IIT-JEE 2011 Paper 2 Offline

Let f \(:\)\(\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]\) be a continuous function such that
\(f\left( x \right) = f\left( {1 - x} \right)\) for all \(x \in \left[ { - 1,2} \right]\)
Let $${R_1} = \int\limits_{ - 1}^2 {xf\left(...

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.

The straight line 2x - 3y = 1 divides the circular region \({x^2}\, + \,{y^2}\, \le \,6\) into two parts.
If $$S = \left\{ {\left( {2,\,{3 \over 4}} \right),\,\left( {{5 \over 2},\,{3 \over 4}} \right),\,\left( {{1 \over 4} - \,{1 \over 4}...

Let \(\omega = {e^{{{i\pi } \over 3}}}\), and a, b, c, x, y, z be non-zero complex numbers such that
\(a + b + c = x\)
\(a + b\omega + c{\omega ^2} = y\)
\(a + b{\omega ^2} + c\omega = z\)
Then the value of $${{{{\left| x \righ...

Let \(y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,\) where \(f'(x)\) denotes \({{df\left( x \right)} \over {dx}}\) and \(g(x)\) is a given non-constant differ...

Let f(x) = x2 and g(x) = sin x for all x \(\in\) R. Then the set of all x satisfying \((f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)\), where \((f \circ g)(x) = f(g(x))\), is

Let \(f:(0,1) \to R\) be defined by \(f(x) = {{b - x} \over {1 - bx}}\), where b is a constant such that \(0 < b < 1\). Then

Match the statements given in Column I with the intervals/union of intervals given in Column II :

Let \(P(6, 3)\) be a point on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal at the point \(P\) intersects the \(x\)-axis at \((9, 0)\), then the eccentricity of the hyperbola is

If \(\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta\), \(b > 0\) and \(\theta \in ( - \pi ,\pi ]\), then the value of \(\theta\) is

If \(f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} < x \le 0} \cr {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.\), then

Let \(\omega\) \(\ne\) 1 be a cube root of unity and S be the set of all non-singular matrices of the form \(\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]\), where each of a,...

Let M be a 3 \(\times\) 3 matrix satisfying \(M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]\), $$M\left[ {\matrix{
1 \cr
{ - 1} \cr
0 \c...

Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by

Let \((x, y)\) be any point on the parabola \({y^2} = 4x\). Let \(P\) be the point that divides the line segment from \((0, 0)\) to \((x, y)\) in the ratio \(1 : 3\). Then the locus of \(P\) is

Let \(E\) and \(F\) be two independent events. The probability that exactly one of them occurs is \(\,{{11} \over {25}}\) and the probability of none of them occurring is \(\,{{2} \over {25}}\). If \(P(T)\) denotes the probability of occurr...

The number of distinct real roots of \({x^4} - 4{x^3} + 12{x^2} + x - 1 = 0\)

A value of \(b\) for which the equations
\($\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr }\)$
have one root in common is

Let \(\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j\) and \(\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k\) be three given vectors. If \(\overrightarrow r\) is a vector such t...

Match the statements given in Column -\(I\) with the values given in Column-\(II.\)
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) Column-\(I\)
(A) \(\,\,\,\,\)If $$\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \wideh...