JEE Main 2013 (Offline)

Distance between two parallel planes \(2x+y+2z=8\) and \(4x+2y+4z+5=0\) is :

If the lines \({{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}\) and \({{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}\) are coplanar, then \(k\) can have :

The real number \(k\) for which the equation, \(2{x^3} + 3x + k = 0\) has two distinct real roots in \(\left[ {0,\,1} \right]\)

The intercepts on \(x\)-axis made by tangents to the curve,
\(y = \int\limits_0^x {\left| t \right|dt,x \in R,}\) which are parallel to the line \(y=2x\), are equal to :

The area (in square units) bounded by the curves \(y = \sqrt {x,}\) \(2y - x + 3 = 0,\) \(x\)-axis, and lying in the first quadrant is :

The term independent of \(x\) in expansion of
\({\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}\) is

The circle passing through \((1, -2)\) and touching the axis of \(x\) at \((3, 0)\) also passes through the point :

If z is a complex number of unit modulus and argument \(\theta\), then arg \(\left( {{{1 + z} \over {1 + \overline z }}} \right)\) equals :

Statement-1 : The value of the integral
\(\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}}\) is equal to \(\pi /6\)
Statement-2 : $$\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)}...

At present, a firm is manufacturing \(2000\) items. It is estimated that the rate of change of production P w.r.t. additional number of workers \(x\) is given by \({{dp} \over {dx}} = 100 - 12\sqrt x .\) If the firm employs \(25\) more wo...

If \(y = \sec \left( {{{\tan }^{ - 1}}x} \right),\) then \({{{dy} \over {dx}}}\) at \(x=1\) is equal to :

The equation of the circle passing through the foci of the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\), and having centre at \((0,3)\) is :

\(ABCD\) is a trapezium such that \(AB\) and \(CD\) are parallel and \(BC \bot CD.\) If \(\angle ADB = \theta ,\,BC = p\) and \(CD = q,\) then AB is equal to:

If \(\int {f\left( x \right)dx = \psi \left( x \right),}\) then \(\int {{x^5}f\left( {{x^3}} \right)dx}\) is equal to

If \(x, y, z\) are in A.P. and \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in A.P., then :

\(\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}\) is equal to

Consider :
Statement − I : \(\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)\) is a fallacy.
Statement − II :\(\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)\) is a tautology.

The number of values of \(k\), for which the system of equations : \($\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \right)y = 3k - 1} \cr }\)$
has no solution, is

If \(P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]\) is the adjoint of a \(3 \times 3\) matrix \(A\) and
\(\left| A \right| = 4,\) then \(\alpha\) is equal to :

Given : A circle, \(2{x^2} + 2{y^2} = 5\) and a parabola, \({y^2} = 4\sqrt 5 x\).
Statement-1 : An equation of a common tangent to these curves is \(y = x + \sqrt 5\).
Statement-2 : If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \n...

Let A and B be two sets containing 2 elements and
4 elements respectively. The number of subsets of
A \(\times\) B having 3 or more elements is :

Let \({T_n}\) be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If \({T_{n + 1}} - {T_n}\) = 10, then the value of n is :

A multiple choice examination has \(5\) questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get \(4\) or more correct answers just by guessing is :

If the equations \({x^2} + 2x + 3 = 0\) and \(a{x^2} + bx + c = 0,\) \(a,\,b,\,c\, \in \,R,\) have a common root, then \(a\,:b\,:c\,\) is

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10
to each of the students. Which of the following statistical measures will not change even after the grace
marks were given?

The \(x\)-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as \((0, 1) (1, 1)\) and \((1, 0)\) is :

A ray of light along \(x + \sqrt 3 y = \sqrt 3\) gets reflected upon reaching \(X\)-axis, the equation of the reflected ray is :

The expression \({{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}\) can be written as:

If the vectors \(\overrightarrow {AB} = 3\widehat i + 4\widehat k\) and \(\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k\) are the sides of a triangle \(ABC,\) then the length of the median through \(A\) is :