If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

(a)

\(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________

(a)

sinx cosx

(b)

1

(c)

2

(d)

none

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

(a)

unique solution

(b)

one parameter family of solution

(c)

two parameter family of solutions

(d)

in consistent

If A = [2 0 1] then the rank of AA^T is ______

(a)

1

(b)

2

(c)

3

(d)

0

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

(a)

\(\cfrac { 1 }{ i+2 } \)

(b)

\(\cfrac { -1 }{ i+2 } \)

(c)

\(\cfrac { -1 }{ i-2 } \)

(d)

\(\cfrac { 1 }{ i-2 } \)

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

(a)

|z| = 1, arg(z) =\(\frac { \pi }{ 4 } \)

(b)

|z| = 1, arg(z) = \(\frac { \pi }{ 6 } \)

(c)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg(z) = \(\frac { 5\pi }{ 24 } \)

(d)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg (z) = tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________

(a)

\(\frac { 1 }{ 13 } \)

(b)

\(\frac { 1 }{ 5} \)

(c)

\(\frac { 1 }{ 12 } \)

(d)

none of these

If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

(a)

-2i sin(α - β)

(b)

2i sin(α - β)

(c)

2 cos(α - β)

(d)

-2 cos(α - β)

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

(a)

a ≥ 0

(b)

a > 0

(c)

a < 0

(d)

a ≤ 0

For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

(a)

0

(b)

1

(c)

4

(d)

3

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 tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________ 

(a)

5

(b)

\(\frac { 1 }{ 5 } \)

(c)

\(\frac { 5 }{ 14 } \)

(d)

\(\frac { 14 }{ 5 } \)

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

(a)

\(\frac { 19 }{ 8 } \)

(b)

\(\frac { 8 }{ 19 } \)

(c)

\(\frac { 19 }{ 12 } \)

(d)

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

The equation of the normal to the circle x² + y² − 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

(a)

x + 2y = 3

(b)

x + 2y + 3 = 0

(c)

2x + 4y + 3 = 0

(d)

x − 2y + 3 = 0

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

The area of the circle (x - 2)² + (y - k)² = 25 is _________

(a)

25ㅠ

(b)

5ㅠ

(c)

10ㅠ

(d)

25

If t₁ and t₂ are the extremities of any focal chord of y² = 4ax then t₁t₂ is ______________

(a)

-1

(b)

0

(c)

±1

(d)

\(\frac12\)

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

(a)

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

(b)

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

(c)

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

(d)

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

The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________

(a)

30^o

(b)

60^o

(c)

45^o

(d)

90^o

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) then \(\overset { \rightarrow }{ a } +\left( -\overset { \rightarrow }{ b } \right) \) will be perpendiculur to \(\overset { \rightarrow }{ c } \) only when t = _________________

(a)

5

(b)

4

(c)

3

(d)

\(\frac { 7 }{ 3 } \)

The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________

(a)

\(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

(b)

4 \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \)

(c)

\(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }-{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

(d)

4 \({ \left| \overset { \rightarrow }{ a } \right| }^{ 2 }{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 }\)

If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is _____________

(a)

土 \(\frac { 1 }{ 3 } \)

(b)

土 \(\frac { 1 }{ 4 } \)

(c)

土 \(\frac { 1 }{ 9 } \)

(d)

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

The unit normal vectors to the plane 2x - y + 2z = 5 are ________________

(a)

\(\overset { \wedge }{ 2i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \)

(b)

\(\frac { 1 }{ 3 } \left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \right) \)

(c)

\(-\frac { 1 }{ 3 } \left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \right) \)

(d)

\(\pm \frac { 1 }{ 3 } \left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \right) \)

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)

\(5\sqrt { 3 } \)