For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\) 

The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3

Find the angle of intersection of the curves 2y² = x³ and y² = 32x.

If the curves 4x=y² and 4xy=k cut at right angles show that k²=512.

Find the local maximum and local minimum values for f(x)=12x²-2x²-x⁴.

Find \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } } \)  at x = 2, y = 3 if f(x,y) = 2x² + 3y² - 2xy

Find \(\frac { \partial w }{ \partial u } ,\frac { \partial w }{ \partial v } \) if w=sin^-1(x,y) where x=u+v,y=u-v

Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3

Find the area of the loop of the curve 3ay²=x(x-a)²

Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\frac { \pi }{ 3 } \) and x- axis are as 2 : 3.

The surface area of a balloon being inflated changes at a constant rate. If initially, its radius 3 units and after 2 seconds it is 5 units, find the radius after t seconds.

Sovle : (x+y+1)²dy=dx,y(-1)=0

Construct the truth table for (p ∧ q) v r.