The shaded region in the adjoining diagram represents.

(a)

A\B

(b)

B\A

(c)

AΔB

(d)

A'

Let X = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (1, 3), (2, 2), (3, 3), (2, 1), (3, 1), (1, 4),(4, 1)}. Then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

If A = {x / x is an integer, x² \(\le\) 4} then elements of A are ___________

(a)

A = {-1, 0, 1}

(b)

A = {-1, 0, 1, 2}

(c)

A = {0, 2, 4}

(d)

A = {- 2, - 1, 0, 1, 2}

Solve \(\sqrt{7+6x-x^2}=x+1\)

(a)

(1, -3)

(b)

(3, -1)

(c)

(1, -1)

(d)

(3, -3)

For the below figure of ax² + bx + c = 0

(a)

a < 0, D > 0

(b)

a > 0, D > 0

(c)

a < 0, D < 0

(d)

a > 0, D = 0

cos 2ፀ cos 2ф + sin²(ፀ - ф) - sin²(ፀ + ф) is equal to

(a)

sin2(ፀ+\(\phi \))

(b)

cos2(ፀ+\(\phi \))

(c)

sin2(ፀ-\(\phi \))

(d)

cos2(ፀ-\(\phi \))

In 3 fingers, the number of ways four rings can be worn is _______ ways.

(a)

4³-1

(b)

3⁴

(c)

68

(d)

64

If n+1 C₃ = 2.nC₂₁ then n = _________

(a)

3

(b)

4

(c)

5

(d)

6

The remainder when 38¹⁵ is divided by 13 is

(a)

12

(b)

1

(c)

11

(d)

5

In the series \(\frac{1}{1+\sqrt 2}+\frac{1}{\sqrt 2+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 4}+...\) some of first 24 number is ______________

(a)

4

(b)

\(\sqrt 24\)

(c)

\(\frac{1}{\sqrt 24}\)

(d)

\(\frac{1}{\sqrt 25-\sqrt 24}\)

Which of the following point lie on the locus of 3x²+ 3y²- 8x - 12y + 17 = 0

(a)

(0, 0)

(b)

(-2, 3)

(c)

(1, 2)

(d)

(0, -1)

Distance between the lines 5x + 3y - 7 = 0 and 15x + 9y + 14 = 0 is ______________

(a)

\(\frac{35}{\sqrt{34}}\)

(b)

\(\frac{1}{3\sqrt{34}}\)

(c)

\(\frac{35}{2\sqrt{34}}\)

(d)

\(\frac{35}{3\sqrt{34}}\)

The lines x + 2y - 3 = 0 and 3x - y + 7 = 0 are ______________

(a)

parallel

(b)

neither parallel nor perpendicular

(c)

perpendicular

(d)

parallel as wellas perpendicular

If A =\(\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}\), B = \(\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}\) and (A + B)² = A² + B², then the values of a and b are

(a)

a = 4, b = 1

(b)

a = 1, b = 4

(c)

a = 0, b = 4

(d)

a = 2, b = 4

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

(a)

45°

(b)

60°

(c)

90°

(d)

30°

The vector in the direction of the vector\(\hat{i}-2\hat{j}+2\hat{k}\) that has magnitude 9 is _________ .

(a)

\(\hat{i}-2\hat{j}+2\hat{k}\)

(b)

\(\frac { \hat { i } -2\hat { j } +2\hat { k } }{ 3 } \)

(c)

3(\(\hat{i}-2\hat{j}+2\hat{k}\))

(d)

9(\(\hat{i}-2\hat{j}+2\hat{k}\))

\(lim_{\theta\rightarrow0}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

(a)

1

(b)

-1

(c)

0

(d)

2

If y = cos (sin x²), then \({dy\over dx}\) at x = \(\sqrt{\pi\over 2}\) is

(a)

-2

(b)

2

(c)

\(-2\sqrt{\pi\over 2}\)

(d)

0

\(\text { If } f(x)= \begin{cases}x-5 & \text { if } x \leq 1 \\ 4 x^2-9 & \text { if } 1<x<2 \\ 3 x+4 & \text { if } x \geq 2\end{cases}\), then the right hand derivative of f(x) at x = 2 is

(a)

0

(b)

2

(c)

3

(d)

4

Choose the correct or the most suitable answer from the given four alternatives.
If f(x) = 4x⁸, then ________

(a)

\(f^{ ' }\left( \frac { 1 }{ 2 } \right) =f^{ ' }\left( \frac { -1 }{ 2 } \right) \)

(b)

\(f\left( \frac { 1 }{ 2 } \right) =-f^{ ' }\left( \frac { -1 }{ 2 } \right) \)

(c)

\(f\left( \frac { 1 }{ 2 } \right) =f\left( \frac { -1 }{ 2 } \right) \)

(d)

\(f\left( \frac { 1 }{ 2 } \right) =f^{ ' }\left( \frac { -1 }{ 2 } \right) \)

If the equations x² - ax + b = 0 and x² - ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2 (b + f).

Find cos(x - y), given that cos x = \(-\frac{4}{5}\) with \(\pi<x<{{3\pi}\over{2}}\) and \(sin \ y = -{{24}\over{25}}\) with \(\pi<x<{{3\pi}\over{2}}\). 

Find the value of \(\frac { 12! }{ 9!\times 3! } \)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\({ \left( x+2 \right) }^{ -\frac { 2 }{ 3 } }\)

lf P(2,-7) is a given point and Q is a point on (2x² + 9y² = 18), then find the equations of the locus of the mid-point of PQ.

Find the sum A + B + C if A, B, C are given by
\(A=\left[\begin{array}{ll} \sin ^2 \theta & 1 \\ \cot ^2 \theta & 0 \end{array}\right], B=\left[\begin{array}{cc} \cos ^2 \theta & 0 \\ -\operatorname{cosec}^2 \theta & 1 \end{array}\right] \text { and } C=\left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right]\)

Verify whether the following ratios are direction cosines of some vector or not \({1\over\sqrt{2}},{1\over 2},{1\over 2}\)

In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow0}{cos x-1\over x}\)

x	-0.1	-0.01	-0.001	0.001	0.01	0.1
f(x)	0.04995	0.0049999	0.0004999	–0.0004999	–0.004999	–0.04995

Find the derivatives of the following : \(\sqrt{x^2+y^2}=tan^{-1}({y\over x})\)

If R is the set of all real numbers, what do the cartesian products R \(\times\) Rand R \(\times\)R \(\times\)R represent?

If \(\left( { x }^{ \frac { 1 }{ 2 } }+{ x }^{ -\frac { 1 }{ 2 } } \right) ^{ 2 }=\frac { 9 }{ 2 } \), then find the value of \(\left( { x }^{ \frac { 1 }{ 2 } }-{ x }^{ -\frac { 1 }{ 2 } } \right) \)for x > 1

Prove that \(cos\frac { B-C }{ 2 }= \frac { b+c }{ a } sin\frac { A }{ 2 } \)

A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total length of the spring was measured each time shown in the following table.

Weight, (kg)	2	4	5	8
Length, (cm)	3	4	4.5	6

(i) Draw a graph showing the results.
(ii) Find the equation relating the length of the spring to the weight on it.
(iii) What is the actual length of the spring.
(iv) If the spring has to stretch to 9 cm long, how much weight should be added?
(v) How long will the spring be when 6 kilograms of weight on it?

The line 2x - y = 5 turns about the point on it, whose ordinate and abscissae are equal, through an angle of 45° in the anti-clockwise direction. find the equation of the line in the new position.

If A =\(\begin{bmatrix} 1&-1 &2 \\ -2 & 1 & 3 \\ 0 &-3 &4 \end{bmatrix}\) and B = \(\begin{bmatrix} 1&-3 \\ -1 & 1 \\ 1 &2 \end{bmatrix}\) find AB and BA if they exist.

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
\(lim_{x\rightarrow3}(4-x)\).

Differentiate : \(y=(x^3-1)^{100}\)

Show that the points whose position vectors are 2\(\hat{i}\) + 3\(\hat{j}\) − 5\(\hat{k}\), 3\(\hat{i}\) + \(\hat{j}\) − 2\(\hat{k}\) and, 6\(\hat{i}\) − 5\(\hat{j}\) + 7\(\hat{k}\) are collinear

1. Write the values of f at -4, 1, -2, 7, 0 if
   \(f(x)=\left\{ \begin{matrix} -x+4& if -\infty <x\leq -3\\ x+4& if -3<x<-2\\ x^{2}-x& if -2\leq x < 1 \\ x-x^{2}& if 1\leq x<7\\ 0& otherwise\\ \end{matrix}\right.\)
   

2. If \(x={{\sqrt{3}-\sqrt{2}}\over{\sqrt{3}+\sqrt{2}}}\) and \(y={{\sqrt{3}+\sqrt{2}}\over{\sqrt{3}-\sqrt{2}}}\) find the value of x²+xy+y².

1. Prove that \(cos({ 30 }^{ 0 }-A)cos({ 30 }^{ 0 }+A)cos({ 45 }^{ 0 }-A)cos(45^{ 0 }+A)=cos2A+\frac { 1 }{ 4 } \)

2. Prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) ...\left( 1+\frac { 1 }{ n } \right) =\left( n+1 \right) \) for all \(n\in N\) by the principle of mathematical induction.

1. Compute the sum of first n terms of the following series 6 + 66 + 666 + .......

2. Consider a hollow cylindrical vessel, with circumference 24 cm and height 10 cm. An ant is located on the outside of vessel 4 cm from the bottom. There is a drop of honey at the diagrammatically opposite inside of the vessel, 3 cm from the top. 
   (i) What is the shortest distance the ant would need to crawl to get the honey drop?
   (ii) Equation of the path traced out by the ant.
   (iii) Where the ant enter in to the cylinder? Here is a picture that illustrates the position of the ant and the honey.
   

1. Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos \(\alpha\) + y sin \(\alpha\) = p is \(\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}\) where p is a constant.

2. Compute all minors, cofactors of A and hence compute |A| if A =\(\begin{bmatrix} 1& 3 &-2 \\4 & -5 &6 \\ -3 & 5 & 2 \end{bmatrix}\) .
   Also check that | A | remains unaltered by expanding along any row or any column.

1. Evaluate the following limits :
   \(lim_{x\rightarrow a}{\sqrt{x-b}-\sqrt{a-b}\over x^2-a^2}(a>b)\)

2. If y = \((cos^{-1}x)^2\) ,prove that \((1-x^2){d^2y\over dx^2}-x{dy\over dx}-2=0.\)  Hence find y₂ when x = 0