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

(a)

3

(b)

4

(c)

2

(d)

5

 The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is

(a)

1+ i

(b)

i

(c)

1

(d)

0

The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

(a)

2

(b)

4

(c)

1

(d)

∞

The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

(a)

[1, 2]

(b)

[-1, 1]

(c)

[0, 1]

(d)

[-1, 0]

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

(a)

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

(b)

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

(c)

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

(d)

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

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

If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to

(a)

2

(b)

-1

(c)

1

(d)

0

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

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

If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

(a)

x^y log x

(b)

y log x

(c)

yx^y-1

(d)

x log y

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 \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)

(a)

cos x - x sin x

(b)

sin x + x cos x

(c)

x cos x

(d)

x sin x

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

(a)

10

(b)

5

(c)

8

(d)

9

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

(a)

3

(b)

2

(c)

1

(d)

0

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 in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is

(a)

0.125

(b)

0.25

(c)

0.375

(d)

0.75

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

The proposition p ∧ (¬p ∨ q) is

(a)

a tautology

(b)

a contradiction

(c)

logically equivalent to p ∧ q

(d)

logically equivalent to p ∨ q

If adj A = \(\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), find A⁻¹.

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

Find the modulus and principal argument of the following complex numbers:
\(-\sqrt { 3 } -i\)

Find the principal value of
 \({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

If y = 4x + c is a tangent to the circle x² + y² = 9, find c 

Prove that \((\vec { a } .(\vec { b } \times \vec { c } ))\vec { a } =(\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )\)

A rectangular page is to contain 24 cm² of print. The margins at the top and bottom of the page are 1.5 cm and the margins at other sides of the page is 1 cm. What should be the dimensions of the page so that the area of the paper used is minimum.

Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

If \(\frac { z+3 }{ z-5i } =\frac { 1+4i }{ 2 } \), find the complex number z in the rectangular form

If z = 2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\frac { 2\pi }{ 3 } \).

Solve the equations
x⁴+ 3x³- 3x - 1 = 0

A circle of area 9π square units has two of its diameters along the lines x + y = 5 and x−y = 1. Find the equation of the circle.

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.

Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z  = −1

Solve the equation (x-2) (x-7) (x-3) (x+2)+19 = 0

Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .

If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

Evaluate: \(\\ \\ \int _{ -1 }^{ 1 }{ { e }^{ -\lambda x }(1-{ x }^{ 2 }) } dx\)

A random variable X has the following probability mass function

x	1	2	3	4	5	6
f(x)	k	2k	6k	5k	6k	10k

Find
(i) P(2 < X < 6)
(ii) P(2 ≤ X < 5)
(iii) P(X ≤4)
(iv) P(3 < X )