If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

(a)

\(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

Cramer's rule is applicable only when ______

(a)

Δ ≠ 0

(b)

Δ = 0

(c)

Δ =0, Δ_x =0

(d)

Δ_x = Δ_y = Δ_z =0

In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________

(a)

trivial solution

(b)

only non - trivial solution

(c)

no solution

(d)

trivial solution and infinitely many non - trivial solutions

 The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is

(a)

1+ i

(b)

i

(c)

1

(d)

0

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

(a)

(1, 0)

(b)

(−1, 1)

(c)

(0, 1)

(d)

(1, 1)

If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________

(a)

\(\frac { 1 }{ 13 } \)

(b)

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

(c)

\(\frac { 1 }{ 12 } \)

(d)

none of these

If z = 1-cos θ + i sin θ, then |z| = _____________

(a)

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

(b)

2 cos\(\frac { \theta }{ 2 } \)

(c)

2|sin\(\frac { \theta }{ 2 } \)|

(d)

2|cos\(\frac { \theta }{ 2 } \)|

If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________

(a)

a > b > 0

(b)

a < b < 0

(c)

b < a < 0

(d)

b > a > 0

\(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =__________

(a)

cosθ + i sinθ

(b)

cosθ - i sinθ

(c)

sinθ - i cosθ

(d)

sinθ + icosθ

If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

(a)

-2i sin(α - β)

(b)

2i sin(α - β)

(c)

2 cos(α - β)

(d)

-2 cos(α - β)

The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

(a)

\(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

(b)

\(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

(c)

\(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

(d)

\(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

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

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

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

(a)

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

(b)

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

(c)

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

(d)

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

Equation of tangent at (-4, -4) on x² = -4y is _____________

(a)

2x - y + 4 = 0

(b)

2x + y - 4 = 0

(c)

2x - y - 12 = 0

(d)

2x + y + 4 = 0

y² - 2x - 2y + 5 = 0 is a _________

(a)

circle

(b)

parabola

(c)

ellipse

(d)

hyperbola

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is __________

(a)

1

(b)

5

(c)

7

(d)

9

The line y = mx +1 is a tangent to the parabola y² = 4x if m = ______________

(a)

1

(b)

2

(c)

3

(d)

4

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)

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

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

Which one of the following is a binary operation on N?

(a)

Subtraction

(b)

Multiplication

(c)

Division

(d)

All the above

If the function f(x) = sin^-1(x² - 3), then x belongs to

(a)

[-1, 1]

(b)

[\(\sqrt2\), 2]

(c)

\(\\ \\ \\ \left[ -2,-\sqrt { 2 } \right] \cup \left[ \sqrt { 2 } ,2 \right] \)

(d)

\([-2,-\sqrt{2}]\)

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1 x + cos^-1 y is equal to

(a)

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

(b)

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

(c)

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

(d)

\(\pi\)

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 \(\theta ={ sin }^{ -1 }\left( sin(-{ 60 }^{ 0 }) \right) \) then one of the possible values of \(\theta\) is _________

(a)

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

(b)

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

(c)

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

(d)

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

\({ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right) \)

(a)

\(\cfrac { 9\pi }{ 8 } \)

(b)

\(\cfrac { -9\pi }{ 8 } \)

(c)

\(\cfrac { \pi }{ 8 } \)

(d)

\(\cfrac { -\pi }{ 8 } \)

For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ\(\sqrt { \frac { 1 }{ g } } \), where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l.

For each of the following differential equations, determine its order, degree (if exists)
\(x={ e }^{ xy\left( \frac { dy }{ dx } \right) }\)

Show that y = e^−x + mx + n is a solution of the differential equation e^x \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.

Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

*	a	b	c
a	b
b	c	b	a
c	a		c

Find the square root of 6−8i .

Find the modulus and principal argument of the following complex numbers.
\(\sqrt { 3 } \)-i

Simplify the following
\(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

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a, b, c

Evaluate the following limits, if necessary use L’Hopitals rule
(i) \(\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ sinx }\)
(ii) \(\underset { x\rightarrow 0 }{ lim } \cfrac { cotx }{ cot2x } \) 
(iii) \(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }\)

If w=xy+z and x=cot, y=sint, z=t then find \(\frac { dw }{ dt } \)

Find the area bounded by x=0,x=6+5y-y²

Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.

A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40 cm wide from rim to rim and 30 cm deep. The bulb is located at the focus.
(1) What is the equation of the parabola used for reflector?
(2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

An equation of the elliptical part of an optical lens system is \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } \) = 1. The parabolic part of the system has a focus in common with the right focus of the ellipse. The vertex of the parabola is at the origin and the parabola opens to the right. Determine the equation of the parabola.

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\)

1. Sovle \(\left( x+2 \right) \frac { dy }{ dx } =x2+4x-9\) .Also find the domain of the function.

2. Solve : \(\frac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx\)

1. Solve : \(\frac { dy }{ dx } =-\frac { x+ycos }{ 1+sinx } \) .Also find the domain of the function.

2. It is given that the rate at which some bacteria multiply is proportional to the instantaneous number present. If the original number of bacteria doubles in two hours, in how many hours will it be five times.

Simplify: (1+i)¹⁸

Show that \(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 5 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

If \(2cos\alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that ​\({ x }^{ m }{ y }^{ n }+\frac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)​

If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64

Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.

Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 = 0 where z is a complex number.

1. Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r