JEE Advanced 2014 Paper 1 Offline

From a point \(P\left( {\lambda ,\lambda ,\lambda } \right),\) perpendicular \(PQ\) and \(PR\) are drawn respectively on the lines \(y=x, z=1\) and \(y=-x, z=-1.\) If \(P\) is such that \(\angle QPR\) is a right angle, then the possible val...

The slope of the tangent to the curve \({\left( {y - {x^5}} \right)^2} = x{\left( {1 + {x^2}} \right)^2}\) at the point \((1, 3)\) is

A circle S passes through the point (0, 1) and is orthogonal to the circles \({(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^2}\, + \,{y^2} = 1\). Then

Let a \(\in\) R and f : R \(\to\) R be given by f(x) = x5 \(-\) 5x + a. Then,

The value of \(\int\limits_0^1 {4{x^3}\left\{ {{{{d^2}} \over {d{x^2}}}{{\left( {1 - {x^2}} \right)}^5}} \right\}dx}\) is

Let \(f:\left( {0,\infty } \right) \to R\) be given by \(f\left( x \right)\)= \(\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt\). Then

For every pair of continuous function f, g : [0, 1] \(\to\) R such that max {f(x) : x \(\in\) [0, 1]} = max {g(x) : x \(\in\) [0, 1]}. The correct statement(s) is (are)

Let \(f:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R\) be given by \(f(x) = {[\log (\sec x + \tan x)]^3}\). Then,

Let f : [0, 4\(\pi\)] \(\to\) [0, \(\pi\)] be defined by f(x) = cos\(-\)1 (cos x). The number of points x \(\in\) [0, 4\(\pi\)] satisfying the equation \(f(x) = {{10 - x} \over {10}}\) is

Let \(f:(a,b) \to [1,\infty )\) be a continuous function and g : R \(\to\) R be defined as $$g(x) = \left\{ {\matrix{
0 & , & {x < a} \cr
{\int_a^x {f(t)dt} } & , & {a \le x \le b} \cr
{\int_a^b {f(t)dt} } & , & {x > b} \cr

...

The largest value of the non-negative integer a for which \(\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x - 1) + a} \over {x + \sin (x - 1) - 1}}} \right\}^{{{1 - x} \over {1 - \sqrt x }}}} = {1 \over 4}\) is

Let f : R \(\to\) R and g : R \(\to\) R be respectively given by f(x) = | x | + 1 and g(x) = x2 + 1. Define h : R \(\to\) R by $$h(x) = \left\{ {\matrix{
{\max \{ f(x),g(x)\} ,} & {if\,x \le 0.} \cr
{\min \{ f(x),g(x)\} ,} & {if\,x ...

Let M be a 2 \(\times\) 2 symmetric matrix with integer entries. Then, M is invertible, if

Let M and N be two 3 \(\times\) 3 matrices such that MN = NM. Further, if M \(\ne\) N2 and M2 = N4, then

Let \({n \ge 2}\) be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue lin...

Let \({n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}\) be positive integers such that \({n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_5}\) = 20. Then the number of such destinct arrangements $$\,({n_1}\,,\,{n_2},\,\,{n_3},\,\,{n_4}\,,{n...

Let a, b, c be positive integers such that \({b \over a}\) is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of \({{{a^2} + a - 14} \over {a + 1}}\) is

For a point \(P\) in the plane, Let \({d_1}\left( P \right)\) and \({d_2}\left( P \right)\) be the distance of the point \(P\) from the lines \(x - y = 0\) and \(x + y = 0\) respectively. The area of the region \(R\) consisting of all point...

Let \(\overrightarrow x ,\overrightarrow y\) and \(\overrightarrow z\) be three vectors each of magnitude \(\sqrt 2\) and the angle between each pair of them is \({\pi \over 3}\). If \(\overrightarrow a\) is a non-zero vector perpendic...

Let \(\overrightarrow a \,\,,\,\,\overrightarrow b\) and \(\overrightarrow c\) be three non-coplanar unit vectors such that the angle between every pair of them is \({\pi \over 3}.\) If $$\overrightarrow a \times \overrightarrow b + \o...