Let $$\omega $$ $$ = - {1 \over 2} + i{{\sqrt 3 } \over 2},$$ then the value of the det. $$\,\left| {\matrix{ 1 &am

Let a complex number $$\alpha ,\,\alpha \ne 1$$, be a root of the equation $${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, whe

Use mathematical induction to show that $${\left( {25} \right)^{n + 1}} - 24n + 5735$$ is divisible by $${\left( {24} \

Let a, b be positive real numbers. If a, $${{A_1},{A_2}}$$, b are in arithmetic progression, a, $${{G_1},{G_2}}$$, b are

A straight line $$L$$ through the origin meets the lines $$x + y = 1$$ and $$x + y = 3$$ at $$P $$ and $$Q$$ respectivel

A straight line $$L$$ with negative slope passes through the point $$(8, 2)$$ and cuts the positive coordinate axes at p

A triangle with vertices $$(4, 0), (-1, -1), (3, 5)$$is

Locus of mid point of the portion between the axes of $$x$$ $$\cos \alpha + y\sin \alpha = p$$ where $$p$$ is constant

If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$ axis then

The pair of lines represented by $$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$ are perpendicular to each othe

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellips

Prove that $$\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}} $$.

For any natural number $$m$$, evaluate $$\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m}

Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies t

A box contains $$N$$ coins, $$m$$ of which are fair and the rest are biased. The probability of getting a head when a fa

Let $$V$$ be the volume of the parallelopiped formed by the vectors $$\overrightarrow a = {a_1}\widehat i + {a_2}\wideh