The relation R defined on a set A= {0,-1, 1, 2} by xRy if |x²+y²| ≤ 2, then which one of the following is true?

(a)

R = {(0,0), (0,-1), (0, 1), (-1, 0), (-1, 1), (1, 2), (1, 0)}

(b)

R^-1 = {(0,0), (0,-1), (0, 1), (-1, 0), (1, 0)}

(c)

Domain of R is {0,-1, 1, 2}

(d)

Range of R is {0,-1, 1}

Domain of the function \(y={x-1\over x+1}\) is __________

(a)

1R

(b)

Q

(c)

R-(-1)

(d)

R-1

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

(a)

0.5

(b)

2.5

(c)

1.5

(d)

1.25

\(\sqrt [ 4 ]{ { \left( -2 \right) }^{ 4 } } \times { \left( -1000 \right) }^{ \frac { 1 }{ 3 } }\) is ___________

(a)

20

(b)

-20

(c)

2^-10

(d)

100

Which of the following is not true?

(a)

​​​​​​sinፀ = \(-\frac { 3 }{ 4 } \)

(b)

cosፀ = -1

(c)

tanፀ = 25

(d)

secፀ = \(\frac { 1 }{ 4 } \)

If A, B, C are in A.P and B = \(\frac{\pi}{4}\) then tan A tan B tan C = _______________

(a)

1

(b)

-1

(c)

0

(d)

None

If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

(a)

45

(b)

40

(c)

10!

(d)

2¹⁰

is _________

(a)

\(\lfloor{n}(n+2)\)

(b)

(c)

(d)

none of these

\(\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...\) is ______________

(a)

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

(b)

\(\frac{e+e^{-1}}{2}\)

(c)

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

(d)

none of these

The locus of a point which moves such that it maintains equal distance from the fixed point is a ______________

(a)

straight line

(b)

line bisector

(c)

circle

(d)

angle bisector

The image of the point (1, 2) with respect to the line y = x is ______________

(a)

(-1, -2)

(b)

(2, 1)

(c)

(2, -1)

(d)

(2, 1)

What must be the matrix X, if 2x +\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

(a)

\(\begin{bmatrix} 1& 3 \\ 2 &-1 \end{bmatrix}\)

(b)

\(\begin{bmatrix} 1& -3 \\ 2 &-1 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 2& 6 \\ 4 &-2 \end{bmatrix}\)

(d)

\(\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}\)

The value of x, for which the matrix A = \(\begin{bmatrix} e^{x-2}& e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}\) is singular

(a)

9

(b)

8

(c)

7

(d)

6

If \(\begin{bmatrix} 4 & 3 \\ -2 & x \end{bmatrix}\) is singular then the value of x is _____________

(a)

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

(b)

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

(c)

3

(d)

-2

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

(a)

\(\overrightarrow { a } =\overrightarrow { b } \)

(b)

\(\overrightarrow { a } =\overrightarrow { -b } \)

(c)

\(\overrightarrow { a } =\pm \overrightarrow { b } \)

(d)

both are null vectors

Assertion (A) : \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are the position vector three collinear points then 2 \(\overset { \rightarrow }{ a }=\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)
Reason (R): Collinear points, have same direction

(a)

Both A and R are true and R is the correct explanation of A

(b)

Both A and R are true and R is not a correct explantion of A

(c)

A is true but R is false

(d)

A is false but R is true

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

(a)

1

(b)

-1

(c)

0

(d)

2

The rate of change of area A of a circle of radius r is

(a)

\(2\pi r\)

(b)

\(2\pi r\frac { dr }{ dt } \)

(c)

\(\pi { r }^{ 2 }\frac { dr }{ dt } \)

(d)

\(\pi \frac { dr }{ dt } \)

If \(f(x)=\left\{\begin{array}{l} x+1, \quad \text { when } x<2 \\ 2 x-1 \text { when } x \geq 2 \end{array}\right.\), then f'(2) is

(a)

0

(b)

1

(c)

2

(d)

does not exist

 Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } \) then \(\frac { dy }{ dx } \) is _____

(a)

1

(b)

\(\pi \)

(c)

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

(d)

0

Let A={1,2,3,4} and B = {a,b,c,d}. Give a function from A\(\rightarrow\)B for each of the following:
neither one- to -one and nor onto.

Given log₂¹⁶ = 4. Find log₁₆²

For each given Angle, find a coterminal angle with a measure of \(\theta\) such that \(0^o\le \theta \le 360°\) 
395⁰ 

In how many ways can the letters of the word PENCIL be arranged so that N is always next to E.

Write the first 6 terms of the exponential series \({ e }^{ \frac { 1 }{ 2 } x }\)

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

Find |A| if A = \(\begin{bmatrix} 0& sin \alpha &cos \alpha \\ sin \alpha & 0 & sin \beta \\ cos \alpha & -sin\beta & 0 \end{bmatrix}\).

Find the value of x such that [1 \(\times\) 1]\(\left[ \begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ x \end{matrix} \right] =0\)

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { { e }^{ 5x }-1 }{ x } \)

Differentiate \(\sin { (\sqrt { 3 } \sin { x } +\cos { x } ) } \) with respect to x.

Compare and contrast the graph y = x² - 1, y = 4(x² - 1) and y = (4x)² = 1.

Find the zeros of the polynomial function f(x) = 9x²- 36

In \(\triangle\)ABC, Prove the following
\(\frac { a+b }{ a-b } =tan\left( \frac { A+B }{ 2 } \right) cot\left( \frac { A-B }{ 2 } \right) \)

If p(h) is the statement "n² + n is even" and if p(r) is true, then p(r + 1) is true.

Prove that \(\sqrt [ 3 ]{ { x }^{ 3 }+6 } -\sqrt [ 3 ]{ { x }^{ 3 }+3 } \) is approximately equal to \(\frac { 1 }{ { x }^{ 2 } } \) when x is sufficiently large.

Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1).

Prove that \(\left| \begin{matrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{matrix} \right| \) = 4(a + b)(b + c)(c + a). Using factor theorem.

Let \(\overrightarrow { a } ,\overrightarrow { b } \) and \(\overrightarrow { c } \)  be non-coplanar vectors. Let A, B and C be the points whose position vectors with respect to the origin O are \(\overrightarrow { a } +2\overrightarrow { b } +3\overrightarrow { c } ,-2\overrightarrow { a } +3\overrightarrow { b } +5\overrightarrow { c } \) and \(7\overrightarrow { a } -\overrightarrow { c } \)  respectively. Then prove that A, B and C are collinear.

Examine the continuity of \(f\left( x \right) =\begin{cases} \frac { \sin { 2x } }{ \sin { 3x } } \quad if\quad x\neq 0 \\ 2\quad \quad \quad if\quad x=0 \end{cases}at\quad x=0\)

If xy = 4, Prove that \(x\left( \frac { dy }{ dx } +{ y }^{ 2 } \right) =3y.\)

1. Show that the statement, "if f and g o f are one-to-one, then g is one-to-one" is not true.

2. Resolve the following rational expressions into partial fractions.
   \({{1}\over{x^4-1}}\)

1. Find the derivatives of the following functions with respect to corresponding independent variables : y = x sin x cos x

2. Differentiate \({ tan }^{ -1 }(secx+tanx),\) \(-\frac{\pi}{ 2 }\) with respect to 'x'.
   

1. Find the equation of the straight lines passing through (8, 3) and having intercepts whose sum is 1.

2. If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p, q, and r.

1. For any natural number n, 7ⁿ - 2ⁿ is divisible by 5.

2. If a, b, c are respectively the p^th q^th and r^th terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

1. The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), 2 \(\hat{i}\) + 5\(\hat{j}\), 3\(\hat{i}\) + 2\(\hat{j}\) - 3\(\hat{k}\), and \(\hat{i}\) - 6\(\hat{j}\) - \(\hat{k}\), respectively. Prove that the line PQ and RS are parallel.

2. Determine k, so that  \(f\left( x \right) =\begin{cases} k{ x }^{ 2 },\quad x\le 2 \\ 3,\quad x>2 \end{cases}\) is continuous.

1. Using the mathematical induction, show that for any natural number n > 2
   \({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)

2. If ²²p_r+1: ²⁰p_r+2 = 11 : 52, find r.

1. Resolve into partial fractions \(\frac { 9 }{ (x-1)(x+2)^{ 2 } } \)

2. Suppose two radar stations located 100 km apart, each detect a fighter aircraft between them. The angle of elevation measured by the first station is 30°, whereas the angle of elevation measured by the second station is 45°. Find the altitude of the aircraft at that instant.