JEE Main 2014 (Offline)

The image of the line \({{x - 1} \over 3} = {{y - 3} \over 1} = {{z - 4} \over { - 5}}\,\) in the plane \(2x-y+z+3=0\) is the line :

The angle between the lines whose direction cosines satisfy the equations \(l+m+n=0\) and \({l^2} = {m^2} + {n^2}\) is :

If \(f\) and \(g\) are differentiable functions in \(\left[ {0,1} \right]\) satisfying
\(f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) = 0\) and \(f\left( 1 \right) = 6,\) then for some \(c \in \left] {0,1} \right[\)

If \(x=-1\) and \(x=2\) are extreme points of \(f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x\) then

The area of the region described by
\(A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.\) and \(\left. {{y^2} \le 1 - x} \right\}\) is :

If the coefficints of \({x^3}\) and \({x^4}\) in the expansion of \(\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}\) in powers of \(x\) are both zero, then \(\left( {a,\,b} \right)\) is equal to:

Let \(C\) be the circle with centre at \((1, 1)\) and radius \(=\) \(1\). If \(T\) is the circle centred at \((0, y)\), passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equal to :

If z is a complex number such that \(\,\left| z \right| \ge 2\,\), then the minimum value of \(\,\,\left| {z + {1 \over 2}} \right|\) :

The integral \(\int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx\) equals:

Let the population of rabbits surviving at time \(t\) be governed by the differential equation \({{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.\) If \(p(0)=100,\) then \(p(t)\) equals:

If \(g\) is the inverse of a function \(f\) and \(f'\left( x \right) = {1 \over {1 + {x^5}}},\) then \(g'\left( x \right)\) is equal to:

The locus of the foot of perpendicular drawn from the centre of the ellipse \({x^2} + 3{y^2} = 6\) on any tangent to it is :

A bird is sitting on the top of a vertical pole \(20\) m high and its elevation from a point \(O\) on the ground is \({45^ \circ }\). It files off horizontally straight away from the point \(O\). After one second, the elevation of the bird ...

The integral \(\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx}\) is equal to

\(\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}\) is equal to :

The statement \(\sim \left( {p \leftrightarrow \sim q} \right)\) is :

If \(A\) is a \(3 \times 3\) non-singular matrix such that \(AA'=A'A\) and
\(B = {A^{ - 1}}A',\) then \(BB'\) equals:

If \(\alpha ,\beta \ne 0,\) and \(f\left( n \right) = {\alpha ^n} + {\beta ^n}\) and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 ...

The slope of the line touching both the parabolas \({y^2} = 4x\) and \({x^2} = - 32y\) is

Let \(A\) and \(B\) be two events such that \(P\left( {\overline {A \cup B} } \right) = {1 \over 6},\,P\left( { {A \cap B} } \right) = {1 \over 4}\) and \(P\left( {\overline A } \right) = {1 \over 4},\) where \(\overline A\) stands for the...

Let \(\alpha\) and \(\beta\) be the roots of equation \(p{x^2} + qx + r = 0,\) \(p \ne 0.\) If \(p,\,q,\,r\) in A.P. and \({1 \over \alpha } + {1 \over \beta } = 4,\) then the value of \(\left| {\alpha - \beta } \right|\) is :

If \(a \in R\) and the equation \(- 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0\) (where [\(x\)] denotes the greater integer \(\le x\)) has no integral solution, then all possible values...

If \({(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},\), then k is equal to :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

The variance of first 50 even natural numbers is

Let \(PS\) be the median of the triangle with vertices \(P(2, 2)\), \(Q(6, -1)\) and \(R(7, 3)\). The equation of the line passing through \((1, -1)\) band parallel to PS is :

Let \(a, b, c\) and \(d\) be non-zero numbers. If the point of intersection of the lines \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\) lies in the fourth quadrant and is equidistant from the two axes then :

Let \(f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) where \(x \in R\) and \(k \ge \,1.\)
Then \({f_4}\left( x \right) - {f_6}\left( x \right)\,\,\) equals :

If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overrightarrow a } \right] = \lambda {\left[ {\overrightarrow a\,\,\,\,\,\,\,\, \overrightarrow ...