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

Find the principal value of -2i.

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

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

Find the number of real solutions of sin (e^x) -5^x + 5^-x

Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)

Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)

If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)

Find the value of p so that 3x + 4y - p = 0 is a tangent to the circle x² +y² - 64 = 0.

Find the equation of the ellipse whose latus rectum is 5 and e = \(\frac { 2 }{ 3 } \)

For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.

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

Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0

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Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

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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.

Find the equation of normal to the curve y⁴=ax²at(a,a)

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) \(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }\)

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 local extrema for the following functions using second derivate test.
(i) x³-x
(ii) \({ x }^{ 3 }-3{ x }^{ 2 }+1,-\frac { 1 }{ 2 } \le x\le 4\)

Find two positive numbers whose product is 100 and whose sum is minimum..