Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).

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:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

Prove the following properties z is real if and only if z = \(\bar { z } \)

Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }-\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is purely imaginary

Evaluate the following if z = 5−2i and w = −1+3i
z² + 2zw + w²

If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

Find the domain of sin⁻¹(2−3x²)

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

A line 3x+4y+10 = 0 cuts a chord of length 6 units on a circle with centre of the circle (2,1). Find the equation of the circle in general form.

Find the equation of the parabola with focus \(\left( -\sqrt { 2 } ,0 \right) \) and directrix x =\(\sqrt { 2 } \).

Find the equation of the hyperbola with vertices (0, ±4) and foci(0, ±6).

Find the equations of tangent and normal to the parabola x²+6x+4y+5 = 0 at (1, -3) .

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

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

\(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =-\hat { i } +2\hat { j } -4\hat { k } ,\vec { c } =\hat { i } +\hat { j } +\hat { k } \) then find the value of \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\).

A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?

If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = \(\sqrt { 3 } x\) then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres.

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x) ≤ 2 for all x. Justify you answer.

Evaluate: \(\underset{x\rightarrow \infty}{lim}(\frac{e^{x}}{x^{m}}), m\in N\)

A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.

Show that \(\int ^\frac{2\pi}{0}_{0}\) g(cos x)dx = 2 \(\int ^{\pi}_{0}\) g(cosx)dx where g(cos x) is a function of cos x