If \(A=\left| \begin{matrix} 6 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{matrix} \right| \), then show that |2A| = 8|A|:

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as: R = {(a, b) : b = a + 1}, is reflexive, symmetric or transitive.

If e^y (x+1) = 1, show that dy/dx = -e^y

Find the general solution of differential equation \(\log { \left( \frac { dy }{ dx } \right) } =x+1\)

Show that : \({ tan }^{ -1 }\left( \frac { 3a^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3a{ x }^{ 2 } } \right) =3tan^{ -1 }\left( \frac { x }{ a } \right) \)

\(\int { \frac { dx }{ 1+sinx } } \)

Solve the system of linear equations, using matrix method in
4x-3y = 3
3x-5y = 7

Draw the graph of the following LLP:\(3x+y\le 17,x,y\ge 0\)

Find matrices X and Y, if X + Y =\(\begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}\) and X - Y =\(\begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}\)

Differentiate \(\log(x^{sin x}+\cot ^{2}x)\) w.r.t x

Prove that: \(2ta{ n }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) ={ sin }^{ -1 }\left( \frac { 31 }{ 25\sqrt { 2 } } \right) \)

Evaluate the integral: \(\int{\sqrt {1+x^2}\over x}dx.\)

Prove that \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\)

Let * be a binary operation defined on Q x Q by (a, b) * (c, d) = (ac, b + ad). where Q is the set of rational numbers. Determine, whether * is commutative and associative. Find the identity element for * and the invertible elements of Q x Q.

If \(\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } \) are unit vectors such that \(\overrightarrow { a } .\overrightarrow { b } =\overrightarrow { a } .\overrightarrow { c } =0\) and the angle between \(\overrightarrow { b } \) and \(\overrightarrow { c } \) is \(\frac{\pi}{6}\) , then prove that:
(i) \(\overrightarrow { a } =\pm 2\left( \overrightarrow { b } \times \overrightarrow { c } \right) \)
(ii) \(\left[ \overrightarrow { a } +\overrightarrow { b } ,\overrightarrow { b } +\overrightarrow { c } ,\overrightarrow { c } +\overrightarrow { a } \right] =\pm 1\)

Determine the product \(\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] ,\)  and use it to solve the system of equations: x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.

Evaluate : \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 3 } }{ \frac { dx }{ 1+\sqrt { \cot { x } } } } \)