IIT-JEE 2010 Paper 2 Offline

If the distance of the point \(P(1, -2, 1)\) from the plane \(x+2y-2z\)\(\, = \alpha ,\) where \(\alpha > 0,\) is \(5,\) then the foot of the perpendicular from \(P\) to the planes is

Match the statement in Column-\(I\) with the values in Column-\(II\)
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) Column-\(I\)
(A)\(\,\,\,\,\) A line from the origin meets the lines $$\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \ov...

Let \(f\) be a function defined on \(R\) (the set of all real numbers)
such that \(f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}\) for all $$x \in...

Consider the polynomial
\(f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.\)
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)
The area bounded by the curve \(y=f(x)\) and the lines \(x=0,\) \(y=0\) ...

Match the statements in Column I with those in Column II.
[Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively, the imaginary part and the real part of z.]
Column I
(A) The set of points z sat...

Let \(f\) be a real-valued function defined on the interval \((-1, 1)\) such that
\({e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,}\) for all \(x \in \left( { - 1,1} \right)\),
and let \({f^{ - 1}}\) be the ...

Tangents are drawn from the point \(P(3, 4)\) to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) touching the ellipse at points \(A\) and \(B\).
The coordinates of \(A\) and \(B\) are

Tangents are drawn from the point \(P(3, 4)\) to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) touching the ellipse at points \(A\) and \(B\).
The orthocentre of the triangle \(PAB\) is

Tangents are drawn from the point \(P(3, 4)\) to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) touching the ellipse at points \(A\) and \(B\).
The equation of the locus of the point whose distances from the point \(P\) and the l...

Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to :

Consider the polynomial
\(f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.\)
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)
The function\(f'(x)\) is

Consider the polynomial
\(f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.\)
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)
The real numbers lies in the interval

For \(r = 0,\,1,....,\) let \({A_r},\,{B_r}\) and \({C_r}\) denote, respectively, the coefficient of \({X^r}\) in the expansions of \({\left( {1 + x} \right)^{10}},\) \({\left( {1 + x} \right)^{20}}\) and \({\left( {1 + x} \right)^{30}}.\)...

Let $k$ be a positive real number and let
$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{a...

A signal which can be green or red with probability \({4 \over 5}\) and \({1 \over 5}\) respectively, is received by station A and then transmitted to station \(B\). The probability of each station receving the signal correctly is $${3 \ove...

Consider a triangle \(ABC\) and let \(a, b\) and \(c\) denote the lengths of the sides opposit to vertices \(A, B\) and \(C\) respectively. Suppose \(a = 6,b = 10\) and the area of the triangle is \(15\sqrt 3\), if \(\angle ACB\) is obtuse...

Let \({a_1},\,{a_{2\,}},\,{a_3}\)......,\({a_{11}}\) be real numbers satisfying \({a_1} = 15,27 - 2{a_2} > 0\,\,and\,\,{a_k} = 2{a_{k - 1}} - {a_{k - 2}}\,\,for\,k = 3,4,........11\). if $$\,\,\,{{a_1^2 + a_2^2 + .... + a_{11}^2} \over {11}...

Two parallel chords of a circle of radius 2 are at a distance \(\sqrt 3 + 1\) apart. If the chords subtend at the center , angles of \({\pi \over k}\) and \({{2\pi } \over k},\) where\(k > 0,\) then the value of \(\left[ k \right]\) is
[N...

Two adjacent sides of a parallelogram \(ABCD\) are given by
\(\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k\) and \(\,\overrightarrow {AD} = -\widehat i + 2\widehat j + 2\widehat k\)
The side \(AD\) is rotated by an a...