Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.

Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

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. 

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form

If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x² = sin 2a

An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

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plane

Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \) 

If Rolle's theorem holds for f (x) = x³ + bx² + ax + 5 on [1,3] with c = \(\left( 2+\frac { 1 }{ \sqrt { 3 } } \right) \) find the values of a and b.

If f(x) = a log x + bx²+ x has entreme values at x = - 1 and x = 2, then find a and b.

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

Gas is escaping from a spherical balloon at the rate of 900 cm³/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?

Find the points of inflection and determine the intervals of concavity of \(y={ e }^{ -{ x }^{ 2 } }\)

Find the local maximum and local minimum values of f(x)=x⁴-3x+3x²-x.

Find the area bounded by x=at²,y=2at between the ordinates corresponding to t = 1 and t = 2.

Find the area bounded by the curve y=xe^x and y=xe^-x and the line x=1.

Find the area bounded by the curve xy2=a2(a-x) and the y-axis