JEE Main 2019 (Online) 10th January Evening Slot

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through which one of the following points ?

On which of the following lines lies the point of intersection of the line,   \({{x - 4} \over 2} = {{y - 5} \over 2} = {{z - 3} \over 1}\)  and the plane,
x + y + z = 2 ?

The tangent to the curve, y = xex2 passing through the point (1, e) also passes through the point

A helicopter is flying along the curve given by y – x3/2 = 7, (x \(\ge\) 0). A soldier positioned at the point \(\left( {{1 \over 2},7} \right)\) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is ...

The positive value of \(\lambda\) for which the co-efficient of x2
in the expression x2 \({\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}\) is 720, is -

If the area of an equilateral triangle inscribed in the circle x2 + y2
+ 10x + 12y + c = 0 is \(27\sqrt 3\) sq units then c is equal to :

Let \(z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.\) If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :

If  \(\int\limits_0^x \,\)f(t) dt = x2 + \(\int\limits_x^1 \,\) t2f(t) dt then f '\(\left( {{1 \over 2}} \right)\) is -

The value of   \(\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,\)  where [t] denotes the greatest integer less than or equal to t, is

The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1, 1) is :

Let f be a differentiable function such that f '(x) = 7 - \({3 \over 4}{{f\left( x \right)} \over x},\) (x > 0) and f(1) \(\ne\) 4. Then \(\mathop {\lim }\limits_{x \to 0'} \,\) xf\(\left( {{1 \over x}} \right)\) :

Let S = \(\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.\) Then S represents :

Let N be the set of natural numbers and two functions f and g be defined as f, g : N \(\to\) N such that
f(n) = $$\left\{ {\matrix{
{{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr
{{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr

...

If  \(\int \,\)x5.e\(-\)4x3 dx = \({1 \over {48}}\)e\(-\)4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -

The value of \(\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} \right)\) is :

Let f : (\(-\)1, 1) \(\to\) R be a function defined by f(x) = max \(\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.\) If K be the set of all points at which f is not differentiable, then K has exactly -

Consider the following three statements :
P : 5 is a prime number
Q : 7 is a factor of 192
R : L.C.M. of 5 and 7 is 35
Then the truth value of which one of the following statements is true ?

The number of values of \(\theta\) \(\in\) (0, \(\pi\)) for which the system of linear equations
x + 3y + 7z = 0
\(-\) x + 4y + 7z = 0
(sin3\(\theta\))x + (cos2\(\theta\))y + 2z = 0.
has a non-trival solution, is -

Let A = \(\left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr } } \right]\) where b > 0.
Then the minimum value of \({{\det \left( A \right)} \over b}\) is -

The length of the chord of the parabola x2 \(=\) 4y having equation x – \(\sqrt 2 y + 4\sqrt 2 = 0\)  is -

If  \(\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r}.{}^{50 - r}{C_{25 - r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} ,\,\,\) then K is equal to :

If the probability of hitting a target by a shooter, in any shot, is \({1 \over 3}\), then the minimum number of independent
shots at the target required by him so that the probability of hitting the target atleast once is greater than $${5...

With the usual notation, in \(\Delta\)ABC, if \(\angle A + \angle B\) = 120o, a = \(\sqrt 3\) \(+\) 1, b = \(\sqrt 3\) \(-\) 1 then the ratio \(\angle A:\angle B,\) is :

The value of \(\lambda\) such that sum of the squares of the roots of the quadratic equation, x2 + (3 – \(\lambda\))x + 2 = \(\lambda\) has the least value is -

Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k \(\in\) N (the set of natural numbers) for which
$$\left| {\matrix{
{{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3...

If mean and standard deviation of 5 observations x1, x2, x3, x4, x5 are 10 and 3, respectively, then the variance of 6 observations x1, x2, ….., x5 and –50 is equal to

Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is :

Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in which quadrant :

The value of \(\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}\) is -

If \(\overrightarrow \alpha\) = \(\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b\)  and  \(\overrightarrow \beta = \left( {4\lambda - 2} \right)\overrightarrow a + 3\overrightarrow b\) be two given vectors $$\ov...