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 A^T is the transpose of a square matrix A, then ___________

(a)

|A| ≠ |A^T|

(b)

|A| = |A^T|

(c)

|A| + |A^T| =0

(d)

|A| = |A^T| only

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 product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is

(a)

-2

(b)

-1

(c)

1

(d)

2

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

(a)

1

(b)

-1

(c)

\(\sqrt { 3i } \)

(d)

\(-\sqrt { 3i } \)

If cot^-1 2 and cot^-1 3 are two angles of a triangle, then the third angle is

(a)

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

(b)

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

(c)

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

(d)

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

An ellipse has OB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

(a)

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

(b)

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

(c)

\(\frac { 1 }{ 4 } \)

(d)

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

If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle \(\frac{\pi}{3}\)

(d)

inclined at an angle  \(\frac{\pi}{6}\)

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 number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

(a)

2

(b)

2.5

(c)

3

(d)

3.5

The minimum value of the function |3 - x| + 9 is

(a)

0

(b)

3

(c)

6

(d)

9

If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

(a)

0.4 cu.cm

(b)

0.45 cu.cm

(c)

2 cu.cm

(d)

4.8 cu.cm

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

(a)

12 x_o+dx

(b)

12x_o dx

(c)

6x_o dx

(d)

6x_o+ dx

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

The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

(a)

2^x + 2^y = C

(b)

2^x - 2^y = C

(c)

\(\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C\)

(d)

x + y = C

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

p	q	(p ∧ q) ⟶ ¬q
T	T	(a)
T	F	(b)
F	T	(c)
F	F	(d)

Which one of the following is correct for the truth value of (p ∧ q)⟶ ¬p?

(a)

(a)	(b)	(c)	(d)
T	T	T	T

(b)

(a)	(b)	(c)	(d)
F	T	T	T

(c)

(a)	(b)	(c)	(d)
F	F	T	T

(d)

(a)	(b)	(c)	(d)
T	T	T	F

Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.

Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.

Use differentials to find \(\sqrt{25.2}\)

If of f(x, y) = x² + y³ + 2xy² find f_xx, f_yy, f_xy and f_yx.

If w=e^xy,x=at²,y=2at, find \(\frac { dw }{ dt } \)

If \(w={ e }^{ { x }^{ 2 }+{ y }^{ 2 } }\) ,x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

Find the volume of the solid y=x³,x=0,y=1 is revolved about the y-axis.

Solve : \(\frac { dy }{ dx } =\frac { { e }^{ x }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

solve: x dy + y dx = xy dx

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).

Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

If z = 2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\frac { \pi }{ 3 } \).

If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1 \((\frac{1-|x|}{4})\)

Find the domain of
g(x) = sin⁻¹x + cos⁻¹x

Find the equation of the ellipse with foci (±2, 0), vertices (±3, 0)  

A room 34m long is constructed to be a whispering gallery. The room has an elliptical ceiling, as shown in Figure. If the maximum height of the ceiling is 8 m, determine where the foci are located.

1. If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3

2. A manufacturer can sell x items at a price of rupees \(\left( 5-\frac { x }{ 100 } \right) \) each. The cost price of x items is Rs.\(\left( \frac { x }{ 5 } +500 \right) \) .Find the numbers of items he should sell to earn maximum profit.

1. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3

2. Find the area bounded by x=at²,y=2at between the ordinates corresponding to t = 1 and t = 2.

1. Find the area bounded by the curve y=xe^x and y=xe^-x and the line x=1.

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. Solve: \(\frac { dy }{ dx } \) = (3x+2y+1)²

2. Solve :x²dy+y(x+y)dx=0 given that y=1 when x=1.

1. Solve :\(x\frac { dy }{ dx } sin\left( \frac { y }{ x } \right) +x-ysin\left( y\frac { y }{ x } \right) =,y(1)=\frac { \pi }{ 2 } \)