IIT-JEE 2011 Paper 1 Offline

Let the straight line \(x=b\) divide the area enclosed by
\(y = {\left( {1 - x} \right)^2},y = 0,\) and \(x=0\) into two parts \({R_1}\left( {0 \le x \le b} \right)\) and
\({R_2}\left( {b \le x \le 1} \right)\) such that $${R_1} - {R_2} =...

If z is any complex number satisfying \(\,\left| {z - 3 - 2i} \right| \le 2\), then the minimum value of \(\left| {2z - 6 + 5i} \right|\) is

The value of \(\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx}\) is

Let \(f:[1,\infty ) \to [2,\infty )\) be a differentiable function such that \(f(1) = 2\). If \(6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5}\) for all \(x \ge 1\), then the value of f(2) is ___________.

Let \(f\left( \theta \right) = \sin \left( {{{\tan }^{ - 1}}\left( {{{\sin \theta } \over {\sqrt {\cos 2\theta } }}} \right)} \right),\) where \(- {\pi \over 4} < \theta < {\pi \over 4}.\)
Then the value of $${d \over {d\left( {\tan \...

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be reciprocal to that of the ellipse \({x^2} + 4{y^2} = 4\). If the hyperbola passes through a focus of the ellipse, then

Let f : R \(\to\) R be a function such that \(f(x + y) = f(x) + f(y),\,\forall x,y \in R\). If f(x) is differentiable at x = 0, then

Let M and N be two 3 \(\times\) 3 non-singular skew symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2N2(MTN)\(-\)1(MN\(-\)1)T is equal to

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Let \(\omega\) be a solution of \({x^3} - 1 = 0\) with \({\mathop{\rm Im}\nolimits} (\omega ) > 0\). If a = 2 with b and c satisfying (E), then the value of \({3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}\) is ...

Let b = 6, with a and c satisfying (E). If \(\alpha\) and \(\beta\) are the roots of the quadratic equation ax2 + bx + c = 0, then \(\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}}\) is

Consider the parabola \({y^2} = 8x\). Let \({\Delta _1}\) be the area of the triangle formed by the end points of its latus rectum and the point \(P\left( {{1 \over 2},2} \right)\) on the parabola and \({\Delta _2}\) be the area of the tria...

The probability of the drawn ball from \({U_2}\) being white is

Given that the drawn ball from \({U_2}\) is white, the probability that head appeared on the coin is

Let \(\alpha\) and \(\beta\) be the roots of \({x^2} - 6x - 2 = 0,\) with \(\alpha > \beta .\) If \({a_n} = {\alpha ^n} - {\beta ^n}\) for \(\,n \ge 1\) then the value of \({{{a_{10}} - 2{a_8}} \over {2{a_9}}}\) is

Let \(\left( {{x_0},{y_0}} \right)\) be the solution of the following equations
\(\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \right)}^{\ell n3}}} \cr {{3^{\ell nx}}\, = {2^{\ell ny}}} \cr }\)
Then \({x_0}\)...

The minimum value of the sum of real numbers \({a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}\) and \({a^{10}}\) where \(a > 0\) is

Let \({{a_1}}\), \({{a_2}}\), \({{a_3}}\)........ \({{a_{100}}}\) be an arithmetic progression with \({{a_1}}\) = 3 and \({S_p} = \sum\limits_{i = 1}^p {{a_i},1 \le } \,p\, \le 100\). For any integer n with \(1\,\, \le \,n\, \le 20\), let m...

A straight line \(L\) through the point \((3, -2)\) is inclined at an angle \({60^ \circ }\) to the line \(\sqrt {3x} + y = 1.\) If \(L\) also intersects the x-axis, then the equation of \(L\) is

Let \(P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \}\) and \(Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \}\) be two sets. Then

The positive integer value of \(n\, > \,3\) satisfying the equation \({1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}\) is

The vector (s) which is/are coplanar with vectors \({\widehat i + \widehat j + 2\widehat k}\) and \({\widehat i + 2\widehat j + \widehat k,}\) and perpendicular to the vector \({\widehat i + \widehat j + \widehat k}\) is/are

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k\) be three vectors. A vector $$\overrightarrow v ...