JEE Main 2015 (Offline)

The distance of the point \((1, 0, 2)\) from the point of intersection of the line \({{x - 2} \over 3} = {{y + 1} \over 4} = {{z - 2} \over {12}}\) and the plane \(x - y + z = 16,\) is :

The equation of the plane containing the line \(2x-5y+z=3; x+y+4z=5,\) and parallel to the plane, \(x+3y+6z=1,\) is :

Let \(f(x)\) be a polynomial of degree four having extreme values
at \(x=1\) and \(x=2\). If \(\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3\), then f\((2)\) is equal to :

The normal to the curve, \({x^2} + 2xy - 3{y^2} = 0\), at \((1,1)\)

The area (in sq. units) of the region described by
\(\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.\) and \(\left. {y \ge 4x - 1} \right\}\) is :

The sum of coefficients of integral power of \(x\) in the binomial expansion \({\left( {1 - 2\sqrt x } \right)^{50}}\) is :

The number of common tangents to the circles \({x^2} + {y^2} - 4x - 6x - 12 = 0\) and \({x^2} + {y^2} + 6x + 18y + 26 = 0,\) is :

Locus of the image of the point \((2, 3)\) in the line \(\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,\) is a :

A complex number z is said to be unimodular if \(\,\left| z \right| = 1\). Suppose \({z_1}\) and \({z_2}\) are complex numbers such that \({{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}\) is unimodular and \({z_2}\) is not unimodula...

The integral
\(\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx}\) is equal to :

Let \(y(x)\) be the solution of the differential equation
\(\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).\) Then \(y(e)\) is equal to :

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1\), is :

If the angles of elevation of the top of a tower from three collinear points \(A, B\) and \(C,\) on a line leading to the foot of the tower, are \({30^ \circ }\), \({45^ \circ }\) and \({60^ \circ }\) respectively, then the ratio, $$AB:BC,$...

The integral \(\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}\) equals :

Let \({\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),\)
where \(\left| x \right| < {1 \over {\sqrt 3 }}.\) Then a value of \(y\) is :

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

If the function.
\(g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr } } \right.\)
is differentiable, then the value of \(k+m\) is :

The negation of \(\sim s \vee \left( { \sim r \wedge s} \right)\) is equivalent to :

If \(A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]\) is a matrix satisfying the equation
\(A{A^T} = 9\text{I},\) where \(I\) is \(3 \times 3\) identity matrix, then the ordered
pair $$(a,...

The set of all values of \(\lambda\) for which the system of linear equations:
$$\matrix{
{2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr
{2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr
{ - {x_1} + 2{x_2} = \lambda {x_3}} \cr

...

Let \(O\) be the vertex and \(Q\) be any point on the parabola, \({{x^2} = 8y}\). If the point \(P\) divides the line segment \(OQ\) internally in the ratio \(1:3\), then locus of \(P\) is :

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

If \(12\) different balls are to be placed in \(3\) identical boxes, then the probability that one of the boxes contains exactly \(3\) balls is :

Let \(\alpha\) and \(\beta\) be the roots of equation \({x^2} - 6x - 2 = 0\). If \({a_n} = {\alpha ^n} - {\beta ^n},\) for \(n \ge 1,\) then the value of \({{{a_{10}} - 2{a_8}} \over {2{a_9}}}\) is equal to :

If m is the A.M. of two distinct real numbers l and n \((l,n > 1)\) and \({G_1},{G_2}\) and \({G_3}\) are three geometric means between \(l\) and n, then \(G_1^4\, + 2G_2^4\, + G_3^4\) equals:

The sum of first 9 terms of the series.
\({{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......\)

Let A and B be two sets containing four and
two elements respectively. Then, the number
of subsets of the set A $\times$ B , each having atleast
three elements are

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted
and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is :

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices \((0, 0)\) \((0, 41)\) and \((41, 0)\) is :

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three non-zero vectors such that no two of them are collinear and $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \ov...