JEE Advanced 2014 Paper 2 Offline

Let \({z_k}\) = \(\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1,2....,9\)
List-I
P. For each \({z_k}\) = there exits as \({z_j}\) such that \({z_k}\).\({z_j}\) = 1
Q. T...

Given that for each \(a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}\) exists. Let this limit be \(g(a).\) In addition, it is given that the ...

Given that for each \(a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}\) exists. Let this limit be \(g(a).\) In addition, it is given that the ...

List - \(I\)
P.\(\,\,\,\,\) The number of polynomials \(f(x)\) with non-negative integer coefficients of degree \(\le 2\), satisfying \(f(0)=0\) and \(\int_0^1 {f\left( x \right)dx = 1,}\) is
Q.\(\,\,\,\,\) The number of points in the in...

The following integral \(\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx}\) is equal to

The function \(y=f(x)\) is the solution of the differential equation
\({{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,\) in \((-1,1)\) satisfying \(f(0)=0\). Then $$\int\limits_{ - {{\sqrt 3 } \ove...

Let \(f:\left[ {0,2} \right] \to R\) be a function which is continuous on \(\left[ {0,2} \right]\) and is differentiable on \((0,2)\) with \(f(0)=1\). Let
\(F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)dt}\) for $...

The common tangents to the circle \({x^2} + {y^2} = 2\) and the parabola \({y^2} = 8x\) touch the circle at the points \(P, Q\) and the parabola at the points \(R\), \(S\). Then the area of the quadrilateral \(PQRS\) is

Let f1 : R \(\to\) R, f2 : [0, \(\infty\)) \(\to\) R, f3 : R \(\to\) R, and f4 : R \(\to\) [0, \(\infty\)) be defined by
$${f_1}\left( x \right) = \left\{ {\matrix{
{\left| x \right|} & {if\,x < 0,} \cr
{{e^x}} & {if\,x \g...

Match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) List-\(I\)
(P.)\(\,\,\,\,\) Let $$y\left( x \right) = \cos \left( {3{{\cos }^{ - 1}}x} \right),x \i...

Coefficient of \({x^{11}}\) in the expansion of \({\left( {1 + {x^2}} \right)^4}{\left( {1 + {x^3}} \right)^7}{\left( {1 + {x^4}} \right)^{12}}\) is

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)\) and \(S\,\,\left( {a{s^2},2as} \right)\) be distinct points on the parabola \({y^2} = 4ax\). Suppose that $$PQ...

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)\) and \(S\,\,\left( {a{s^2},2as} \right)\) be distinct points on the parabola \({y^2} = 4ax\). Suppose that $$PQ...

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1...

Box \(1\) contains three cards bearing numbers \(1,2,3;\) box \(2\) contains five cards bearing numbers \(1,2,3,4,5;\) and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7.\) A card is drawn from each of the boxes. Let $${x_i}...

Box \(1\) contains three cards bearing numbers \(1,2,3;\) box \(2\) contains five cards bearing numbers \(1,2,3,4,5;\) and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7.\) A card is drawn from each of the boxes. Let $${x_i}...

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

In a triangle the sum of two sides is \(x\) and the product of the same sides is \(y\). If \({x^2} - {c^2} = y\), where \(c\) is the third side of the triangle, then the ratio of the in radius to the circum-radius of the triangle is

The quadratic equation \(p(x)\) \(= 0\) with real coefficients has purely imaginary roots. Then the equation \(p(p(x))=0\) has

For \(x \in \left( {0,\pi } \right),\) the equation \(\sin x + 2\sin 2x - \sin 3x = 3\) has