IIT-JEE 2012 Paper 1 Offline

The point \(P\) is the intersection of the straight line joining the points \(Q(2, 3, 5)\) and \(R(1, -1, 4)\) with the plane \(5x-4y-z=1.\) If \(S\) is the foot of the perpendicular drawn from the point \(T(2, 1, 4)\) to \(QR,\) then the l...

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p'(0)\) is

Let \(f:IR \to IR\) be defined as \(f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.\) The total number of points at which \(f\) attains either a local maximum or a local minimum is

Let \(S\) be the area of the region enclosed by \(y = {e^{ - {x^2}}}\), \(y=0\), \(x=0\), and \(x=1\); then

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle \({x^2}\, + \,{y^2} = 9\) is

Let z be a complex number such that the imaginary part of z is non-zero and \(a\, = \,{z^2} + \,z\, + 1\) is real. Then a cannot take the value

If \(y(x)\) satisfies the differential equation \(y' - y\,tan\,x = 2x\,secx\) and \(y(0)=0,\) then

The ellipse \({E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \({E_2}\) passing through the point \((0, 4)\) circumscribes the rectangle $$...

The function \(f:[0,3] \to [1,29]\), defined by \(f(x) = 2{x^3} - 15{x^2} + 36x + 1\), is

Tangents are drawn to the hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,\) parallel to the straight line \(2x - y = 1,\) The points of contact of the tangents on the hyperbola are

The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant \(K\))

If \(\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4\), then

Let \(f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.\)
x\(\in\)R, then f is

Let \(P = [{a_{ij}}]\) be a 3 \(\times\) 3 matrix and let \(Q = [{b_{ij}}]\), where \({b_{ij}} = {2^{i + j}}{a_{ij}}\) for \(1 \le i,j \le 3\). If the determinant of P is 2, then the determinant of the matrix Q is

Let \(S\) be the focus of the parabola \({y^2} = 8x\) and let \(PQ\) be the common chord of the circle \({x^2} + {y^2} - 2x - 4y = 0\) and the given parabola. The area of the triangle \(PQS\) is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

A ship is fitted with three engines \({E_1},{E_2}\) and \({E_3}\). The engines function independently of each other with respective probabilities \({1 \over 2},{1 \over 4}\) and \({1 \over 4}\). For the ship to be operational at least two o...

The value of \(6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}...} } } } \right)\) is __________.

Let \(\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]\) be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \le...

If \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) are unit vectors satisfying
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\o...