If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A = 

(a)

\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right] \)

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

(a)

Only (i)

(b)

(ii) and (iii)

(c)

(iii) and (iv)

(d)

(i), (ii) and (iv)

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

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to

(a)

\(\frac { 1 }{ 2 } \ { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

(b)

\(\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

(c)

\(\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

(d)

\({ tan}^{ -1 }\left( \frac { 1}{ 2 } \right) \)

The centre of the circle inscribed in a square formed by the lines x² − 8x − 12 = 0 and y² − 14y + 45 = 0 is

(a)

(4, 7)

(b)

(7, 4)

(c)

(9, 4)

(d)

(4, 9)

The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is

(a)

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

(b)

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

(c)

\(\frac { 1 }{ 3\sqrt { 2 } } \)

(d)

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

If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to

(a)

81

(b)

9

(c)

27

(d)

18

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)

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

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

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

(a)

\(\frac { 11 }{ 243 } \)

(b)

\(\frac { 3 }{ 8 } \)

(c)

\(\frac { 1 }{ 243 } \)

(d)

\(\frac { 5 }{ 243 } \)