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 domain of cos^-1(x² - 4) is______

(a)

[3, 5]

(b)

[-1, 1]

(c)

\(\left[ -\sqrt { 5 } ,-\sqrt { 3 } \right] \cup \left[ \sqrt { 3 } ,\sqrt { 5 } \right] \)

(d)

[0, 1]

The auxiliary circle of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 16 } \) = 1 is __________

(a)

x² + y² = 25

(b)

x² + y² = 16

(c)

x² + y² = 41

(d)

x² + y² = 5

The locus of the point of intersection of perpendicular tangents of the parabola y² = 4ax is

(a)

latus rectum

(b)

directrix

(c)

tangent at the vertex

(d)

axis of the parabola

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

The unit normal vector to the plane 2x + 3y + 4z = 5 is _______________

(a)

\(\frac { 2 }{ \sqrt { 29 } } \overset { \wedge }{ i } +\frac { 3 }{ \sqrt { 29 } } \overset { \wedge }{ j } +\frac { 4 }{ \sqrt { 29 } } \overset { \wedge }{ k } \)

(b)

\(\frac { 2 }{ \sqrt { 29 } } \overset { \wedge }{ i } -\frac { 3 }{ \sqrt { 29 } } \overset { \wedge }{ j } +\frac { 4 }{ \sqrt { 29 } } \overset { \wedge }{ k } \)

(c)

\(\frac { 2 }{ \sqrt { 29 } } \overset { \wedge }{ i } -\frac { 3 }{ \sqrt { 29 } } \overset { \wedge }{ j } -\frac { 4 }{ \sqrt { 29 } } \overset { \wedge }{ k } \)

(d)

\(\frac { 2 }{ 5 } \overset { \wedge }{ i } +\frac { 3 }{ 5 } \overset { \wedge }{ j } +\frac { 4 }{ 5 } \overset { \wedge }{ k } \)

For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

(a)

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

(b)

\(\pi\)

(c)

0

(d)

2

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is

(a)

1

(b)

2

(c)

3

(d)

4

Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q

Which one of the following statements has truth value F?

(a)

Chennai is in India or \(\sqrt 2\) is an integer

(b)

Chennai is in India or \(\sqrt 2\) is an irrational number

(c)

Chennai is in China or \(\sqrt 2\) is an integer

(d)

Chennai is in China or \(\sqrt 2\) is an irrational number