IIT-JEE 2004

\({P_1}\) and \({P_2}\) are planes passing through origin. \({L_1}\) and \({L_2}\) are two line on \({P_1}\) and \({P_2}\) respectively such that their intersection is origin. Show that there exists points \(A, B, C,\) whose permutation $$A...

A parallelopiped \('S'\) has base points \(A, B, C\) and \(D\) and upper face points \(A',\) \(B',\) \(C'\) and \(D'.\) This parallelopiped is compressed by upper face \(A'B'C'D'\) to form a new parallelopiped \('T'\) having upper face poin...

Find the equation of plane passing through \((1, 1, 1)\) & parallel to the lines \({L_1},{L_2}\) having direction ratios \((1,0,-1),(1,-1,0).\) Find the volume of tetrahedron formed by origin and the points where these planes intersect the ...

Using Rolle's theorem, prove that there is at least one root
in \(\left( {{{45}^{1/100}},46} \right)\) of the polynomial
\(P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035\).

Prove that for \(x \in \left[ {0,{\pi \over 2}} \right],\) \(\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }\). Explain
the identity if any used in the proof.

Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, -1) and cutting orthogonally the circle having line segment joining (0, 3) and (- 2, -1) as diameter.

Find the centre and radius of circle given by \(\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,\)
where, $${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ ...

Find the value of \(\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx}\)

If \(y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,}\) then find \({{dy} \over {dx}}\) at \(x = \pi\)

A curve \('C''\) passes through \((2,0)\) and the slope at \((x,y|)\) as \(\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}\). Find the equation of the curve. Find the area bounded by curve and \(x\)-axis in fourth ...

Tangent is drawn to parabola \({y^2} - 2y - 4x + 5 = 0\) at a point \(P\) which cuts the directrix at the point \(Q\). \(A\) point \(R\) is such that it divides \(QP\) externally in the ratio \(1/2:1\). Find the locus of point \(R\)

Prove by permulation or otherwise \({{({n^2})!} \over {{{(n!)}^n}}}\) is an integer \((n \in {1^ + })\).

A box contains \(12\) red and \(6\) white balls. Balls are drawn from the box one at a time without replacement. If in \(6\) draws there are at least \(4\) white balls, find the probability that exactly one white is drawn in the next two dr...

\(A\) and \(B\) are two independent events. \(C\) is even in which exactly one of \(A\) or \(B\) occurs. Prove that \(P\left( C \right) \ge P\left( {A \cup B} \right)P\left( {\overline A \cap \overline B } \right)\)

If \(a,\,b,c\) are positive real numbers. Then prove that
\(${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left( {c + 1} \right)^7} > {7^7}\,{a^4}{b^4}{c^4}\)$

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) are distinct vectors such that
\(\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d\) and $$\overrightarr...