If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

(a)

-40

(b)

-80

(c)

-60

(d)

-20

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) =  \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \), then k =

(a)

0

(b)

sin θ

(c)

cos θ

(d)

1

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

(a)

Only (i)

(b)

(ii) and (iii)

(c)

(iii) and (iv)

(d)

(i), (ii) and (iv)

The principal argument of (sin 40°+i cos 40°)⁵ is

(a)

−110°

(b)

−70°

(c)

70°

(d)

110°

The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \)  = 1 lies on _________

(a)

circle x²+ y² = 1

(b)

x-axis

(c)

y-axis

(d)

the lines x+y = 1

(1+i)³ = ______

(a)

3 + 3i

(b)

1 + 3i

(c)

3 - 3i

(d)

2i - 2

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

\(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

If x_r = \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2, x₃ ... x_∞ is _________

(a)

-∞

(b)

-2

(c)

-1

(d)

0

According to the rational root theorem, which number is not possible rational zero of 4x⁷ + 2x⁴ - 10x³ - 5?

(a)

-1

(b)

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

(c)

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

(d)

5

For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

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

\(cot\left( \frac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

(a)

7

(b)

6

(c)

5

(d)

none

If tan^-1(cot \(\theta\)) = 2\(\theta\), then\(\theta\) = _____________

(a)

\(\pm 3\)

(b)

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

(c)

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

(d)

none

The value of \({ sin }^{ -1 }\left( cos\frac { 33\pi }{ 5 } \right) \) is________

(a)

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

(b)

\(\frac { -\pi }{ 10 } \)

(c)

\(\frac { \pi }{ 10 } \)

(d)

\(\frac { 7\pi }{ 5 } \)

The circle x² + y² = 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85 < m < −35

(d)

−35 < m < 15

The equation of the circle passing through the foci of the ellipse  \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) having centre at (0, 3) is

(a)

x² + y² − 6y − 7 = 0

(b)

x² + y² − 6y + 7 = 0

(c)

x²+y²−6y−5 = 0

(d)

x²+y²−6y+5 = 0

y² - 2x - 2y + 5 = 0 is a _________

(a)

circle

(b)

parabola

(c)

ellipse

(d)

hyperbola

When the eccentricity of a ellipse becomes zero, then it becomes a __________

(a)

straight line

(b)

circle

(c)

point

(d)

parabola

If y = 2x + c is a tangent to the circle x² + y² = 5, then c is __________

(a)

土5

(b)

土\(\sqrt5\)

(c)

土5\(\sqrt2\)

(d)

±2\(\sqrt5\)

The point of contact of y² = 4ax and the tangent y = mx + c is _________

(a)

\(\left( \frac { 2a }{ { m }^{ 2 } } ,\frac { a }{ m } \right) \)

(b)

\(\left( \frac { a }{ { m }^{ 2 } } ,\frac { 2a }{ m } \right) \)

(c)

\(\left( \frac { a }{ m} ,\frac { 2a }{ { m }^{ 2 } } \right) \)

(d)

\(\left( \frac {-a }{ { m }^{ 2 }} ,\frac {- 2a }{ m } \right) \)

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)

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

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

(a)

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

(b)

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

(c)

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

(d)

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

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector ___________

(a)

\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)

(b)

\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

(c)

2\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)

(d)

2\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

The p.v, OP of a point P make angles 60^o and 45^o with X and Y axis respectively. The angle of inclination of  \(\overset { \rightarrow }{ OP } \) with z-axis is ___________

(a)

75^o

(b)

60^o

(c)

45^o

(d)

3

If the work done by a force \(\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k } \) in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m is _____________

(a)

5

(b)

2

(c)

3

(d)

6

For any three vectors \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \), \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) \times \left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) \) is _____________

(a)

0

(b)

\(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)

(c)

2\(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)

(d)

\({ \left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] }^{ 2 }\)

The distance from the origin to the plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +5\overset { \wedge }{ k } \right) =7\) is ______________ 

(a)

\(\frac { 7 }{ \sqrt { 30 } } \)

(b)

\(\frac { \sqrt { 30 } }{ 7 } \)

(c)

\(\frac { 30 }{ 7 } \)

(d)

\(\frac { 7 }{ 30 } \)

If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are mutually 丄^r unit vectors, then \(\left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right| \) is _______________ 

(a)

3

(b)

9

(c)

3 \(\sqrt { 3 } \)

(d)

\(\sqrt { 3 } \)

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

(a)

4

(b)

\(2\sqrt { 3 } \)

(c)

\(4\sqrt { 3 } \)

(d)

\(5\sqrt { 3 } \)

The least value of a when f f(x) = x² + ax + 1 is increasing on (1, 2) is __________

(a)

-2

(b)

2

(c)

1

(d)

-1

The function f(x) = x⁹ + 3x⁷+ 64 is increasing on ________

(a)

R

(b)

(-∞, 0)

(c)

(0, ∞)

(d)

None of these

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

lf u = (x-y)⁴+(y-z)⁴ +(z-x)⁴ then \(\sum { \frac { \partial u }{ \partial x } } \) = _____________

(a)

4

(b)

1

(c)

0

(d)

-4

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 cube root of 127 is ............

(a)

5.026

(b)

5.26

(c)

5.028

(d)

5.075

If is a homogeneous function of x and y of degree n, then \(x\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +y\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ...... \(\frac { { \partial }u }{ \partial { x } } \)

(a)

n

(b)

0

(c)

1

(d)

n - 1

The volume generated by the curve y² = 16x from x = 2 to x = 3 rotating about x - axis ......... cu. units

(a)

72π

(b)

\(\frac { 256\times 19 }{ 3 } \)ㅠ

(c)

40ㅠ

(d)

80ㅠ

 

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

(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)

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

\(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { sinx }{ 2+cosx } dx= } \) __________

(a)

0

(b)

2

(c)

log 2

(d)

log 4

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

(a)

y = cx

(b)

x²+ y² = c

(c)

x²- y² = c

(d)

y = c^x

Using y = vx, the differential equation \(\frac { dy }{ dx } =\frac { y }{ x+\sqrt { xy } } \) is reduced to ________.

(a)

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

(b)

x(1-\(\sqrt{v}\))dv = v\(\sqrt{v}\)dx

(c)

x(1+\(\sqrt{v}\))dv = -v\(\sqrt{v}\)dx

(d)

v(1+\(\sqrt{v}\))dx - v\(\sqrt{v}\)dv = 0

If \(f(x)={ Cx }^{ 2 }={ cx }^{ 2 },0 is the p.d.f, of x then c is _____________

(a)

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

(b)

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

(c)

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

(d)

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

Var (2x ± 5) is =________

(a)

5

(b)

var (2x) ± 5

(c)

4 var (X)

(d)

0

If the mean and variance of a binomial variate are 2 and 1 respectively, the probability that X takes a value greater than one is equal to__________.

(a)

\(\frac { 5 }{ 16 } \)

(b)

\(\frac { 11 }{ 16 } \)

(c)

\(\frac { 10 }{ 16 } \)

(d)

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

A die is thrown 10 times. Getting a number greater than 3 is considered a success. The S.D of the number of successes is _________

(a)

2.5

(b)

1.56

(c)

5

(d)

25

A random variable X has the following probapality mass function?

X	-2	3	1
P(X = x)	\(\frac { \lambda }{ 6 } \)	\(\frac { \lambda }{ 4 } \)	\(\frac { \lambda }{ 12 } \)

Then __________ 

(a)

\(\frac { \lambda }{ 6 } +\frac { \lambda }{ 4 } +\frac { \lambda }{ 12 } =1\)

(b)

\(\lambda =2\)

(c)

\(P(X=3)=\frac { 1 }{ 8 } \)

(d)

\(P(1\le X\le 3)=\frac { 2 }{ 3 } \)

If a * b = a²b² - ab then 3 * (1 * 1)

(a)

0

(b)

1

(c)

2

(d)

4

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

(a)

7

(b)

-7

(c)

0

(d)

-49