Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

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

Find the length of the tangent from (2, -3) to the circle x² + y² - 8x - 9y + 12 = 0.

Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

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∆=4

At what point on the curve y = x² on [-2, 2] is the tangent parallel to X-axis?

Verify Rolle ’s Theorem for \(f(x)=\left| x-1 \right| ,O\le x\le 2\) 

Use differentials to find \(\sqrt{25.2}\)

If w=xye^xy find \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \)

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)

Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

Find the area bounded by y=cosx,y=x+1,y=0.

Solve: \(\frac{dy}{dx}=1+e^{x-y}\)

Show that p v (q ∧ r) is a contingency.