If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.

If z = x + iy, find the following in rectangular form.
\(Re\left( \frac { 1 }{ z } \right) \)

If z_i = 2− i and z₂ = -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

If p is real, discuss the nature of the roots of the equation 4x²+ 4px + p + 2 = 0 in terms of p.

Find the principal value of
 \({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

Find the principal value of cos⁻¹\(\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

Find the general equation of a circle with centre (-3, -4) and radius 3 units.

Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter.

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a² = b² + c² − 2bc cos A
(ii) b² = c² + a² − 2ca cos B
(iii) c² = a² + b² − 2ab cos C

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a = b cos C + c cos B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A

For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2

A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

Consider g(x,y) = \(\frac { 2{ x }^{ 2 }y }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x, y) ≠ (0, 0) and g(0, 0) = 0 Show that g is continuous on R²

Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

Evaluate \(\int _{ 0 }^{ 1 }{ xdx } \), as the limit of a sum.

Evaluate \(\int _{ 0 }^{ 1 }{ x^3dx } \), as the limit of a sum.

For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)

For each of the following differential equations, determine its order, degree (if exists)
\({ { \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) } }^{ 2 }+{ \left( \frac { dy }{ dx } \right) }^{ 2 }=xsin\left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) \)

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.

Compute P(X = k) for the binomial distribution, B(n, p) where
\(n=10, p=\frac{1}{5}, k=4\)

Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z

Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)

Determine whether ∗ is a binary operation on the sets given below.
a*b = min (a, b) on A = {1, 2, 3, 4, 5}

Find the values of the following:
\(\int ^\frac{\pi}{2}_{0}\)sin ⁵x cos⁴xdx

Evaluate the following definite integrals:
\(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)

Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

Find the area bounded by y=x²+2,x-x-axis, x=1 and x=2

Evaluate \(\int_{0}^{4}|x-1| d x\)