JEE Main 2016 (Online) 10th April Morning Slot

ABC is a triangle in a plane with vertices
A(2, 3, 5), B(−1, 3, 2) and C(\(\lambda\), 5, \(\mu\)).
If the median through A is equally inclined to the coordinate axes, then the value of (\(\lambda\)3 + \(\mu\)3 + 5) is :

The number of distinct real values of \(\lambda\) for which the lines
\({{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over {{\lambda ^2}}}\) and \({{x - 3} \over 1} = {{y - 2} \over {{\lambda ^2}}} = {{z - 1} \over 2}\) are coplanar ...

Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :

Let C be a curve given by y(x) = 1 + \(\sqrt {4x - 3} ,x > {3 \over 4}.\) If P is a point
on C, such that the tangent at P has slope \({2 \over 3}\), then a point through which the normal at P passes, is :

If the coefficients of x−2 and x−4 in the expansion of \({\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x > 0} \right),\) are m and n respectively, then \({m \over n}\) is equal to :

Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 is :

For x \(\in\) R, x \(\ne\) 0, if y(x) is a differentiable function such that
x \(\int\limits_1^x y\) (t) dt = (x + 1) \(\int\limits_1^x ty\) (t) dt,  then y (x) equals :
(where C is a constant.)

The value of the integral
\(\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,\)
where [x] denotes the greatest integer less than or
equal to x, is :

The solution of the differential equation
\({{dy} \over {dx}}\, + \,{y \over 2}\,\sec x = {{\tan x} \over {2y}},\,\,\)
where 0 \(\le\) x < \({\pi \over 2}\), and y (0) = 1, is given by :

The angle of elevation of the top of a vertical tower from a point A, due east of it is 45o. The angle of elevation of the top of the same tower from a point B, due south of A is 30o. If the distance between A and B is \(54\sqrt 2 \,m,\) th...

A hyperbola whose transverse axis is along the major axis of the conic, \({{{x^2}} \over 3} + {{{y^2}} \over 4} = 4\) and has vertices at the foci of this conic. If the eccentricity of the hyperbola is \({3 \over 2},\) then which of the fo...

The integral \(\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}}\) is equal to :
(where C is a constant of integration.)

\(\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}\) is :

Let a, b \(\in\) R, (a \(\ne\) 0). If the function f defined as
$$f\left( x \right) = \left\{ {\matrix{
{{{2{x^2}} \over a}\,\,,} & {0 \le x < 1} \cr
{a\,\,\,,} & {1 \le x < \sqrt 2 } \cr
{{{2{b^2} - 4b} \over {{x^3}}},} & ...

The contrapositive of the following statement,
“If the side of a square doubles, then its area increases four times”, is :

If    A = \(\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\),
then the determinant of the matrix (A2016 − 2A2015 − A2014) is :

Let A be a 3 \(\times\) 3 matrix such that A2 \(-\) 5A + 7I = 0
Statement - I :  
A\(-\)1 = \({1 \over 7}\) (5I \(-\) A).
Statement - II :
The polynomial A3 \(-\) 2A2 \(-\) 3A + I can be reduced to 5(A \(-\) 4I).
Then :

P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum value of \(t_1^2\) is :

If    \({{{}^{n + 2}C{}_6} \over {{}^{n - 2}{P_2}}}\) = 11, then n satisfies the
equation :

The sum \(\sum\limits_{r = 1}^{10} {\left( {{r^2} + 1} \right) \times \left( {r!} \right)}\) is equal to :

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is :

If x is a solution of the equation, \(\sqrt {2x + 1}\) \(- \sqrt {2x - 1} = 1,\) \(\,\,\left( {x \ge {1 \over 2}} \right),\) then \(\sqrt {4{x^2} - 1}\) is equal to :

Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number.
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :

If   A > 0, B > 0   and    A + B = \({\pi \over 6}\), then the minimum value of tanA + tanB is :

Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P.
If a3 + a7 + a11 + a15 = 72,
then the sum of its first 17 terms is equal to :

Let P = {\(\theta\) : sin\(\theta\) \(-\) cos\(\theta\) = \(\sqrt 2 \,\cos \theta\)} and Q = {\(\theta\) : sin\(\theta\) + cos\(\theta\) = \(\sqrt 2 \,\sin \theta\)} be two sets. Then

The mean of 5 observations is 5 and their variance is 124. If three of the observations
are 1, 2 and 6 ; then the mean deviation from the mean of the data is :

A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is :

A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \({{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}\) ...