IIT-JEE 2002

Find the area of the region bounded by the curves \(y = {x^2},y = \left| {2 - {x^2}} \right|\) and \(y=2,\) which lies to the right of the line \(x=1.\)

Let \(\omega\) \(= - {1 \over 2} + i{{\sqrt 3 } \over 2},\) then the value of the det.
\(\,\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - 1 - {\omega ^2}} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^4}} \cr } } \right|\) ...

Let a complex number \(\alpha ,\,\alpha \ne 1\), be a root of the equation \({z^{p + q}} - {z^p} - {z^q} + 1 = 0\), where p, q are distinct primes. Show that either $$1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha...

Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

For any natural number \(m\), evaluate
\(\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m} + 6} \right)}^{l/m}}dx,x > 0.}\)

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

Use mathematical induction to show that
\({\left( {25} \right)^{n + 1}} - 24n + 5735\) is divisible by \({\left( {24} \right)^2}\) for all \(= n = 1,2,...\)

A box contains \(N\) coins, \(m\) of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is \(1/2\), while it is \(2/3\) when a biased coin is tossed. A coin is drawn from the box at random a...

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 in geometric progression and a, \({{H_1},{H_2}}\), b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {...

A straight line \(L\) through the origin meets the lines \(x + y = 1\) and \(x + y = 3\) at \(P\) and \(Q\) respectively. Through \(P\) and \(Q\) two straight lines \({L_1}\) and \({L_2}\) are drawn, parallel to \(2x - y = 5\) and $$3x + y...

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 other for

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 is

A straight line \(L\) with negative slope passes through the point \((8, 2)\) and cuts the positive coordinate axes at points \(P\) and \(Q\). Find the absolute minimum value of \(OP + OQ,\) as \(L\) varies, where \(O\) is the origin.

Let \(V\) be the volume of the parallelopiped formed by the vectors \(\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,\) \(\,\,\,\,\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k,\) $$\,\,...