Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.

Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y +  z = 10.

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

Solve: 2x+2^x-1+2^x-2 = 7x+7^x-1+7^x-2

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

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

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

Solve: cos(tan^-1x) = \(sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right) \) 

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.

Show that the line x + y + 1 = 0 touches the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 15 } \) = 1 and find the co-ordinates of the point of contact

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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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 \(\overset { \rightarrow }{ c } \)

Find the equation of normal to the cure y = sin²x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).

The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?

Verify Rolle’s theorem for f(x)=ex sinx,\(0\le x\le \pi \)

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) \({ x }^{ 3 }-3{ x }^{ 2 }+1,-\frac { 1 }{ 2 } \le x\le 4\)

Evaluate : \(\underset { \left( x,y,z \right) \rightarrow \left( -1,0,4 \right) }{ lim } \frac { { x }^{ 2 }-{ ze }^{ zy } }{ 6x+2y-2z } \)