JEE Main 2019 (Online) 12th April Morning Slot

If the line \({{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 1} \over { - 1}}\)
intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane
3x + y + 4z = 16 at a point Q, then PQ is equal to :

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate
25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the
horizontal ground when t...

If m is the minimum value of k for which the function f(x) = x\(\sqrt {kx - {x^2}}\) is increasing in the interval [0,3]
and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :

If the area (in sq. units) of the region {(x, y) : y2
\(\le\) 4x, x + y \(\le\) 1, x \(\ge\) 0, y \(\ge\) 0} is a \(\sqrt 2\) + b, then a – b is equal
to :

The coefficient of x18 in the product
(1 + x) (1 – x)10 (1 + x + x2)9
is :

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then
the length (in cm) of their common chord is :

The equation |z – i| = |z – 1|, i = \(\sqrt { - 1}\), represents :

If \(\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx\) = m(\(\pi\) + n), then m.n is equal to

Let f : R \(\to\) R be a continuously differentiable function such that f(2) = 6 and f'(2) = \({1 \over {48}}\). If \(\int\limits_6^{f\left( x \right)} {4{t^3}} dt\) = (x - 2)g(x), then $$\mathop {\lim }\limits_{x \to 2} g\left( x \right)...

Consider the differential equation, \({y^2}dx + \left( {x - {1 \over y}} \right)dy = 0\), If value of y is 1 when x = 1, then the value of x
for which y = 2, is :

If ey
+ xy = e, the ordered pair \(\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)\) at x = 0 is equal to :

If the normal to the ellipse 3x2
+ 4y2
= 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent
to the ellipse at P passes through Q(4,4) then PQ is equal to :

For x \(\in\) (0, 3/2), let f(x) = \(\sqrt x\) , g(x) = tan x and h(x) = \({{1 - {x^2}} \over {1 + {x^2}}}\). If \(\phi\) (x) = ((hof)og)(x), then \(\phi \left( {{\pi \over 3}} \right)\)
is equal to :

The integral \(\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx\) is equal to :
(Here C is a constant of integration)

The value of \({\sin ^{ - 1}}\left( {{{12} \over {13}}} \right) - {\sin ^{ - 1}}\left( {{3 \over 5}} \right)\) is equal to :

If \(\alpha\) and \(\beta\) are the roots of the equation 375x2
– 25x – 2 = 0, then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\alpha ^r}} + \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\beta ^r}} ...

If the truth value of the statement p \(\to\) (~q \(\vee\) r) is false (F), then the truth values of the statements p, q, r are
respectively :

If \(B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1} \cr } } \right]\) is the inverse of a 3 × 3 matrix A, then the sum of all values of \(\alpha\) for which
det(A) + 1 = 0, is :

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = \(\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right]\), then AB is equal
to :

Let P be the point of intersection of the common tangents to the parabola y2
= 12x and the hyperbola
8x2
– y2
= 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS'
in a ratio :

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21
are distinct, is :

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle
formed with these chosen vertices is equilateral is :

Let a random variable X have a binomial distribution with mean 8 and variance 4. If \(P\left( {X \le 2} \right) = {k \over {{2^{16}}}}\), then k
is equal to :

Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6= – 48, then S10 is equal to :

For x \(\varepsilon\) R, let [x] denote the greatest integer \(\le\) x, then the sum of the series
$$\left[ { - {1 \over 3}} \right] + \left[ { - {1 \over 3} - {1 \over {100}}} \right] + \left[ { - {1 \over 3} - {2 \over {100}}} \right] ...

If the data x1, x2,......., x10 is such that the mean of first four of these is 11, the mean of the remaining six is
16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :

The number of solutions of the equation
1 + sin4
x = cos23x, \(x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right]\) is :

The equation y = sinx sin (x + 2) – sin2
(x + 1) represents a straight line lying in :

If the volume of parallelopiped formed by the vectors \(\widehat i + \lambda \widehat j + \widehat k\), \(\widehat j + \lambda \widehat k\) and \(\lambda \widehat i + \widehat k\) is minimum, then \(\lambda\) is
equal to :

Let \(\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k\) be two vectors. If a vector perpendicular to both the vectors
$$\overrightarrow a + \overrightarrow b ...