If x < 0, y < 0 such that xy = 1, then tan^-1(x) + tan^-1(y) =_____

(a)

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

(b)

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

(c)

\(-\pi \)

(d)

none

The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________

(a)

x² - y² = 32

(b)

y² - x² = 32

(c)

x² - y² = 16

(d)

y² - x² = 16

If B, B¹ are the ends of minor axis, F₁, F₂ are foci of the ellipse \(\frac { { x }^{ 2 } }{ 8 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 then area of F₁BF₂B¹ is __________

(a)

16

(b)

8

(c)

16\(\sqrt2\)

(d)

32\(\sqrt2\)

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

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

(a)

1

(b)

2

(c)

3

(d)

\(\infty\)

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 the curves y = 2e^x and y = ae^-x intersect orthogonally, then a = _________

(a)

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

(b)

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

(c)

2

(d)

2e²

The function -3x+12 is ________ function on R.

(a)

decreasing

(b)

strictly decreasing

(c)

increasing

(d)

strictly increasing

The approximate value of (627)^\(\frac14\) is ................

(a)

5.002

(b)

5.003

(c)

5.005

(d)

5.004

The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is __________

(a)

4

(b)

8\(\frac23\)

(c)

13

(d)

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

The area bounded by the parabola y = x² and the line y = 2x is __________

(a)

\(\frac43\)

(b)

\(\frac23\)

(c)

\(\frac{51}{3}\)

(d)

\(\frac{30}{3}\)

The area of the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) __________

(a)

6π

(b)

36π

(c)

6π²

(d)

36π²

The solution of \(\frac{dy}{dx}+y\) cot x = sin 2x is ___________

(a)

y sin x = \(\frac{2}{3}\)sin³x+c

(b)

y sec x = \(\frac{x^2}{2}+c\)

(c)

y sin x = c+x

(d)

2y sin x = sin x - \(\frac{sin\ 3x}{3}+c\)

The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________

(a)

\(y={ e }^{ \frac { x }{ 2 } }\left[ A\ cos\frac { x }{ 2 } +B\ sin\frac { x }{ 2 } \right] \)

(b)

\(y={ e }^{ \frac { x }{ 2 } }\left[ A\ cos\frac { x }{ 2 } -B\ sin\frac { x }{ 2 } \right] \)

(c)

\(y=Acos\frac { x }{ 2 } +Bsin\frac { x }{ 2 } \)

(d)

\(y={ Ae }^{ \frac { x }{ 2 } }+B{ e }^{ \frac { -x }{ 2 } }\)

If F(x) is the probability distribution function then \(F\left( -\infty \right) \) is is _____________

(a)

1

(b)

2

(c)

\(\infty \)

(d)

0

The random variable X has variance 4 and E(x²), then the mean of x is _____________

(a)

\(2\sqrt { 3 } \)

(b)

4

(c)

2

(d)

\(\sqrt { 2 } \)

In a binomial distribution, if the mean is 8 and the variance is 6, then the number of trials is _____________

(a)

32

(b)

48

(c)

16

(d)

12

In a binomial distribution,\(n=4,P(X=0)=\frac { 16 }{ 81 } \),then \(P(X=4)\) _____________

(a)

\(\frac { 1 }{ 16 } \)

(b)

\(\frac { 1 }{ 81 } \)

(c)

\(\frac { 1 }{ 27 } \)

(d)

\(\frac { 1 }{ 8 } \)

A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is _____________

(a)

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

(b)

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

(c)

\(\frac { 4 }{ 5 } \)

(d)

\(\frac { 5 }{ 4 } \)

The sum of the mean and variance of a binomial distribution for 6 total is 2.16. Then the probability of success p =__________

(a)

0.4

(b)

0.6

(c)

0.8

(d)

0.2

If F(x) is a distribution function of a random variable then the false statement is ____________

(a)

\(F(\infty )=1\)

(b)

\(F(-\infty )=-1\)

(c)

\({ F }^{ ' }\left( x \right) =f(x)\)

(d)

\(0<\mathrm{F}(x)<1\)

The identity element in the group {R - {1},x} where a * b = a + b - ab is __________

(a)

0

(b)

1

(c)

\(\frac { 1 }{ a-1 } \)

(d)

\(\frac { a }{ a-1 } \)

Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z is ________

(a)

1

(b)

0

(c)

1

(d)

-1

A binary operation * is defined on the set of positive rational numbers Q⁺ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is _____________

(a)

\(\frac { 3 }{ 160 } \)

(b)

\(\frac { 5 }{ 160 } \)

(c)

\(\frac { 3 }{ 10 } \)

(d)

\(\frac { 3 }{ 40 } \)

The number of commutative binary operations which can be defined on a set containing n elements is _________

(a)

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

(b)

\({ n }^{ { n }^{ 2 } }\)

(c)

\(n^{ \frac { n }{ 2 } }\)

(d)

n²