The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

(a)

-8

(b)

-4

(c)

-2

(d)

0

The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________

(a)

x + 4y + 12 = 0

(b)

4x + y + 12 = 0

(c)

4x - y - 14 = 0

(d)

x + 4y - 12 = 0

In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

(a)

<b

(b)

≤b

(c)

=b

(d)

≠b

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

If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

(a)

x^y log x

(b)

y log x

(c)

yx^y-1

(d)

x log y

If u = x^y + y^x then u_x + u_y at x = y = 1 is _____________

(a)

0

(b)

2

(c)

1

(d)

∞

If u = sin-1 \(\left( \frac { { x }^{ 4 }+{ y }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) and f = sin u then f is a homogeneous function of degree ..................

(a)

0

(b)

1

(c)

2

(d)

4

The area between y² = 4x and its latus rectum is

(a)

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

(b)

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

(c)

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

(d)

\(\frac{5}{3}\)

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

\(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ 1+{ x }^{ 2 } } } \) is __________

(a)

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

(b)

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

(c)

\(\frac { \pi }{ 12 } \)

(d)

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

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

(a)

\(\frac43\)

(b)

\(\frac23\)

(c)

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

(d)

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

\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x-cos \ x }{ 1+sin \ x cos \ x } dx= } \) ............

(a)

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

(b)

0

(c)

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

(d)

\( \pi\)

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

(a)

1, 2

(b)

2, 2

(c)

1, 1

(d)

2, 1

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

(a)

2^x + 2^y = C

(b)

2^x - 2^y = C

(c)

\(\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C\)

(d)

x + y = C

If sin x is the integrating factor of the linear differential equation \(\frac { dy }{ dx } +Py=Q,\) then P is

(a)

log sin x

(b)

cos x

(c)

tan x

(d)

cot x

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)

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

The solution of log \(\left( \frac { dy }{ dx } \right) \) = ax + by is______.

(a)

\(\frac { { e }^{ ax } }{ a } +\frac { { e }^{ -by } }{ b } +c=0\)

(b)

ae^ax- be^-by + c = 0

(c)

ae^x + be^y = k

(d)

be^ax + ae^-by = k

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)

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

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

(a)

i + 2n, i = 0,1,2... n

(b)

2i- n, i = 0,1,2... n

(c)

n - i, i = 0,1,2... n

(d)

2i + 2n, i = 0, 1, 2...n

Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.

(a)

I and II

(b)

II only

(c)

III only

(d)

II and III

If \(f(x)=\frac { 1 }{ 2 } \) ,\(E\left( { x }^{ 2 } \right) =\frac { 1 }{ 4 } \) then var(x) is _____________

(a)

0

(b)

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

(c)

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

(d)

1

In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?

(a)

y = \(\frac{2}{3}\)

(b)

y = \(\frac{-2}{3}\)

(c)

y = \(\frac{-3}{2}\)

(d)

y = 4

If a * b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is

(a)

commutative but not associative

(b)

associative but not commutative

(c)

both commutative and associative

(d)

neither commutative nor associative

Which one of the following is not a statement?

(a)

2 + 3 =5

(b)

How beautiful is this flower?

(c)

Delhi is the capital of Tamil Nadu

(d)

A triangle has found angles.

Which of the following is a statement?

(a)

7+2<10

(b)

Wish you all success

(c)

All the best

(d)

How old are you?