JEE Advanced 2016 Paper 2 Offline

Let \(P\) be the image of the point \((3,1,7)\) with respect to the plane \(x-y+z=3.\) Then the equation of the plane passing through \(P\) and containing the straight line \({x \over 1} = {y \over 2} = {z \over 1}\) is

Let f: R \(\to \left( {0,\infty } \right)\) and g : R \(\to\) R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'\((2)\) \(=\) g\((2)=0\), f''\((2)\)\(\ne 0\) and g'\((2)\) \(\ne 0\). If ...

Area of the region \(\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} \right\}\)
is equal to

Let \(a,\,b \in R\,and\,{a^{2\,}} + {b^2} \ne 0\). Suppose \(S = \left\{ {Z \in C:Z = {1 \over {a + ibt}}, + \in R,t \ne 0} \right\}\), where \(i = \sqrt { - 1}\). Ifz = x + iy and z \(\in\) S, then (x, y) lies on

Let $$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \ove...

The value of \(\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx}\) is equal to

Let \({F_1}\left( {{x_1},0} \right)\) and \({F_2}\left( {{x_2},0} \right)\) for \({{x_1} < 0}\) and \({{x_2} > 0}\), be the foci of the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 8} = 1\). Suppose a parabola having vertex at the origin an...

Let \({F_1}\left( {{x_1},0} \right)\) and \({F_2}\left( {{x_2},0} \right)\) for \({{x_1} < 0}\) and \({{x_2} > 0}\), be the foci of the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 8} = 1\). Suppose a parabola having vertex at the origin an...

Let a, b \(\in\) R and f : R \(\to\) R be defined by \(f(x) = a\cos (|{x^3} - x|) + b|x|\sin (|{x^3} + x|)\). Then f is

Let \(f:\left[ { - {1 \over 2},2} \right] \to R\) and \(g:\left[ { - {1 \over 2},2} \right] \to R\) be function defined by \(f(x) = [{x^2} - 3]\) and \(g(x) = |x|f(x) + |4x - 7|f(x)\), where [y] denotes the greatest integer less than or equ...

Let \(P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]\) and I be the identity matrix of order 3. If \(Q = [{q_{ij}}]\) is a matrix such that \({P^{50}} - Q = I\) and $${{{q_{31}} + {q_{32}}} \o...

Let a, \(\lambda\), m \(\in\) R. Consider the system of linear equations
ax + 2y = \(\lambda\)
3x \(-\) 2y = \(\mu\)
Which of the following statements is(are) correct?

Let \(P\) be the point on the parabola \({y^2} = 4x\) which is at the shortest distance from the center \(S\) of the circle \({x^2} + {y^2} - 4x - 16y + 64 = 0\). Let \(Q\) be the point on the circle dividing the line segment \(SP\) interna...

Football teams \({T_1}\) and \({T_2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \({T_1}\) winning, drawing and losing a game against \({T_2}\) are $${1...

Football teams \({T_1}\) and \({T_2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \({T_1}\) winning, drawing and losing a game against \({T_2}\) are $${1...

Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) with the common difference loge2. Suppose a1, a2, ......, a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 ...

The value of \(\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}}\) is equal to

Let \(\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k\) be a unit vector in \({{R^3}}\) and \(\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).\) Given that there exists a vector $${...