IIT-JEE 2009 Paper 2 Offline

A line with positive direction cosines passes through the point P(2, \(-\)1, 2) and makes equal angles with the coordinate axes. The line meets the plane \(2x + y + z = 9\) at point Q. The length of the line segment PQ equals

Let \(p(x)\) be a polynomial of degree \(4\) having extremum at
\(x = 1,2\) and \(\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2\).
Then the value of \(p (2)\) is

For the function
\($f\left( x \right) = x\cos \,{1 \over x},x \ge 1,\)$

The maximum value of the function \(f(x) = 2{x^3} - 15{x^2} + 36x - 48\) on the set \(A = \{ x|{x^2} + 20 \le 9x|\}\) is __________.

The centres of two circles \({C_1}\) and \({C_2}\) each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segement joining the centres of \({C_1}\) and \({C_2}\) and C a circle touching circles ...

If \({I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,}\) .... then

Let \(f:R \to R\) be a continuous function which satisfies \(f(x) = \int\limits_0^x {f(t)dt}\). Then, the value of \(f(\ln 5)\) is ____________.

Match the statements/expressions in Column I with the values given in Column II:

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
...

The normal at a point \(P\) on the ellipse \({x^2} + 4{y^2} = 16\) meets the \(x\)- axis \(Q\). If \(M\) is the mid point of the line segment \(PQ\), then the locus of \(M\) intersects the latus rectums of the given ellipse at the points

An ellipse intersects the hyperbola \(2{x^2} - 2{y^2} = 1\) orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then

If the function \(f(x) = {x^3} + {e^{x/2}}\) and \(g(x) = {f^{ - 1}}(x)\), then the value of \(g'(1)\) is _________.

The locus of the orthocentre of the triangle formed by the lines \((1 + p)x - py + p(1 + p) = 0,\)
\((1 + q)x - qy + q(1 + q) = 0\)
and \(y = 0\), where \(p \ne q\), is :

The tangent \(PT\) and the normal \(PN\) to the parabola \({y^2} = 4ax\) at a point \(P\) on it meet its axis at points \(T\) and \(N\), respectively. The locus of the centroid of the triangle \(PTN\) is a parabola whose

Let \(\left( {x,\,y,\,z} \right)\) be points with integer coordinates satisfying the system of homogeneous equation:
\($\matrix{ {3x - y - z = 0} \cr { - 3x + z = 0} \cr { - 3x + 2y + z = 0} \cr }\)$
Then the number of su...

Let ABC and ABC' be two non-congruent triangles with sides AB = 4, AC = AC' = 2\(\sqrt2\) and angle B = 30\(^\circ\). The absolute value of the difference between the areas of these triangles is ___________.

The smallest value of \(k\), for which both the roots of the equation
\(${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right) = 0\)$
are real, distinct and have values at least 4, is

If the sum of first \(n\) terms of an A.P. is \(c{n^2}\), then the sum of squares of these \(n\) terms is

Match the statements/expressions in Column I with the values given in Column II:

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
...

For \(0 < \theta < {\pi \over 2},\) the solution (s) of
\($\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \over 4}} \right)\,\cos ec\,\left( {\theta + {{m\pi } \over 4}} \right) = 4\sqrt 2 }\)$ is (are)...