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

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\) 

(a)

0

(b)

1

(c)

2

(d)

∞

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

If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to

(a)

\(e^{x^{2}+y^{2}}\)

(b)

2xu

(c)

x²u

(d)

y²u

If log_e4 = 1.3868, then log_e4.01 = _____________

(a)

1.3968

(b)

1.3898

(c)

1.3893

(d)

none

If f(x, y, z) = sin (xy) + sin (yz) + sin (zx) then f_xx is _____________

(a)

-y sin (xy) + z² cos (xz)

(b)

y sin (xy) - z² cos (xz)

(c)

y sin (xy) + z² cos (xz)

(d)

-y² sin (xy) - z² cos (xz)

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

(a)

5.002

(b)

5.003

(c)

5.005

(d)

5.004

If u = \((\frac{y}{x})\) then x \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) = .....................

(a)

0

(b)

1

(c)

2u

(d)

u

If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)

(a)

cos x - x sin x

(b)

sin x + x cos x

(c)

x cos x

(d)

x sin x

The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is

(a)

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

(b)

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

(c)

\(\frac{5}{2}\)

(d)

\(\frac{7}{2}\)

The value of \(\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \sqrt { \frac { 1-cos2x }{ 2x } } } \) dx is __________

(a)

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

(b)

2

(c)

0

(d)

1

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

\(\int _{ a }^{ b }{ f(x) } dx=\) ..............

(a)

\(2\int _{ 0 }^{ a }{ f(x) } dx\)

(b)

\(\int _{ a }^{ b }{ f(a-x) } dx\)

(c)

\(\int _{ b }^{ a }{ f(b-x) } dx\)

(d)

\(\int _{ a }^{ b }{ f(a+b-x) } dx\)

The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

(a)

\(\frac { x }{ { e }^{ \lambda } } \)

(b)

\(\frac { { e }^{ \lambda} }{ x } \)

(c)

\({ \lambda e }^{ x }\)

(d)

e^x

If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________

(a)

-cot x

(b)

cot x

(c)

tan x

(d)

-tan x

The transformation y = vx reduces \(\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x } \) __________ 

(a)

\(\frac { 3av }{ 4v+1 } =\frac { dx }{ x } \)

(b)

\(\frac { 3dv }{ v+1 } =\frac { dx }{ x } \)

(c)

\(2x\frac { dv }{ dx } =v\)

(d)

\(\frac { 3dv }{ 1-2v } ==\frac { dx }{ x } \)

The differential equation corresponding to xy = c² where c is an arbitrary constant is ________.

(a)

xy"+ x = 0

(b)

y" = 0

(c)

xy'+ y = 0

(d)

xy"-x = 0

The general solution of x \(\frac{dy}{dx}\) = y is _________.

(a)

y = cx

(b)

x²+ y² = c

(c)

x²- y² = c

(d)

y = c^x

The differential equation associated with the family of concentric circles having their centres at the origin is _________.

(a)

\(\frac { dy }{ dx } =\frac { -x }{ y } \)

(b)

\(\frac { dy }{ dx } =\frac { -y }{ x } \)

(c)

\(\frac { dy }{ dx } =\frac { x }{ y } \)

(d)

\(\frac { dy }{ dx } =\frac { y }{ x } \)

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

(a)

1

(b)

2

(c)

3

(d)

4

If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is 

(a)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 6 }\left( \frac { 2 }{ 5 } \right) ^{ 4 }\) 

(b)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 10 }\)

(c)

\(\left( \frac { 10 }{ 5 } \right) { \left( \frac { 3 }{ 5 } \right) }^{ 4 }\left( \frac { 2 }{ 5 } \right) ^{ 6 }\)

(d)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 5 }\left( \frac { 2 }{ 5 } \right) ^{ 5 }\)

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

Which of the following is a contradiction?

(a)

p v q

(b)

p ∧ q

(c)

q v ~ q

(d)

q ∧ ~ q

Let p: Kamala is going to school
q: There are 20 students in the class. Then Kamala is not going to school or there are 20 students in the class is represented by

(a)

p v q

(b)

p ∧ q

(c)

~ p

(d)

~ p v q

In (S, *), is defined by x * y = x where x, y \(\in \) S, then

(a)

associative

(b)

Commutative

(c)

associative and commutative

(d)

neither associative nor commutative