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

The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

(a)

0

(b)

1

(c)

2

(d)

infinitely many

In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________

(a)

trivial solution

(b)

only non - trivial solution

(c)

no solution

(d)

trivial solution and infinitely many non - trivial solutions

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)

\(\sqrt { 5 } +2\)

The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \)  = 1 lies on _________

(a)

circle x²+ y² = 1

(b)

x-axis

(c)

y-axis

(d)

the lines x+y = 1

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

\(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

(a)

0

(b)

n

(c)

< n

(d)

r

The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________

(a)

no solution

(b)

one solution

(c)

two solution

(d)

more than one solution

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) 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 number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi  is ___________

(a)

0

(b)

1

(c)

2

(d)

infinte

If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________

(a)

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

(b)

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

(c)

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

(d)

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

The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

(a)

10

(b)

\( {2} \sqrt {5}\)

(c)

6

(d)

4

Equation of tangent at (-4, -4) on x² = -4y is _____________

(a)

2x - y + 4 = 0

(b)

2x + y - 4 = 0

(c)

2x - y - 12 = 0

(d)

2x + y + 4 = 0

If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

(a)

\(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)

(b)

\(\frac{1}{3}\)\(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)

(c)

1

(d)

-1

If  \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _______________

(a)

45^o

(b)

60^o

(c)

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

(d)

cos^-1 \(\left( \frac { 2 }{ 7 } \right) \)

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

(a)

4

(b)

\(2\sqrt { 3 } \)

(c)

\(4\sqrt { 3 } \)

(d)

\(5\sqrt { 3 } \)

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

(a)

1

(b)

-1

(c)

0

(d)

∞

The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

(a)

\(\frac{1}{31}\)

(b)

\(\frac15\)

(c)

5

(d)

31

The cube root of 127 is ............

(a)

5.026

(b)

5.26

(c)

5.028

(d)

5.075

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

(a)

10

(b)

5

(c)

8

(d)

9

The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is

(a)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0\)

(b)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(c)

\(\frac { { d }y }{ { dx } } +y=0\)

(d)

\(\frac { { d }y }{ { dx } } -y=0\)

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)

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

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

The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

(a)

Q⁺

(b)

Z

(c)

R

(d)

C