IIT-JEE 2012 Paper 2 Offline

The equation of a plane passing through the line of intersection of the planes \(x+2y+3z=2\) and \(x-y+z=3\) and at a distance \({2 \over {\sqrt 3 }}\) from the point \((3, 1, -1)\) is

If the straight lines \(\,{{x - 1} \over 2} = {{y + 1} \over k} = {z \over 2}\) and \({{x + 1} \over 5} = {{y + 1} \over 2} = {z \over k}\) are coplanar, then the plane (s) containing these two lines is (are)

Let \(f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}\) for all \(x \in IR\) and let
\(g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}\) ...

If \(f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt\) for all \(x \in \left( {0,\infty } \right),\) then

Let \(f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}\) for all \(x \in IR\) and let
\(g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt}\) ...

A tangent PT is drawn to the circle \({x^2}\, + {y^2} = 4\) at the point P \(\left( {\sqrt 3 ,1} \right)\). A straight line L, perpendicular to PT is a tangent to the circle \({(x - 3)^2}\) + \({y^2}\) = 1.
A possible equation of L is

A tangent PT is drawn to the circle \({x^2}\, + {y^2} = 4\) at the point P \(\left( {\sqrt 3 ,1} \right)\). A straight line L, perpendicular to PT is a tangent to the circle \({(x - 3)^2}\) + \({y^2}\) = 1
A common tangent of the two circl...

The value of the integral \(\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx}\) is

Let \(f:( - 1,1) \to R\) be such that \(f(\cos 4\theta ) = {2 \over {2 - {{\sec }^2}\theta }}\) for \(\theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 4},{\pi \over 2}} \right)\). Then the value(s) of $$f\left( {{1 \o...

For every integer n, let an and bn be real numbers. Let function f : R \(\to\) R be given by
$$f(x) = \left\{ {\matrix{
{{a_n} + \sin \pi x,} & {for\,x \in [2n,2n + 1]} \cr
{{b_n} + \cos \pi x,} & {for\,x \in (2n - 1,2n)} \cr

} ...

If P is a 3 \(\times\) 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 \(\times\) 3 identity matrix, then there exists a column matrix $$X = \left[ {\matrix{
x \cr
y \cr
z \cr

} } \right] \ne \...

If the ad joint of a 3 \(\times\) 3 matrix P is \(\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]\), then the possible value(s) of the determinant of P is(are)

Let \({{a_n}}\) denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let \({{b_n}}\) = the number of such n-digit integers ending with digit 1 and \({{c_n}}\) =t...

Let \({{a_n}}\) denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let \({{b_n}}\) = the number of such n-digit integers ending with digit 1 and \({{c_n}}\) =t...

Let \(X\) and \(Y\) be two events such that \(P\left( {X|Y} \right) = {1 \over 2},\) \(P\left( {Y|X} \right) = {1 \over 3}\) and \(P\left( {X \cap Y} \right) = {1 \over 6}.\) Which of the following is (are) correct ?

Four fair dice \({D_1,}\) \({D_2,}\) \({D_3}\) and \({D_4}\) ; each having six faces numbered \(1, 2, 3, 4, 5\) and \(6\) are rolled simultaneously. The probability that \({D_4}\) shows a number appearing on one of \({D_1},\) \({D_2}\) and ...

Let \(PQR\) be a triangle of area \(\Delta\) with \(a=2\), \(b = {7 \over 2}\) and \(c = {5 \over 2}\); where \(a, b,\) and \(c\) are the lengths of the sides of the triangle opposite to the angles at \(P.Q\) and \(R\) respectively. Then $...

Let \(\alpha\)(a) and \(\beta\)(a) be the roots of the equation \((\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1) = 0\) where \(a > - 1\). Then \(\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a)\) and ...

Let \({a_1},{a_2},{a_3},.....\) be in harmonic progression with \({a_1} = 5\) and \({a_{20}} = 25.\) The least positive integer \(n\) for which \({a_n} < 0\) is

If \(\overrightarrow a\) and \(\overrightarrow b\) are vectors such that \(\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29}\) and $$\,\overrightarrow a \times \left( {2\widehat i + 3\widehat j + 4\widehat k} \right...