JEE Advanced 2015 Paper 1 Offline

In \({R^3},\) consider the planes \(\,{P_1}:y = 0\) and \({P_2}:x + z = 1.\) Let \({P_3}\) be the plane, different from \({P_1}\) and \({P_2}\), which passes through the intersection of \({P_1}\) and \({P_2}.\) If the distance of the point ...

In \({R^3},\) let \(L\) be a straight lines passing through the origin. Suppose that all the points on \(L\) are at a constant distance from the two planes \({P_1}:x + 2y - z + 1 = 0\) and \({P_2}:2x - y + z - 1 = 0.\) Let \(M\) be the loc...

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of \(V\) \(m{m^3}\), has a \(2\) mm thick solid wall and is open at the top. The bottom of the container is a soli...

Let \(F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)}\) for all \(x \in R\) and \(f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]\) be a continuous function. For $$a \in \left[ ...

Let \(f:R \to R\) be a function defined by \(f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.\) where \(\left[ x \right]\) is the greatest integer less than or equal to $$...

Let \(y(x)\) be a solution of the differential equation \(\left( {1 + {e^x}} \right)y' + y{e^x} = 1.\)
If \(y(0)=2\), then which of the following statement is (are) true?

Consider the family of all circles whose centres lie on the straight line \(y=x,\) If this family of circle is represented by the differential equation \(Py'' + Qy' + 1 = 0,\) where \(P, Q\) are functions of \(x,y\) and \(y'\) $$\left( {her...

Let \(f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)\) for all \(x \in R\) and g(x) = \({{\pi \over 2}\sin x}\) for all x\(\in\)R. Let \((f \circ g)(x)\) denote f(g(x)) and \((g \circ f)(x)\) denote g...

Let \(g:R \to R\) be a differentiable function with \(g(0) = 0\), \(g'(0) = 0\) and \(g'(1) \ne 0\). Let
\(f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.\)
and \(h(x) = {e^{|x|}}\)...

Let X and Y be two arbitrary, 3 \(\times\) 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 \(\times\) 3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?

Which of the following values of \(\alpha\) satisfy the equation
$$\left| {\matrix{
{{{(1 - \alpha )}^2}} & {{{(1 + 2\alpha )}^2}} & {{{(1 + 3\alpha )}^2}} \cr
{{{(2 + \alpha )}^2}} & {{{(2 + 2\alpha )}^2}} & {{{(2 + 3\alpha )}^2}} ...

If the normals of the parabola \({y^2} = 4x\) drawn at the end points of its latus rectum are tangents to the circle \({\left( {x - 3} \right)^2} + {\left( {y + 2} \right)^2} = {r^2}\), then the value of \({r^2}\) is

Let \(P\) and \(Q\) be distinct points on the parabola \({y^2} = 2x\) such that a circle with \(PQ\) as diameter passes through the vertex \(O\) of the parabola. If \(P\) lies in the first quadrant and the area of the triangle $$\Delta OPQ$...

Let the curve \(C\) be the mirror image of the parabola \({y^2} = 4x\) with respect to the line \(x+y+4=0\). If \(A\) and \(B\) are the points of intersection of \(C\) with the line \(y=-5\), then the distance between \(A\) and \(B\) is

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that ...

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least \(0.96,\) is

Match the following :

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overflow:hidden;padding:10px 5px;word-break:no...

The number of distinct solutions of the equation
\({5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2\) in the interval \(\left[ {0,\,2\pi } \right]\) is

Let \(\Delta PQR\) be a triangle. Let \(\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP}\) and \(\overrightarrow c = \overrightarrow {PQ} .\) If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right...

Match the following :




Column I

Column II







(A)

In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector

$ \alpha \hat{i} + \beta \hat{j} $ on

$ \sqrt{3}\hat{i} + \h...