If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) = 

(a)

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

(b)

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

(c)

\(\frac { 1 }{ 4 } \)

(d)

1

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 x^a y^b = e^m, x^c y^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

(a)

e^{(Δ₂  / Δ₁)}, e^{(Δ₃ / Δ₁)}

(b)

log (Δ₁ / Δ₃), log (Δ₂ / Δ₃)

(c)

log (Δ₂ / Δ₁), log(Δ₃ / Δ₁)

(d)

e^{(Δ₁ / Δ₃)},e^{(Δ₂ / Δ₃)}

If |z - 2 + i | ≤ 2, then the greatest value of |z| is

(a)

\(\sqrt { 3 } -2\)

(b)

\(\sqrt { 3 } +2\)

(c)

\(\sqrt { 5 } -2\)

(d)

\(\sqrt { 5 } +2\)

If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

(a)

1

(b)

2

(c)

3

(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

The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x² + y² − 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to

(a)

\(0,-\frac { 40 }{ 9 } \)

(b)

0

(c)

\(\frac { 40 }{ 9 } \)

(d)

\(\frac { -40 }{ 9 } \)

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is

(a)

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

(b)

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

(c)

\(\frac { \pi }{ 4 } \)

(d)

\( { \pi }\)

If the planes \(\vec { r } .(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } .(4\hat{i}+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are

(a)

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

(b)

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

(c)

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

(d)

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

If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\), then the value of λ is

(a)

\(2\sqrt { 3 } \)

(b)

\(3\sqrt { 2 } \)

(c)

0

(d)

1

The number given by the Rolle's theorem for the functlon x³ - 3x², x ∈ [0, 3] is

(a)

1

(b)

\(\sqrt { 2 } \)

(c)

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

(d)

2

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

(a)

2

(b)

2.5

(c)

3

(d)

3.5

If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

(a)

xy + yz + zx

(b)

x(y + z)

(c)

y(z + x)

(d)

0

If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

(a)

z - x

(b)

y - z

(c)

x - z

(d)

y - x

If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is

(a)

3

(b)

6

(c)

9

(d)

5

The value of \(\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx\) is

(a)

\(\frac { { \pi }^{ 2 } }{ 4 } -1\)

(b)

\(\frac { { \pi }^{ 2 } }{ 4 } +2\)

(c)

\(\frac { { \pi }^{ 2 } }{ 4 } +1\)

(d)

\(\frac { { \pi }^{ 2 } }{ 4 } -2\)

The number of arbitrary constants in the general solutions of order n and n +1 are respectively

(a)

n-1,n

(b)

n,n+1

(c)

n+1,n+2

(d)

n+1,n

The number of arbitrary constants in the particular solution of a differential equation of third order is

(a)

3

(b)

2

(c)

1

(d)

0

Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

(a)

9

(b)

8

(c)

6

(d)

3

Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

Write \(\frac { 3+4i }{ 5-12i } \) in the x + iy form, hence find its real and imaginary parts.

Find the modulus of the following complex numbers
\(\frac { 2i }{ 3+4i } \)

If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)

Evaluate the following if z = 5−2i and w = −1+3i
2z + 3w

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)

Show that the equation x⁹- 5x⁵+ 4x⁴+ 2x²+ 1 = 0 has atleast 6 imaginary solutions.

If w (x, y, z) = x² + y² + y², x = e^t, y = e^t sin t, z = e^t cos t, find \(\frac{dw}{dt}\)

Let z(x, y) = x³ - 3x²y³, where x = se^t, y = se^-t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \)

Evaluate the following
\(\int _{ 0 }^{ \pi /4 }{ { sin}^{ 6}2x\ dx } \)

Solve the differential equation:
x cos y dy = e^x(x log x + 1)dx

The probability density function of X is 
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.2 ≤ X< 0.6) 

If the probability mass function f(x) of a random variable X is

x	1	2	3	4
f (x)	\(\cfrac { 1 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 1 }{ 12 } \)

find (i) its cumulative distribution function, hence find
(ii) P(X ≤ 3) and,
(iii) P(X ≥ 2)

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.

Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\frac { \pi }{ 3 } \) and x- axis are as 2 : 3.

Solve : \(\left( 1+{ x }^{ 2 } \right) \frac { dy }{ dx } -x={ 2tan }^{ -1 }x\)