisfy both the equations

245.

If lAI = 0 and (adj A) B ≠ - 0, then system of equations has no solution.

246.

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|} $

247.

$A^{-1} \text {exist iff }|A| \neq 0$

248.

249.

We know, if A is non-singular matrix of order n, then
| adj(A)| = lA|^n-1

250.

△ Sum of product of elements of any row (or colunm) with their corresponding cofactors

251.

$ \Delta=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \\ =a_{11} M_{11}-a_{12} M_{12}+a_{13} M_{13} \\ =1 (-40)-3(-10)+(-2)(35) \\ =-40+30-70=-80 $

252.

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

253.

(d)
c(a²-b²)

254.

By defmation of minor

255.

By definition of collinearity.

256.

$\frac{1}{2}\left|\begin{array}{ccc} 2 & -6 & 1 \\ 5 & 4 & 1 \\ k & 4 & 1 \end{array}\right|=\pm 35$

257.

Area of triangle, $\Delta=\frac{1}{2}\left|\begin{array}{lll} a & b+c & 1 \\ b & c+a & 1 \\ c & a+b & 1 \end{array}\right|$

258.

By formula of area of triangle

259.

$\text { Let } \Delta=\left|\begin{array}{rrr} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{array}\right| $
$ =x\left(-x^{2}-1\right)-\sin \theta(-x \sin \theta-\cos \theta) +\cos \theta(-\sin \theta+x \cos \theta)$
$ =-x^{3}-x+x \sin ^{2} \theta+\sin \theta \cos \theta-\sin \theta \cos \theta+x \cos ^{2} \theta \\ =-x^{3}-x+x\left(\sin ^{2} \theta+\cos ^{2} \theta\right)=-x^{3}-x+x \\ \left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right] $

260.

Clearly,
$ f(a) =\left|\begin{array}{ccc} 0 & 0 & a-b \\ 2 a & 0 & a-c \\ a+b & a+c & 0 \end{array}\right| $
$=[(a-b)\{2 a \cdot(a+c)\}] \neq 0 $
$ \therefore \ f(b) =\left|\begin{array}{ccc} 0 & b-a & 0 \\ b+a & 0 & b-c \\ 2 b & b+c & 0 \end{array}\right|$
$ =-(b-a)[2 b(b-c)] $
$ =-2 b(b-a)(b-c) \neq 0 $

261.

$ \Delta_{1} =\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right| =\left|\begin{array}{ccc} A & x & z y \\ B & y & z x \\ C & z & x y \end{array}\right| \\ =\frac{1}{x y z}\left|\begin{array}{lll} A x & x^{2} & x y z \\ B y & y^{2} & x y z \\ C z & z^{2} & x y z \end{array}\right|=\frac{x y z}{x y z}\left|\begin{array}{ccc} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right|=\Delta $

262.

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

263.

(a)
skew-symmetric matrix

264.

(d)
None of these

265.

266.

(a)
I

267.

(b)
x = 2, y = 3

268.

$ A B=B I$
$\Rightarrow \frac{1}{3}(A B)=I$
$\Rightarrow A\left(\frac{1}{3} B\right)=I$
$ A^{-1}=\frac{1}{3} B $

269.

By definition of invertible matrix

270.

According to rule of elementary row operations, we apply these operations simultaneously on X and on the first matrix A of the product AB on RHS.

271.

Given, $\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$
On applying $R_{1} \rightarrow R_{1}-3 R_{2},$ we get
$\left[\begin{array}{cc} 4-9 & 2-9 \\ 3 & 3 \end{array}\right]=\left[\begin{array}{cc} 1-0 & 2-9 \\ 0 & 3 \end{array}\right]\left[\begin{array}{cc} 2 & 0 \\ 1 & 1 \end{array}\right]$
$\Rightarrow$$\left[\begin{array}{rr} -5 & -7 \\ 3 & 3 \end{array}\right]=\left[\begin{array}{rr} 1 & -7 \\ 0 & 3 \end{array}\right] \cdot\left[\begin{array}{ll} 2 & 0 \\ 1 & 1 \end{array}\right]$

272.

Given, $\left[\begin{array}{rr}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]$
On applying $C_{2} \rightarrow C_{2}-2 C_{1},$
we get $\left[\begin{array}{rr}1 & -3-2 \\ 2 & 4-4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1-6 \\ 2 & 4-4\end{array}\right]$

$\Rightarrow \left[\begin{array}{rr}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{rr}3 & -5 \\ 2 & 0\end{array}\right]$

273.

$\left[\begin{array}{rr} 1 & -3 \\ 2 & 4 \end{array}\right]=\left[\begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array}\right]\left[\begin{array}{ll} 3 & 1 \\ 2 & 4 \end{array}\right]$

274.

$\text { Let } A=\left[\begin{array}{ccc} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{array}\right] \text { . Then, } A^{\prime}=-A$

275.

Hint $A+A^{\prime}=I$I
$\Rightarrow\left[\begin{array}{cc}2 \cos \alpha & 0 \\ 0 & 2 \cos \alpha\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow$$2 \cos \alpha=1$
$\Rightarrow \cos \alpha=\frac{1}{2}$
$\Rightarrow \alpha=\frac{\pi}{3}$

276.

(d)
$\left[\begin{array}{cc} p & q \\ q & p-q \end{array}\right]$

277.

(A + B) (A - B) = A(A - B) + B(A - B)
= A² - AB + BA - B²

278.

Let A = [aij]₂ x 3 and B = [bij]₃ _{x 2}
Since, number of columns of A = number of rows of B
$\therefore$ AB is defined
Also, as number of columns of B = number of rows of A.
$\therefore$ BA is defined.
Hence, both AB and BA are defined.

279.

280.

(a)
$\left[\begin{array}{cc}a^{2}+b^{2} & 0 \\ 0 & a^{2}+b^{2}\end{array}\right]$

281.

Hint The order of SA is 3 x m and 2B is 3 x n, where m = 7L
$\therefore$ The order of SA - 2B is 3 x m or 3 x 7L

282.

We have, $\left[\begin{array}{cc}1 & 2 \\ -2 & -b\end{array}\right]+\left[\begin{array}{ll}a & 4 \\ 3 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 0\end{array}\right]$
= $\left[\begin{array}{cc}a+1 & 6 \\ 1 & 2-b\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 0\end{array}\right]$
$\Rightarrow$a + 1 = 5, 2 - b = 0
$\Rightarrow$a = 4, b = 2
$\Rightarrow$$a^{2}+b^{2}=20$

283.

Only B + D is defined because matrices of the same order can only be added

284.

If square matrix in which all diagonals elements are 1 and rest are 0, is called unit matrix.

285.

Number of entries in 3 x 3 matrix is 9. Since, each entry has 2 choices, namely 2 or 0. Therefore, number of
$\text { possible matrices }=\underbrace{2 \times 2 \times 2 \ldots \times 2=2^{9}}_{9 \text { times }}=512$

286.

We know that if a matrix is of order m x n, then it has mn elements. Thus, to find all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural numbers, whose product is 8. Thus, all possible ordered pair are (1,8), (8, I), (2, 4), (4, 2).

287.

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

288.

289.

(a)
$\frac{5 \pi^{2}}{4} and \frac{\pi^{2}}{8}$

290.

291.

(b)
$\alpha=0, \beta=\pi$

292.

(b)
unique solution

293.

294.

(a)
1

295.

(a)
0

296.

(a)
0

297.

(d)
Both 'a' and 'b'

298.

$\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$

299.

$ \tan \left(\cos ^{-1} \frac{3}{5}+\tan ^{-1} \frac{1}{4}\right) $
$\text { Let } \cos ^{-1} \frac{3}{5}=x \Rightarrow \cos x=\frac{3}{5} $
$\therefore \ \tan x=\frac{4}{3} \Rightarrow x=\tan ^{-1} \frac{4}{3} $
$\therefore \ \tan \left(\tan ^{-1} \frac{4}{3}+\tan ^{-1} \frac{1}{4}\right) $
$\therefore \ \tan \left[\tan ^{-1}\left\{\frac{\left(\frac{4}{3}+\frac{1}{4}\right)}{1-\frac{4}{3} \times \frac{1}{4}}\right\}\right]=\frac{19}{8}$

300.

$ 2 \tan ^{-1} x=\sin ^{-1} \frac{2 x}{1+x^{2}}$
$\therefore \ 2 \tan ^{-1} \frac{2}{3}=\sin ^{-1} \frac{2\left(\frac{2}{3}\right)}{1+\left(\frac{2}{3}\right)^{2}}=\sin ^{-1} \frac{12}{13}$
$\cos \left(2 \tan ^{-1} x\right)=\cos \left(\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right) $

301.

$ \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) $

302.

$\begin{array}{l} \text { Given, } \tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right] \\ =\tan ^{-1}\left[2 \sin \left(2 \times \frac{\pi}{6}\right)\right]=\tan ^{-1}\left(2 \sin \frac{\pi}{3}\right) \\ =\tan ^{-1}\left(2 \times \frac{\sqrt{3}}{2}\right)=\tan ^{-1} \sqrt{3}=\frac{\pi}{3} \end{array}$

303.

(d)
1

304.

Range of $\sin ^{-1} x is \left(\frac{-\pi}{2}, \frac{\pi}{2}\right]$
$\therefore\ \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

305.

Cosine functions respected to any interval $\left[-\pi, 0]\right.$

306.

Let $y=r(x)=x^{2}+5$
$\Rightarrow$ Then, $x^{3}=y-5 \rightarrow x=(y-5)^{\prime \prime}$
$\Rightarrow f^{-1}(y)=(y-5)^{3} \quad\left[\because y=f(x) \rightarrow x=f^{-1}\langle y\}\right.$
or $f^{-1}(x)=\{x-5)^{10}$

307.

By the theorem, $(\text { gof })^{-1}=f^{-1}$ og $^{-1}$ For this, we will show that $\left(f^{-1} o g^{-1}\right) o(g o f)=I_{X}$
and $(g \circ f) o\left(f^{-1} o g^{-1}\right)=I_{Z}$
Now, $\left(f^{-1} o g^{-1}\right) o(g o f)=\left\{\left(f^{-1} o g^{-1}\right) o g\right\} o f$
[by using theorem}
discussed in Topic 3 $=\left(f^{-1} o I_{Y}\right. -of \left[\because g\right. is invertible \left.\Rightarrow g^{-1} o g=I_{Y}\right]$
$=I_{X} $ $\left[\because f\right. is invertible \left.\Rightarrow f^{-1} o f=I_{X}\right]$
Similarly, $(g \circ f) o\left(f^{-1}{O g}^{-1}\right)=I_{Z} \mid$

308.

According to given condition f is invertible and hence one-one onto.

309.

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

310.

Let $f(f(y))=y$, for all $y \neq-1$ $\Rightarrow f\left(\frac{a y}{y+1}\right)=y$, for all $y \neq-1$
$\Rightarrow \frac{a\left(\frac{a y}{y+1}\right)}{\frac{a y}{y+1}+1}=y,for\ all\ y \neq-1$
$\Rightarrow \frac{a^{2} y}{a y+y+1}=y, for\ all \ y \neq-1$
$\Rightarrow a^{2} y=(a+1) y^{2}+y, for\ all \ y \neq-1$
$\Rightarrow (a+1) y^{2}+\left(1-a^{2}\right) y=0, for \ all \ y \neq-1$
$\Rightarrow a+1=0 and 1-a^{2}=0 \Rightarrow a=-1$

311.

By the properties, gof is ene-one $\Rightarrow$ is one-one.

312.

Consider $((f \cdot g) o h)(x)$
$=(f \cdot g)(h(x))=f(h(x)) \cdot g(h(x))$
$=(f o h)(x) \cdot(g o h)(x)$
$=\{(f o h) \cdot(g o h)\}(x)$
$\text { Hence, }(f \cdot g) o h=(f o h) \cdot(g o h)$
Now, consider $((f+g) o h)(x)=(f+g)(h(x))$
$=f(h(x))+g(h(x))=(\text { foh })(x)+(\text { goh })(x)$
$=\{(f o h)+(g o h)\}(x) $
Hence $(f+g) o h=f o h+g o h$

313.

$\operatorname{gcg} f(x)=3 \cos ^{2} x_{i} f_{0}(x)=\cos 3 x^{2}$
At X = 0, gof(x) ≠ fog(x)

314.

Range (f) = Integers ee Rand [2,3] = [2,4] = 2
⇒ f is not one-one.

315.

One-one onto mapping is possible only ifn(A) = n(B)

316.

Total number of functions = (n(B)^n(A) = 2ⁿ. Clearly a function will not be onto if all elements of A map to either a or b.

317.

If n(A) = x and n(B) = y, then number of one-one functions from A to B is given by ^y P_x, where x ≤y.

318.

A function f : A ➝ B is said to be onto, if for every b ∈ B, there exists an element a in A such that f(a) = b

319.

The set of all elements related to 1 is { a ∈ A : a = 1}.

320.

Clearly R is reflexive and transitive. For R to be symmetric we should add (3, 1) in R.

321.

R = {(I,6), (2, 7), (3, 8)}

322.

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

323.

324.

(b)
f'(x).∆x

325.

(a)
564.06

326.

(d)
2.0025

327.

328.

329.

330.

(b)
0.12x² m²

331.

332.

(d)
π cm³/s

333.

334.

(d)
Not continuous at x = 0

335.

(d)
(0, 1)

336.

337.

338.

339.

(a)
a = 2, b = 0

340.

(b)
Discontinuous at x = 2

341.

(d)
Discontinuous exactly at three points

342.

(a)
[x]

343.

(b)
i = 1, 2 ; j = 1, 2, 3

344.

(b)
-2

345.

(a)
0

346.

(a)
2 x 5

347.

(d)
diagonal matrix.

348.

(a)
The number of Men in BPO II

349.

(d)
1

350.

(b)
No real value

351.

352.

(b)
3

353.

354.

355.

(b)
(nπ, (n + 1) π)

356.

(d)
-π/4

357.

(d)
π/4

358.

(b)
0

359.

(a)
$\frac { 3\pi }{ 4 } $

360.

(d)
not defined

361.

(a)
y = sin^-1x

362.

(b)
not defined

363.

(d)
{…..-5,-2,1,4,7,.}

364.

(d)
Reflexive and transitive only

365.

366.

(d)
2^nxn

367.

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

368.

369.

370.

(d)
Symmetric

371.

(a)
Equivalence relation

372.

(a)
{4,5,6}

373.

(d)
R is reflexive and transitive but not symmetric

374.

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

375.

376.

(d)
4

377.

(d)
0

378.

(b)
Transitive

379.

(b)
Symmetric

380.

381.

(a)
{(x, y): x² + y²= 5}

382.

(a)
$\left( 4,\pm \frac { 8 }{ 3 } \right) $

383.

(a)
x + y = 3

384.

(b)
x – y = 0

385.

(a)
1

386.

(b)
$\frac67$

387.

388.

(d)
77.66

389.

(a)
(1, 2)

390.

(d)
-$\frac13$

391.

(d)
126

392.

393.

394.

(b)
16

395.

(b)
Is * commutative but not associative?

396.

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

397.

398.

As y’ = 3x² – 3, for a point of absolute maximum or minimum y’=0 ⇒ x = ± 1.
y]_x=0 = 2,
y]_x=1= 1 – 3 + 2 = 0,
y]_x=-1= -1 +3+ 2 = 4,
y]_x=2 = 8 – 6 + 2 = 4

399.

As for y² = x, 2yy^' = 1⇒ y']_(1,1) = $\frac12$
for x² = y, y' = 2x ⇒ y']_(1,1)= 2
$\therefore tan\theta =\left| \frac { \frac { 1 }{ 2 } -2 }{ 1+\frac { 1 }{ 2 } .2 } \right| $ $\Rightarrow tan\theta =\left| \frac { -\frac { 3 }{ 2 } }{ 2 } \right| $
$\Rightarrow \theta ={ tan }^{ -1 }\left( \frac { 3 }{ 4 } \right) $ or ${ cot }^{ -1 }\left( \frac { 4 }{ 3 } \right) $

400.

As $\frac { dy }{ dx } ={ 2e }^{ 2x }\Rightarrow \frac { dy }{ dx } $]_(0,1) = 2e² = 2
Equation of tangent is y-1=2(x-0)
⇒ 2x-y+1=0, if it meets the x-axis, then y = 0 ⇒ x = -$\frac12$, point is ($-\frac12$, 0)

401.

As for curve ay + x² = 7
⇒ a$\frac { dy }{ dx } $ + 2x =0
⇒ $\frac { dy }{ dx } $ = $-\frac { 2x }{ a } \Rightarrow \frac { dy }{ dx } $]_(1,1) = -$\frac2a$
and for curve x³= y, 3x² = $\frac { dy }{ dx } $
⇒ $\frac { dy }{ dx } $ ]_(1,1) = 3
If curves cut orthogonally then
$\left( -\frac { 2 }{ a } \right) $ x 3 = -1 ⇒ a = 6

402.

As for y = ae^-x
$\frac { dy }{ dx } ={ ae }^{ -x }$ and for the curve
$y={ be }^{ x },\frac { dy }{ dx } ={ be }^{ x }$
If coures are orthogonal then
-ae^-xx be^x = -1
⇒ ab = 1

403.

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) $

404.

405.

As $\frac{dy}{dt}$ = 2x.$\frac{dx}{dt}$
⇒ 6.$\frac{dx}{dt}$ = 2x.$\frac{dx}{dt}$ ⇒ x = 3
From curve, y = 9. Point is (3, 9)

406.

As $\frac { dx }{ dt } =2$ cm/s, x is side of equiolatral triangle.
A = $\frac { \sqrt { 3 } }{ 4 } { x }^{ 2 }$
$\Rightarrow \frac { dA }{ dx } =\frac { \sqrt { 3 } }{ 2 } { x }\frac { dx }{ dt } $
= $\frac { \sqrt { 3 } }{ 2 } x2=\sqrt { 3 } x$
∴$|\frac{dA}{dx}|$x=10 = 10$\sqrt3$ cm²/s

407.

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

408.

As y = sin x, t
= log $x\frac { dy }{ dx } =\frac { dy }{ dx } \div \frac { dt }{ dx } $
$=\frac { dy }{ dx } (sin \ x)\div \frac { dt }{ dx } (log \ x)$
= cos x $\div $ $\frac1x$ = x cos x

409.

As y = xy ⇒ log y = y log x
⇒ $\frac { 1 }{ y } .{ y }^{ ' }=\frac { y }{ x } +logx.{ y }^{ ' }$
$\Rightarrow { y }^{ ' }\left[ \frac { 1 }{ y } -log \ x \right] $
$=\frac { y }{ x } \Rightarrow x(1-y \ log \ x){ y }^{ ' }={ y }^{ 2 }$

410.

y = tan^-1 $\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) $
= tan^-1 $\left( \frac { \pi }{ 4 } \right) -{ tan }^{ -1 }{ x }^{ 2 }$
${ y }^{ ' }=0-\frac { -1 }{ 1+{ x }^{ 4 } } 2x=\frac { -2x }{ 1+{ x }^{ 4 } } $

411.

As (gof) (x) = g[f(x)] = g(e^x)
= loge e^x = x
∴ (gof)' (x) = 1

412.

As $f(x)=\frac { log(logx) }{ 2logx } $
$\Rightarrow { f }^{ ' }(x)=\frac { 1 }{ 2 } \left[ \frac { logx.\frac { 1 }{ logx } .\frac { 1 }{ x } -log(logx).\frac { 1 }{ x } }{ { (logx) }^{ 2 } } \right] $
$\Rightarrow { f }^{ ' }(x)=\frac { 1 }{ 2 } \left[ \frac { 1-log(logx) }{ x{ (logx) }^{ 2 } } \right] $
$\Rightarrow { f }^{ ' }(e)=\frac { 1 }{ 2 } \left[ \frac { 1-log(loge) }{ x{ e(loge) }^{ 2 } } \right] $=$\frac{1}{2e}$

413.

As y = sin^-1 $y={ sin }^{ 2 }\left\{ 3.\frac { x }{ 2 } -4.{ \left( \frac { x }{ 2 } \right) }^{ 3 } \right\} $
= 3 sin^-1 $\frac { x }{ 2 } $
$\therefore \ \frac { dy }{ dx } =3.\frac { 1 }{ \sqrt { 1-\frac { { x }^{ 2 } }{ 4 } } } .\frac { 1 }{ 2 } $
$=\frac { 3 }{ \sqrt { 4-{ x }^{ 2 } } } $

414.

As x^o = $\frac { \pi }{ 180 } { x }^{ c }$
$\therefore \frac { d }{ dx } (cot{ x }^{ o })=$$\frac { d }{ dx } \left( cot\frac { \pi }{ 180 } x \right) $
$=-\frac { \pi }{ 180 } { cosec }^{ 2 }\frac { \pi }{ 180 } x$
$=-{ 1 }^{ o }{ cosec }^{ 2 }\frac { \pi }{ 180 } x$
$=-{ 1 }^{ o }{ cosec }^{ 2 }{ x }^{ 0 }$

415.

As f(x) = |x – a| is continuous at x = a but not differentiable thereat.

416.

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

417.

As differentiable functions is continuous also

418.

As $\lim _{ x\rightarrow 2 }{ \frac { sin({ e }^{ x-2 }-1) }{ log(x-1) } } =\lim _{ h\rightarrow 0 }{ \frac { sin({ e }^{ h }-1) }{ log(1+h) } } $
On substituting h = x - 2
= $\lim _{ h\rightarrow 0 }{ \frac { sin({ e }^{ h }-1) }{ log(1+h) } } .\frac { { e }^{ h }-1 }{ h } $.$\frac { h }{ log(1+h) } $
= 1.1.1
= 1 nad f(2) = k

419.

As $\lim _{ x\rightarrow 0 }{ \frac { \sqrt { \frac { 1 }{ 2 } (1-cosx) } }{ x } } $
$=\lim _{ x\rightarrow 0 }{ \frac { |sin \ x| }{ x } } $
and LHL ≠ RHL at x = 0

420.

As f(2) is not defined so / is not continuous at x = 2 ‘g’ is a polynomial function, so continuous at x = 2.

421.

As $\left| { (AB) }^{ -1 } \right| =\frac { 1 }{ |AB| } =\frac { 1 }{ |A||B| } $
$\Rightarrow 8=\frac { 1 }{ 2|B| } \Rightarrow B=\frac { 1 }{ 16 } $

422.

Performing ${ R }_{ 3 }\rightarrow { R }_{ 3 }-{ cosyR }_{ 1 }+{ siny }{ R }_{ 2 }$
we get on simplification
$\triangle =sin \ y-cos \ y=\sqrt { 2 } .sin\left( y-\frac { \pi }{ 4 } \right) \le \sqrt { 2 } $
As -1 $\le sin\left( y-\frac { \pi }{ 4 } \right) \le 1$
⇒ $-\sqrt { 2 } \le \sqrt { 2 } $ $sin\left( y-\frac { \pi }{ 4 } \right) \le \sqrt { 2 } $
∴ $[-\sqrt { 2 } ,\sqrt { 2 } ]$

423.

As C₂⟶ C₂ - 2C₃ gives
$\left| \begin{matrix} cos\ x & 2sinx & sinx \\ x & x & x \\ 1 & 2x & x \end{matrix} \right| $ = -x(x cos x - sin x)
= -x² cos x + x sin x
$\lim _{ x\rightarrow 0 }{ \frac { f(x) }{ { x } } } $ = $\lim _{ x\rightarrow 0 }{ \left( \frac { -{ x }^{ 2 }cos \ x+x \ sin \ x }{ { x }^{ 2 } } \right) } $
= $\lim _{ x\rightarrow 0 }{ (-cosx) } +\lim _{ x\rightarrow 0 }{ \left( \frac { sinx }{ x } \right) } $
= - 1 + 1 = 0

424.

As if A = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ then $\left| A \right| =\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
$KA=\begin{bmatrix} Ka & Kb \\ Kc & Kd \end{bmatrix}$ and $\left| KA \right| =\begin{bmatrix} Ka & Kb \\ Kc & Kd \end{bmatrix}$
$={ K }^{ 2 }\begin{vmatrix} a & b \\ c & d \end{vmatrix}={ K }^{ 2 }|A|$

425.

${ \triangle }_{ 1 }=\left| \begin{matrix} Ax & By & Cz \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ yz & zx & xy \end{matrix} \right| $
C₁ ⟶ xC₁, C₂ ⟶ yC₂ and C₃ ⟶ zC₃ and dividing determinat by xyz
= $\frac { 1 }{ xyz } \left| \begin{matrix} { Ax }^{ 2 } & { By }^{ 2 } & { Cz }^{ 2 } \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ xyz & xyz & xyz \end{matrix} \right| $
Taking xyz common from R₃, we get
$\frac { xyz }{ xyz } \left| \begin{matrix} { Ax }^{ 2 } & { By }^{ 2 } & { Cz }^{ 2 } \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ 1 & 1 & 1 \end{matrix} \right| $
= $\left| \begin{matrix} { Ax }^{ 2 } & { By }^{ 2 } & { Cz }^{ 2 } \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ 1 & 1 & 1 \end{matrix} \right| =\left| \begin{matrix} { Ax }^{ 2 } & x^{ 3 } & 1 \\ { By }^{ 2 } & { y }^{ 3 } & 1 \\ { Cz }^{ 2 } & { z }^{ 3 } & 1 \end{matrix} \right| $=Δ
⇒ Δ₁ - Δ = 0

426.

Δ = 6(-1)- 1(1) = -7

427.

As $\begin{vmatrix} 2x & -1 \\ 4 & 2 \end{vmatrix}=\begin{vmatrix} 3 & 0 \\ 2 & 1 \end{vmatrix}$
⇒ 4x + 4 = 3 - 0
⇒ x = $-\frac { 1 }{ 4 } $

428.

As $\begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}=\begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}\Rightarrow x=y$

429.

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.

430.

As A² = $\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}+\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}=\begin{bmatrix} 0 & 4 \\ 4 & 0 \end{bmatrix}$

431.

By definition.

432.

(a)
I

433.

As total elements are 6 and each entry can be done in 2 ways. Hence, total possibilities = 2⁶ = 64.

434.

As diag (3, -1) is a diagonal matrix. Its order is 2 × 2 with diagonal elements 3 and (-1).

435.

As a₂₃ = $\frac { { (-2+6) }^{ 2 } }{ 5 } =\frac { 16 }{ 5 } $

436.

As 6 → 1 × 6, 2 × 3, 3 × 2, 6 × 1.

437.

As y = cos^-1 (x² - 4)
⇒ cos y = x² - 4
Since - 1 ≤ cos y ≤ 1
i.e - 1 ≤ x²-4 ≤1
⇒ 3 ≤ x² ≤ 5
⇒ $\sqrt3$≤ |x|≤ $\sqrt5$
⇒ x ∈ [-$\sqrt5$ ,-$\sqrt3$] ∪ [-$\sqrt5$,$\sqrt3$]

438.

As sin^-1 x = $\frac { \pi }{ 2 } $, sin^-1 y = $\frac { \pi }{ 2 } $, sin^-1z = $\frac { \pi }{ 2 } $
⇒ x = 1, y = 1, z = 1
∴ x + y²+ z³= 1 + 1 + 1 = 3

439.

As 2 sin^-1 x = $\pi -\frac { \pi }{ 2 } =\frac { \pi }{ 2 } $
$\Rightarrow { sin }^{ -1 }x=\frac { \pi }{ 4 } \Rightarrow x=\frac { 1 }{ \sqrt { 2 } } $

440.

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.

441.

as sec^-1x + sec^-1y = $\frac{\pi}{2}$
⇒ $\frac{\pi}{2}$ - cosec^-1 x + $\frac{\pi}{2}$ - cosec^-1 y = $\frac{\pi}{2}$
⇒ cosec^-1 x + cosec^-1 y = $\frac{\pi}{2}$

442.

As -1 ≤ -x² < 1
⇒ 1 ≥ x² ≥ -1
⇒ 0 ≤ x² ≤ 1
⇒ |x| ≤ 1
⇒ -1 ≤ x ≤ 1.

443.

As cos(-680°) = cos 680°
= cos(720° – 40°) = cos 40°
∴ cos-1[cos(-680°)J = cos-1 (cos 40°)
= 40° = $\frac{2\pi}{9}$.

444.

As sec $\left( { tan }^{ -1 }\frac { y }{ 3 } \right) $ $=\sqrt { 1+{ tan }^{ 2 }\left( { tan }^{ -1 }\frac { y }{ 3 } \right) } $
$=\sqrt { 1+\frac { { y }^{ 2 } }{ 9 } } =\frac { \sqrt { 9+{ y }^{ 2 } } }{ 3 } $

445.

As sin ($-\frac{\pi}{2}$)= -1, and tan-1(-1) = $-\frac{\pi}{4}$.

446.

Let θ = sin^-1 $-(\frac12)$
⇒ sin^-1 = $-\frac12$ = sin $(-\frac{\pi}{6})$ = θ = $-\frac{\pi}{6}$

447.

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

448.

Total injective mappings/functions
= ⁴ P₃ = 4! = 24.

449.

Reflexive, true as x s x ⇒ x - x + $\sqrt3$
= $\sqrt3$ is an irrational number
Symmetric, fase e.g x =  $\sqrt3$, y= 2
x S y ⇒ $\sqrt3$ - 2 + $\sqrt3$ = 2 $\sqrt3$ -2 is an irrational number
but ySx ⇒ 2 - $\sqrt3$ + $\sqrt3$ = 2 $\sqrt3$ -2 is not irrational number
transitive, false e.g = x = 1 + $\sqrt3$, y = 5 z = 2 $\sqrt3$
x S y ⇒ 1 + $\sqrt3$ - 5 + $\sqrt3$ = 2 $\sqrt3$ - 4 is an irrational number
y S z ⇒ 5 -2 $\sqrt3$ + $\sqrt3$ = 5 - $\sqrt3$  is an irrational number
But x S z ⇒ 1 + $\sqrt3$ - 2 $\sqrt3$ + $\sqrt3$ = 1  is an irrational number