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

If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________

(a)

a² + b² + c² = 1

(b)

abc ≠ 1

(c)

a + b + c =0

(d)

a² + b² + c² + 2abc =1

If A is a non-singular matrix then IA^-1| = ______

(a)

\(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)

(b)

\(\frac { 1 }{ |A^{ 2 }| } \)

(c)

\(\left| \frac { 1 }{ A } \right| \)

(d)

\(\frac { 1 }{ |A| } \)

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

(a)

1+ i

(b)

i

(c)

1

(d)

0

.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

(a)

Vertices of a right angled triangle

(b)

Vertices of an equilateral triangle

(c)

Vertices of an isosceles

(d)

Collinear

The amplitude of \(\frac{1}{i}\) is equal to _______

(a)

0

(b)

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

(c)

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

(d)

\(\pi \)

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_o g)(x), then the degree of h is

(a)

mn

(b)

m+n

(c)

mⁿ

(d)

n^m

If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

(a)

0, 2

(b)

0, 1

(c)

0, 3

(d)

0, √3

\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for

(a)

\(-\pi \le x\le 0\)

(b)

\(0 \le x\le \pi\)

(c)

\(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

(d)

\(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

(a)

\(\frac { \pi }{ 6 } \)

(b)

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

(c)

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

(d)

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

\(cot\left( \frac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

(a)

7

(b)

6

(c)

5

(d)

none

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 latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________

(a)

\(\frac { 98 }{ 6 } \)

(b)

\(\frac { 72 }{ 7 } \)

(c)

\(\frac { 72 }{ 14 } \)

(d)

\(\frac { 98 }{ 12 } \)

The angle between the tangents drawn from (1, 4) to the parabola y² = 4x is __________

(a)

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

(b)

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

(c)

\(\frac { \pi }{ 5 } \)

(d)

\(\frac { \pi }{ 5 } \)

If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then

(a)

\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 1

(b)

\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = -1

(c)

\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 0

(d)

\([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 2

Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let  \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________

(a)

0

(b)

1

(c)

2

(d)

3

If \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are any three vectors, then \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \) if and only if __________

(a)

\(\overset { \rightarrow }{ b } \), \(\overset { \rightarrow }{ c } \) are collinear

(b)

\(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) are collinear

(c)

\(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are collinear

(d)

none

The straight lines \(\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 } \) and \(\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 } \) are _____________

(a)

parallel

(b)

perpendicular

(c)

inclined at 45^o

(d)

none

Let \(\overset { \rightarrow }{ u } ,\overset { \rightarrow }{ v } ,\overset { \rightarrow }{ w } \) be vectors such that \(\overset { \rightarrow }{ u } +\overset { \rightarrow }{ v } +\overset { \rightarrow }{ w } =\overset { \rightarrow }{ 0 } \). If \(\left| \overset { \rightarrow }{ u } \right| \) = 3, \(\left| \overset { \rightarrow }{ v } \right| \) = 4, \(\left| \overset { \rightarrow }{ w } \right| \) = 5  then \(\overset { \rightarrow }{ u } .\overset { \rightarrow }{ v } +\overset { \rightarrow }{ v } .\overset { \rightarrow }{ w } +\overset { \rightarrow }{ w } .\overset { \rightarrow }{ u } \) is ______________

(a)

25

(b)

-25

(c)

5

(d)

\(\sqrt { 5 } \)

If the slope of the curve 2y² = ax²+b at (1,-1) is - 1, then the values of a, b is __________

(a)

2, 0

(b)

0, 2

(c)

0, 0

(d)

2, 2

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 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 solution of the differential equation \(\frac { dy }{ dx } =2xy\) is

(a)

y = Ce^_x2

(b)

y = 2x² + C

(c)

y = Ce^_−x2 + C

(d)

y = x² + C

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.

(a)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 3 } \)

(b)

\( \frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 6 } \)

(c)

\(l,\frac { { l }^{ 2 } }{ 12 } \)

(d)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 12 } \)

Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q