IIT-JEE 2005

Find the equation of the plane containing the line \(2x-y+z-3=0,3x+y+z=5\) and at a distance of \({1 \over {\sqrt 6 }}\) from the point \((2, 1, -1).\)

If \(\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},\) for all \({x_1},{x_2} \in R\). Find the equation of tangent to the cuve \(y = f\left( x \right)\) at the point \((1, 2)\).

If \(p(x)\) be a polynomial of degree \(3\) satisfying \(p(-1)=10, p(1)=-6\) and \(p(x)\) has maxima at \(x=-1\) and \(p'(x)\) has minima at \(x=1\). Find the distance between the local maxima and local minima of the curve.

Find the area bounded by the curves \({x^2} = y,{x^2} = - y\) and \({y^2} = 4x - 3.\)

If $$\left[ {\matrix{
{4{a^2}} & {4a} & 1 \cr
{4{b^2}} & {4b} & 1 \cr
{4{c^2}} & {4c} & 1 \cr

} } \right]\left[ {\matrix{
{f\left( { - 1} \right)} \cr
{f\left( 1 \right)} \cr
{f\left( 2 \right)} \cr

} } \r...

Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circles at their points of contact, find the distance of P from the points of contact.

If one the vertices of the square circumscribing the circle \(\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i\). Find the other vertices of the square.

Evaluate \(\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx\)

If length of tangent at any point on the curve \(y=f(x)\) intecepted between the point and the \(x\)-axis is length \(1.\) Find the equation of the curve.

\(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable
function such that \(\left| {f\left( x \right)} \right| \le 1\) and \(f'(x)=g(x).\)
If \({f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.\) Prove that there e...

Find the equation of the common tangent in \({1^{st}}\) quadrant to the circle \({x^2} + {y^2} = 16\) and the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over 4} = 1\). Also find the length of the intercept of the tangent between the coordin...

Tangents are drawn from any point on the hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) to the circle \({x^2} + {y^2} = 9\).Find the locus of mid-point of the chord of contact.

If total number of runs scored in n matches is \(\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,\) where \(n > 1\), and the runs scored in the \({k^{th}}\) match are given by k. \(\,{2^{n + 1 - k}}\), where \(1 \le k \le n\). ...

A person goes to office either by car, scooter, bus or train, the probability of which being \({1 \over 7},{3 \over 7},{2 \over 7}\) and \({1 \over 7}\) respectively. Probability that he reaches office late, if he takes car, scooter, bus o...

In an equilateral triangle, \(3\) coins of radii \(1\) unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

The area of the triangle formed by intersection of a line parallel to \(x\)-axis and passing through \(P (h, k)\) with the lines \(y = x\) and \(x + y = 2\) is \(4{h^2}\). Find the locus of the point \(P\).

Find the range of values of \(\,t\) for which \($2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].\)$

If the incident ray on a surface is along the unit vector \(\widehat v\,\,,\) the reflected ray is along the unit vector \(\widehat w\,\,\) and the normal is along unit vector \(\widehat a\,\,\) outwards. Express \(\widehat w\,\,\) in term...