Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

Find the real solutions of the equation
\({ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi }{ 2 } \)

Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)

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s, t

If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)

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lies in the plane containing \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \)

The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

The volume of a cube is increasing at the rate of 8cm³/s.How fast is the surface area increasing when the length of an edge is 12cm?

Find the intervals of monotonicities and find the local extremum for the following functions
i) f(x) = 20 - x - x²
ii) f(x) = x(x-1) (x+1) on [0, 2]

Find the intervals of concavity and the point of inflection of the function f(x) = 2x² + 5x² - 4x

Find the approximate value of f (3.02) where f(x) = 3x² + 5x +3.

If f = \(\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } \) then show that = \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) = -f

Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)

Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 0,0 \right) }{ lim } \frac { { x }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \)

Find the area between the curve y=1-|x| and x-axis

Solve: \(\frac{dy}{dx}+y=cos x\)

Obtain the D.E of all circles of radius ‘r’

Solve : cos x(1 + cos y) dx - sin y(1 + sinx) dy = 0.