If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A = 

(a)

A^-1

(b)

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

(c)

3A^-1

(d)

2A^-1

If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a)

adj A = |A|A^-1

(b)

adj(AB) = (adj A)(adj B)

(c)

det A^-1 = (det A)^-1

(d)

(ABC)^-1 = C^-1B^-1A^-1

If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 = 

(a)

\(\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right] \)

If x^a y^b = e^m, x^c y^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

(a)

e^{(Δ₂  / Δ₁)}, e^{(Δ₃ / Δ₁)}

(b)

log (Δ₁ / Δ₃), log (Δ₂ / Δ₃)

(c)

log (Δ₂ / Δ₁), log(Δ₃ / Δ₁)

(d)

e^{(Δ₁ / Δ₃)},e^{(Δ₂ / Δ₃)}

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 system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________

(a)

k ≠ 0

(b)

-1 < k < 1

(c)

-2 < k < 2

(d)

k = 0

If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to 

(a)

0

(b)

1

(c)

2

(d)

3

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

(a)

−110°

(b)

−70°

(c)

70°

(d)

110°

If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is

(a)

1

(b)

i

(c)

x²+y²

(d)

1+n²

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

(a)

|z| = 1, arg(z) =\(\frac { \pi }{ 4 } \)

(b)

|z| = 1, arg(z) = \(\frac { \pi }{ 6 } \)

(c)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg(z) = \(\frac { 5\pi }{ 24 } \)

(d)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg (z) = tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

If zⁿ = \(cos\frac { n\pi }{ 3 } +isin\frac { n\pi }{ 3 } \), then z₁, z₂ ..... z₆ is _________

(a)

1

(b)

-1

(c)

i

(d)

-i

If α, β and γ are the zeros of x³ + px² + qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is

(a)

\(-\frac { q }{ r } \)

(b)

\(-\frac { p }{ r } \)

(c)

\(\frac { q }{ r } \)

(d)

\(-\frac { q }{ p } \)

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

The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________

(a)

no solution

(b)

one solution

(c)

two solution

(d)

more than one solution

If the equation ax²+ bx+c = 0(a > 0) has two roots ∝ and β such that ∝ <- 2 and β > 2, then __________

(a)

b²-4ac = 0

(b)

b² - 4ac <0

(c)

b² - 4ac >0

(d)

b² - 4ac ≥ 0

The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

(a)

[1, 2]

(b)

[-1, 1]

(c)

[0, 1]

(d)

[-1, 0]

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

(a)

\(-\sqrt { \frac { 24 }{ 25 } } \)

(b)

\(\sqrt { \frac { 24 }{ 25 } } \)

(c)

\(\frac{1}{5}\)

(d)

\(-\frac{1}{5}\)

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

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

(a)

4(a²+b²)

(b)

2(a²+b²)

(c)

a² +b²

(d)

\(\frac { 1 }{ 2 } \)(a²+b²)

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² , then the value of r² is

(a)

2

(b)

3

(c)

1

(d)

4

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

(a)

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

(b)

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

(c)

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

(d)

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

If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \), then the value of \(\lambda +\mu \) is

(a)

0

(b)

1

(c)

6

(d)

3

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

If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________

(a)

\(\left| \overset { \rightarrow }{ d } \right| \)

(b)

\(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)

(c)

\(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ 0 } \)

(d)

a, b, c are coplanar

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 \({ \left| \overset { \rightarrow }{ a } \right| }^{ 2 }{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 }\)

The straight lines \(\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 } \) and \(\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 } \) are _____________

(a)

parallel

(b)

perpendicular

(c)

inclined at 45^o

(d)

none

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)

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

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)

\(\sqrt { 3 } \)

A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

(a)

\(\frac{3}{25} \text { radians } / \mathrm{sec}\)

(b)

\(\frac{4}{25} \text { radians } / \mathrm{sec}\)

(c)

\(\frac{1}{5} \text { radians } / \mathrm{sec}\)

(d)

\(\frac{1}{3} \text { radians } / \mathrm{sec}\)

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 percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

(a)

\(\frac{1}{31}\)

(b)

\(\frac15\)

(c)

5

(d)

31

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 the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________

(a)

9.72 cm³

(b)

0.972 cm³

(c)

0.972π cm³

(d)

9.72π cm³

If log_e4 = 1.3868, then log_e4.01 = _____________

(a)

1.3968

(b)

1.3898

(c)

1.3893

(d)

none

The percentage error in the 11^th root of the number 28 is approximately .......... times the percentage error in 28.

(a)

\(\frac{1}{28}\)

(b)

\(\frac{1}{11}\)

(c)

11

(d)

28

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 \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is

(a)

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

(b)

2

(c)

1

(d)

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

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

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 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 I.F. of (1+y²) dx = (tan^-1-t-x) dy is ________.

(a)

e^tan-1 y

(b)

e^tan-1 x

(c)

tan^-1 y

(d)

tan^-1x

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

The variance of a binomial distribution is________.

(a)

equal to its mean

(b)

less than its mean

(c)

greater than its mean

(d)

none

If x is a continuous random variable then P(x ≥ a) = ________

(a)

\(P(x \ < \ a)\)

(b)

\(P(a\le x\le b)\)

(c)

\(P\left( x>a \right) \)

(d)

\(1-P\left( x\le a-1 \right) \)

A binary operation on a set S is a function from

(a)

S ⟶ S

(b)

(SxS) ⟶ S

(c)

S⟶ (SxS)

(d)

(SxS) ⟶ (SxS)

The binary operation * defined on a set s is said to be commutative if ______

(a)

a*b \(\in \) S ∀ a, b \(\in \) S

(b)

a*b = b*a ∀ a, b \(\in \) S

(c)

(a*b) * c = a*(b*c) ∀ a, b \(\in \) S

(d)

a*b = e ∀ a, b \(\in \) S

'+' is not a binary operation on ___________

(a)

~

(b)

z

(c)

c

(d)

Q- {0}

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?