If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is

(a)

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

(b)

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

(c)

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

(d)

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

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

The polynomial x³ - kx² + 9x has three real zeros if and only if, k satisfies

(a)

|k| ≤ 6

(b)

k = 0

(c)

|k| > 6

(d)

|k| ≥ 6

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]

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 angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

(a)

0°

(b)

30°

(c)

45°

(d)

90°

The tangent to the curve y² - xy + 9 = 0 is vertical when 

(a)

y = 0

(b)

\(\\ \\ y=\pm \sqrt { 3 } \)

(c)

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

(d)

\(y=\pm 3\)

The curve y= ax⁴ + bx² with ab > 0

(a)

has, no horizontal tangent

(b)

is concave up

(c)

is concave down

(d)

has no points of inflection

The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is

(a)

0.3xdx m³

(b)

0.03x m³

(c)

0.03x² m³

(d)

0.03x³ m³

If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is 

(a)

10

(b)

5

(c)

8

(d)

9

For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

(a)

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

(b)

\(\pi\)

(c)

0

(d)

2

The solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}\) is

(a)

\(x\phi \left( \frac { y }{ x } \right) =k\)

(b)

\(\phi \left( \frac { y }{ x } \right) =kx\)

(c)

\(y\phi \left( \frac { y }{ x } \right) =k\)

(d)

\(\phi \left( \frac { y }{ x } \right) =ky\)

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

Which one of the following is incorrect? For any two propositions p and q, we have

(a)

¬ (p∨q) ≡ ¬ p ∧ ¬q

(b)

¬ ( p∧ q)≡¬p ∨ ¬q

(c)

¬ (p ∨ q)≡¬p∨¬q

(d)

¬(¬p)≡ p

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

(a)

a ≥ 0

(b)

a > 0

(c)

a < 0

(d)

a ≤ 0

If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

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 \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

(a)

0, 2

(b)

0, 1

(c)

0, 3

(d)

0, √3

If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to

(a)

tan^-1x

(b)

sin^-1x

(c)

0

(d)

\(\pi\)

Expand the polynomial f(x)=x²-3x+2 in power of (x-2)

Without using any kind of computational aid use linear approximation to find the value of e^0.1

Form the differential equation satisfied by are the straight lines in my-plane.

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

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.

Obtain the Cartesian form of the locus of z in each of the following cases.
|z| = |z - i|

Find the value of
\(tan\left( { sin }^{ -1 }\frac { 3 }{ 5 } +{ cot }^{ -1 }\frac { 3 }{ 2 } \right) \)

Find the equation of the parabola with vertex (-1, -2), axis parallel to y-axis and passing through (3, 6)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { y }^{ 2 } }{ 16 } -\frac { { x }^{ 2 } }{ 9 } =1\)

For any vector \(\vec { a } \), prove that \(\hat { i } \times (\vec { a } \times \hat { i } )+\hat { j } \times (\vec { a } \times \hat { j } )+\hat { k } \times \vec { a } \times \hat { k } =2\vec { a } \).

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.

Find the domain of cos^-1\((\frac{2+sinx}{3})\)

Find the real solutions of the equation
\({ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi }{ 2 } \)

1. missle fired from ground level rises x metres vertically upwards in t seconds and \(x=100t-\frac { 25 }{ 2 } { t }^{ 2 }\). Find the 
   (i) initial velocity of the missile
   (ii) the time when the height of the missile is maximum
   (iii) the maximum height reached
   (iv) the velocity which the missile strikes the ground.

2. Find the area of the loop of the curve 3ay²=x(x-a)²

1. A population grows at the rate of 2% per year. How long does it take for the population to double?

2. Verify 
   (i) closure property 
   (ii) commutative property 
   (iii) associative property 
   (iv) existence of identity and
   (v) existence of inverse for the operation +₅ on Z₅ using table corresponding to addition modulo 5.

1. Verify (p ∧ ~p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.

2. If U(x, y, z) = log (x³ + y³ + z³), find \(\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z } \)

1. If V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0

2. W(x, y, z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u ∈ R. Find \(\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v } \), and evaluate them at \(\left( \frac { 1 }{ 2 } ,1 \right) \)

1. Prove that f(x, y) = x³ - 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.