IIT-JEE 2003

(i) Find the equation of the plane passing through the points \((2, 1, 0), (5, 0, 1)\) and \((4, 1, 1).\)
(ii) If \(P\) is the point \((2, 1, 6)\) then find the point \(Q\) such that \(PQ\) is perpendicular to the plane in (i) and the mid ...

Using the relation \(2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0\) or otherwise,
prove that \(\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]\)

If the function \(f:\left[ {0,4} \right] \to R\) is differentiable then show that
(i)\(\,\,\,\,\,\) For \(a, b\)\(\,\,\)$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left(...

Find a point on the curve \({x^2} + 2{y^2} = 6\) whose distance from
the line \(x+y=7\), is minimum.

If \(P(1)=0\) and \({{dp\left( x \right)} \over {dx}} > P\left( x \right)\) for all \(x \ge 1\) then prove that
\(P(x)>0\) for all \(x>1\).

For the circle \({x^2}\, + \,{y^2} = {r^2}\), find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.

Prove that there exists no complex number z such that \(\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1\) where \(\left| {{a_r}} \right| < 2\).

If \({z_1}\) and \({z_2}\) are two complex numbers such that \(\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,\) then prove that \(\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1\).

If \(f\) is an even function then prove that
\(\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.}\)

A right circular cone with radius \(R\) and height \(H\) contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant \(= k > 0\). Find the time after which the come is empty.

Prove that
$${2^k}\left( {\matrix{
n \cr
0 \cr

} } \right)\left( {\matrix{
n \cr
k \cr

} } \right) - {2^{^{k - 1}\left( {\matrix{
n \cr
2 \cr

} } \right)}}\left( {\matrix{
n \cr
1 \cr

} } \ri...

Normals are drawn from the point \(P\) with slopes \({m_1}\), \({m_2}\), \({m_3}\) to the parabola \({y^2} = 4x\). If locus of \(P\) with \({m_1}\) \({m_2}\)\(= \alpha\) is a part of the parabola itself then find \(\alpha\).

\(A\) is targeting to \(B, B\) and \(C\) are targeting to \(A.\) Probability of hitting the target by \(A,B\) and \(C\) are \({2 \over 3},{1 \over 2}\) and \({1 \over 3}\) respectively. If \(A\) is hit then find the probability that \(B\) ...

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is \(p.\) If he fails in one of the exams then the probability of his passing in the next exam is \({p \over 2}\) otherwi...

If \({I_n}\) is the area of \(n\) sided regular polygon inscribed in a circle of unit radius and \({O_n}\) be the area of the polygon circumscribing the given circle, prove that
$$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {...

If \({x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0\) where \(a,\,b\, \in \,R\) then find the values of a for which equation has unequal real roots for all values of \(b\).

If a, b, c are in A.P., \({a^2}\), \({b^2}\), \({c^2}\) are in H.P., then prove that either a = b = c or a, b, \({ - {c \over 2}}\) form a G.P.

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,\) are three non-coplanar unit vectors and \(\alpha ,\beta ,\gamma\) are the angles between \(\overrightarrow u\) and \(\overrightarrow v\) and \(\overrightarrow w ,\) $$\ove...