Which of the following is not an elementary transformation?

(a)

R_i ↔️ R_j

(b)

R_i ⟶ 2R_i + R_j

(c)

C_j ⟶ C_j + C_i

(d)

R_i ⟶ R_i + C_j

If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________

(a)

1

(b)

-1

(c)

i

(d)

0

If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________

(a)

a > b > 0

(b)

a < b < 0

(c)

b < a < 0

(d)

b > a > 0

If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

(a)

2 cos nθ

(b)

2 i sin nθ

(c)

2ⁿ cosθ

(d)

2ⁿ i sinθ

If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

(a)

\(\frac{-7}{9}\)

(b)

\(\frac{7}{9}\)

(c)

0

(d)

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

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

In a \(\Delta ABC\)  if C is a right angle, then  \({ tan }^{ -1 }\left( \frac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \frac { b }{ c+a } \right) =\) ________

(a)

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

(b)

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

(c)

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

(d)

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

The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

(a)

parabola

(b)

ellipse

(c)

hyperbola

(d)

rectangular hyperbola

Th point of curve y = 2x² - 6x - 4 at which the targent is parallel to x - axis is __________

(a)

\(\left( \frac { 5 }{ 2 } ,\frac { -7 }{ 12 } \right) \)

(b)

\(\left( \frac { -5 }{ 2 } ,\frac { -7 }{ 2 } \right) \)

(c)

\(\left( \frac { -5 }{ 2 } ,\frac { 17 }{ 12 } \right) \)

(d)

\(\left( \frac { 3 }{ 2 } ,\frac { -7 }{ 2 } \right) \)

If θ is the angle between the vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow}{b} \), then sin θ is ___________ 

(a)

\(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \)

(b)

\(\frac { \left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| }{ \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } \)

(c)

\(\sqrt { 1-{ \left( \frac { \overset { \rightarrow }{ a. } \overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \right) }^{ 2 } } \)

(d)

0

For what value of \(\left( \overset { \rightarrow }{ a } \right) \) will the straight lines \(\frac { x+2 }{ a } =\frac { y }{ 3 } =\frac { z-1 }{ 4 } \)and \(\frac { x-3 }{ a } =\frac { y-1 }{ 4 } =\frac { z-7 }{ a } \)be perpendicular?

(a)

1

(b)

2

(c)

3

(d)

-3

Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\) and \(\overset { \rightarrow }{ c } \) be three vectors having magnitudes 1, 1, 2 respectively. If \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } =0\) then the acute angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) is ___________ 

(a)

0

(b)

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

(c)

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

(d)

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

If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________

(a)

2 < t < 5

(b)

2 ≤ t ≤ 5

(c)

t ≥ 2

(d)

t ≤ 2

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

(a)

1

(b)

-1

(c)

0

(d)

∞

If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________

(a)

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

(b)

0

(c)

u

(d)

2u

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

(a)

xy^x-1

(b)

yx^y-1

(c)

0

(d)

1

The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________

(a)

4 : 9

(b)

9 : 4

(c)

2 : 3

(d)

3 : 2 

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

(a)

f(2a -x) = f(x)

(b)

f(a -x) = f(x)

(c)

f(x) = - f(-x)

(d)

f(-x) = f(x)

The value of \(\int _{ 0 }^{ \frac { \pi }{ 3 } } { tan } x \ dx\) __________

(a)

-log 2

(b)

log 2

(c)

-log 3

(d)

log 3

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 I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________

(a)

sec x

(b)

cos x

(c)

e^{tan x}

(d)

cot 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)

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

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

In (N, *), x * y = max(x, y), x, y \(\in \) N then 7 * (-7)

(a)

7

(b)

-7

(c)

0

(d)

-49