If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T = 

(a)

A

(b)

B

(c)

I₃

(d)

B^T

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

(a)

\(\cfrac { 1 }{ i+2 } \)

(b)

\(\cfrac { -1 }{ i+2 } \)

(c)

\(\cfrac { -1 }{ i-2 } \)

(d)

\(\cfrac { 1 }{ i-2 } \)

If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to 

(a)

0

(b)

1

(c)

2

(d)

3

According to the rational root theorem, which number is not possible rational zero of 4x⁷ + 2x⁴ - 10x³ - 5?

(a)

-1

(b)

\(\frac { 5 }{ 4 } \)

(c)

\(\frac { 4 }{ 5 } \)

(d)

5

The polynomial x³ - kx² + 9x has three real zeros if and only if, k satisfies

(a)

|k| ≤ 6

(b)

k = 0

(c)

|k| > 6

(d)

|k| ≥ 6

\(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) is equal to

(a)

2\(\pi\)

(b)

\(\pi\)

(c)

0

(d)

tan^-1\(\frac{12}{65}\)

If sin⁻¹x = 2sin⁻¹ \(\alpha\) has a solution, then

(a)

\(|\alpha |\le \frac { 1 }{ \sqrt { 2 } } \)

(b)

\(|\alpha |\ge \frac { 1 }{ \sqrt { 2 } } \)

(c)

\(|\alpha |<\frac { 1 }{ \sqrt { 2 } } \)

(d)

\(|\alpha |>\frac { 1 }{ \sqrt { 2 } } \)

The circle x² + y² = 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85 < m < −35

(d)

−35 < m < 15

The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).

(a)

\(\frac { 6 }{ 5 } \)

(b)

\(\frac { 5 }{ 3 } \)

(c)

\(\frac { 10 }{ 3 } \)

(d)

\(\frac { 3 }{ 5 } \)

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

(a)

\(\vec { a } \)

(b)

\(\vec { b} \)

(c)

\(\vec { c } \)

(d)

\(\vec { 0 } \)

If \([\vec{a}, \vec{b}, \vec{c}]=1\), then the value of \(\frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) \cdot \vec{c}}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) \cdot \vec{a}}\) is

(a)

1

(b)

-1

(c)

2

(d)

3

A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

(a)

2

(b)

2.5

(c)

3

(d)

3.5

The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is 

(a)

(4, 11)

(b)

(4, -11)

(c)

(-4, 11)

(d)

(-4,-11)

If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to

(a)

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

(b)

2xu

(c)

x²u

(d)

y²u

If v (x, y) = log (e^x + e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to

(a)

e^x + e^y

(b)

\(\frac{1}{e^x + e^y}\)

(c)

2

(d)

1

If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is 

(a)

10

(b)

5

(c)

8

(d)

9

The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

(a)

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

(b)

\(\frac{2}{9}\)

(c)

\(\frac{1}{9}\)

(d)

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

The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

(a)

1, 2

(b)

2, 2

(c)

1, 1

(d)

2, 1

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

6

(b)

4

(c)

3

(d)

2

If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

(a)

0 and 12

(b)

5 and 17

(c)

7 and 19

(d)

16 and 24

Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q

Which one of the following is a binary operation on N?

(a)

Subtraction

(b)

Multiplication

(c)

Division

(d)

All the above

In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

(a)

1

(b)

2

(c)

3

(d)

4

If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 12, then the value of |z₁+z₂+z₃| is 

(a)

1

(b)

2

(c)

3

(d)

4

If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

(a)

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

(b)

1

(c)

2

(d)

3

If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is

(a)

1

(b)

i

(c)

x²+y²

(d)

1+n²

The volume of solid of revolution of the region bounded by y² = x(a − x) about x-axis is

(a)

\(\pi a^3\)

(b)

\(\frac { \pi { a }^{ 3 } }{ 4 } \)

(c)

\(\frac { \pi { a }^{ 3 } }{ 5 } \)

(d)

\(\frac { \pi { a }^{ 3 } }{ 6 } \)

If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is

(a)

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

(b)

2

(c)

1

(d)

\(\frac{3}{4}\)

The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is

(a)

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

(b)

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

(c)

0

(d)

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

Which one of the following statements has the truth value T?

(a)

sin x is an even function

(b)

Every square matrix is non-singular

(c)

The product of complex number and its conjugate is purely imaginary

(d)

\(\sqrt 5\) is an irrational number

Which one is the inverse of the statement (pVq)➝(pΛq)?

(a)

(p∧q)➝(pVq)

(b)

ᄀ(pvq)➝(p∧q)

(c)

(ᄀpvᄀq)➝(ᄀp∧ᄀq)

(d)

(ᄀp∧ᄀq)➝(ᄀpVᄀq)

Which one of the following is not true?

(a)

Negation of a negation of a statement is the statement itself

(b)

If the last column of the truth table contains only T then it is a tautology.

(c)

If the last column of its truth table contains only F then it is a contradiction

(d)

If p and q are any two statements then p↔️q is a tautology.

If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I₂, then B = 

(a)

\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A\)

(b)

\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }\)

(c)

\(\left( \cos ^{ 2 }{ \theta } \right) I\)

(d)

(Sin²\(\frac { \theta }{ 2 } \))A

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

(a)

1

(b)

2

(c)

4

(d)

3

If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 = 

(a)

\(\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right] \)

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

(a)

3

(b)

-1

(c)

1

(d)

9

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

(a)

2ab

(b)

ab

(c)

\( \sqrt{ ab}\)

(d)

\(\frac { a }{ b } \)

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

(a)

a parabola

(b)

a hyperbola

(c)

an ellipse

(d)

a circle

Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct 

(a)

both mean and variance exist

(b)

mean exists but variance does not exist

(c)

both mean and variance do not exist

(d)

variance exists but Mean does not exist

A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

(a)

\(\frac{3}{25} \text { radians } / \mathrm{sec}\)

(b)

\(\frac{4}{25} \text { radians } / \mathrm{sec}\)

(c)

\(\frac{1}{5} \text { radians } / \mathrm{sec}\)

(d)

\(\frac{1}{3} \text { radians } / \mathrm{sec}\)

Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is

(a)

\(\cfrac { 19 }{ 6 } \)

(b)

\(-\cfrac { 19 }{ 6 } \)

(c)

\(\cfrac { 3 }{ 2 } \)

(d)

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

The random variable X has the probability density function 
\(f(x)=\left\{\begin{array}{lr} a x+b & 0<x<1 \\ 0 & \text { otherwise } \end{array}\right.\) and \(E(X)=\frac { 7 }{ 12 } \), then a and b are respectively

(a)

1 and \(\frac { 1 }{ 2 } \)

(b)

\(\frac { 1 }{ 2 } \) and 1

(c)

2 and 1

(d)

1 and 2

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

(a)

\(\frac { 57 }{ { 20 }^{ 3 } } \)

(b)

\(\frac { 57 }{ { 20 }^{ 2 } } \)

(c)

\(\frac { { 19 }^{ 3 } }{ { 20 }^{ 3 } } \)

(d)

\(\frac { 57 }{ 20 } \)

The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is

(a)

-2

(b)

-1

(c)

1

(d)

2

The value of \(\left( \cfrac { 1+\sqrt { 3 } i}{ 1-\sqrt { 3}i } \right) ^{ 10 }\) is

(a)

\(cis\cfrac { 2\pi }{ 3 } \)

(b)

\(cis\cfrac { 4\pi }{ 3 } \)

(c)

\(-cis\cfrac { 2\pi }{ 3 }\)

(d)

\(-cis\cfrac { 4\pi }{ 3 }\)

The general solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) is

(a)

e^x + e^y = C

(b)

e^x + e^-y = C

(c)

e^-x + e^y = C

(d)

e^-x + e^-y = C

P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then

(a)

P = Ce^kt

(b)

P = Ce^-kt

(c)

P = Ckt

(d)

Pt = C

The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is

(a)

y = x³ + 2

(b)

y = 3x² + 4

(c)

y = 3x⁴ + 4

(d)

y = 3x² + 5