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 = [2 0 1] then the rank of AA^T is ______

(a)

1

(b)

2

(c)

3

(d)

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

A polynomial equation in x of degree n always has

(a)

n distinct roots

(b)

n real roots

(c)

n complex roots

(d)

at most one root

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

\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for

(a)

\(-\pi \le x\le 0\)

(b)

\(0 \le x\le \pi\)

(c)

\(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

(d)

\(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

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)

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

If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is

(a)

2x + 1 = 0

(b)

x = −1

(c)

2x −1 = 0

(d)

x = 1

Distance from the origin to the plane 3x − 6y + 2z + 7 = 0 is

(a)

0

(b)

1

(c)

2

(d)

3

The number given by the Rolle's theorem for the functlon x³ - 3x², x ∈ [0, 3] is

(a)

1

(b)

\(\sqrt { 2 } \)

(c)

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

(d)

2

The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

(a)

the extreme value theorem

(b)

Fermat's theorem

(c)

Law of mean

(d)

Rolle's theorem

If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

(a)

xy + yz + zx

(b)

x(y + z)

(c)

y(z + x)

(d)

0

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 value of \(\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx\) is

(a)

\(\frac { { \pi }^{ 2 } }{ 4 } -1\)

(b)

\(\frac { { \pi }^{ 2 } }{ 4 } +2\)

(c)

\(\frac { { \pi }^{ 2 } }{ 4 } +1\)

(d)

\(\frac { { \pi }^{ 2 } }{ 4 } -2\)

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

The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

(a)

x

(b)

\(\frac { { x }^{ 2 } }{ 2 } \)

(c)

\(\frac{1}{x}\)

(d)

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

The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________

(a)

tan x+tan y = c

(b)

sec x + sec y = c

(c)

tan x tan y = c

(d)

sec x- sec y = c

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

(a)

\(\frac { 57 }{ { 20 }^{ 3 } } \)

(b)

\(\frac { 57 }{ { 20 }^{ 2 } } \)

(c)

\(\frac { { 19 }^{ 3 } }{ { 20 }^{ 3 } } \)

(d)

\(\frac { 57 }{ 20 } \)

In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

(a)

1

(b)

2

(c)

3

(d)

4

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

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

A particle moves in a line so that x =\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

Find df for f(x) = x² + 3x and evaluate it for
x = 3 and dx = 0.02

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)

Show that x² + y² = r², where r is a constant, is a solution of the differential equation \(\frac { dy }{ dx } \) = -\(\frac { x }{ y } \).

For the random variable X with the given probability mass function as below, find the mean and variance 

Let G = {1, w, w²) where w is a complex cube root of unity. Then find the universe of w². Under usual multiplication.

Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).

Find the fourth roots of unity.

It is known that the roots of the equation x³- 6x²- 4x + 24 = 0 are in arithmetic progression. Find its roots.

Solve: 2x+2^x-1+2^x-2 = 7x+7^x-1+7^x-2

Prove that \({ tan }^{ -1 }\sqrt { x } =\frac { 1 }{ 2 } { cos }^{ -1 }={ \frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) ,x\in \left| 0,1 \right| }\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( y-2 \right) }^{ 2 } }{ 25 } \frac { { \left( x+1 \right) }^{ 2 } }{ 16 } =1\)

Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)

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s, t

Suppose f(x) is a differentiable function for all x with f'(x) ≤ 29 and f(2) = 17. What is the maximum value of f(7)?

Evaluate \(\int _{ 0 }^{ 1 }{ { xe }^{ -2x } } dx\)

In (z, *) where * is defined as a * b = a + b + 2. Verify the commutative and associative axiom.

1. Find the domain of the following functions
   (i) f(x) = sin^-1(2x - 3)
   (ii) f(x) = sin^-1x + cos x

2. Let S be a non-empty set and 0 be a binary operation on s defined by x 0 y = x; x, Y \(\in \) s. Determine whether 0 is commutative and association.

1. Using integration, find the area of the triangle with sides y = 2x + 1, y = 3x + 1 and x = 4.

2. Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function 
   \(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
   Find
   (i) the value of k
   (ii) the distribution function
   (iii) the probability that daily sales will fall between 300 litres and 500 litres?

1. Solve the differential equation (y²-2xy) dx = (x²-2xy) dy

2. The slope of the tangent at p(x,y) on the curve is -\(\left( \frac { y+3 }{ x+2 } \right) \). If the curve passes through the origin, find the equation of the curve.

1. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.

2. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

1. If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)

2. A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

1. Discuss the maximum possible number of positive and negative roots of the polynomial equations x²−5x+6 and x²−5x+16 . Also draw rough sketch of the graphs

2. ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

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   plane

1. A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shore line. How fast is the beam moving along the shore line when it makes an angle of 45° with the shore?

2. Let w(x, y, z) = \(\frac { 1 }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } } } ,(x,y,z)\neq (0,0,0)\). Show that \(\frac { { \partial }^{ 2 }w }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }w }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 }w }{ \partial { z }^{ 2 } } =0\)