JEE Main 2019 (Online) 10th April Evening Slot

A perpendicular is drawn from a point on the line \({{x - 1} \over 2} = {{y + 1} \over { - 1}} = {z \over 1}\) to the plane x + y + z = 3 such that the
foot of the perpendicular Q also lies on the plane x – y + z = 3. Then the co-ordinates...

If the plane 2x – y + 2z + 3 = 0 has the distances
\({1 \over 3}\)
and
\({2 \over 3}\)
units from the planes 4x – 2y + 4z + \(\lambda\) = 0 and
2x – y + 2z + \(\mu\) = 0, respectively, then the maximum value of \(\lambda\) + \(\mu\) i...

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of
50 cm3
/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice
decreases, is :...

If the tangent to the curve \(y = {x \over {{x^2} - 3}}\)
, \(x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)\), at a point (\(\alpha\), \(\beta\)) \(\ne\) (0, 0) on it is parallel to the line
2x + 6y – 11 = 0, then :

The area (in sq.units) of the region bounded by the curves y = 2x
and y = |x + 1|, in the first quadrant is :

The smallest natural number n, such that the coefficient of x in the expansion of \({\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}\) is nC23, is :

The locus of the centres of the circles, which touch the circle, x2
+ y2
= 1 externally, also touch the y-axis and
lie in the first quadrant, is :

If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = \({\pi \over 2}\)
, then :

The integral \(\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx\) is equal to :

Let y = y(x) be the solution of the differential equation,
\({{dy} \over {dx}} + y\tan x = 2x + {x^2}\tan x\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\), such that
y(0) = 1. Then :

Let f(x) = loge(sin x), (0 < x < \(\pi\)) and g(x) = sin–1
(e–x
), (x \(\ge\) 0). If \(\alpha\) is a positive real number such that
a = (fog)'(\(\alpha\)) and b = (fog)(\(\alpha\)), then :

The tangent and normal to the ellipse 3x2
+ 5y2
= 32 at the point P(2, 2) meet the x-axis at Q and R,
respectively. Then the area (in sq. units) of the triangle PQR is :

If 5x + 9 = 0 is the directrix of the hyperbola 16x2
– 9y2
= 144, then its corresponding focus is :

If \(\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c\), where c is a constant of integration, then \(g\)(–1) is equal to :

If \({\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha\),where –1 \(\le\) x \(\le\) 1, – 2 \(\le\) y \(\le\) 2, x \(\le\) \({y \over 2}\)
, then for all x, y, 4x2
– 4xy cos \(\alpha\) + y2
is equal
to :

If \(\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5\), then a + b is equal to :

The negation of the Boolean expression ~ s \(\vee\) (~r \(\wedge\) s) is equivalent to :

The sum of the real roots of the equation
\(\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} & {2x} & {x + 2} \cr } } \right| = 0\), is equal to :

Let \(\lambda\) be a real number for which the system of linear equations x + y + z = 6, 4x + \(\lambda\)y – \(\lambda\)z = \(\lambda\) – 2,
3x + 2y – 4z = – 5 has infinitely many solutions. Then \(\lambda\) is a root of the quadratic ...

If the line ax + y = c, touches both the curves x2
+ y2
= 1 and y2
= 4\(\sqrt 2\)x , then |c| is equal to :

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the
top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total
number of beams is :

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is
more than 99% is :

The angles A, B and C of a triangle ABC are in A.P. and a : b = 1 : \(\sqrt 3\). If c = 4 cm, then the area (in sq. cm)
of this triangle is :

The number of real roots of the equation
5 + |2x
– 1| = 2x
(2x
– 2) is

The sum
\(1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} + {3^3} + ... + {{15}^3}} \over {1 + 2 + 3 + ... + 15}}\)\(- {1 \over 2}\left( {1 + 2 + 3 + ... + 15} \right)\) is equal...

Let \(a\), b and c be in G.P. with common ratio r, where \(a\) \(\ne\) 0 and 0 < r \(\le\) \({1 \over 2}\)
. If 3\(a\), 7b and 15c are the first three
terms of an A.P., then the 4th term of this A.P. is :

Let a1, a2, a3,......be an A.P. with a6 = 2. Then the common difference of this A.P., which maximises the
product a1a4a5, is :

If both the mean and the standard deviation of 50 observations x1, x2,..., x50 are equal to 16, then the mean of (x1 – 4)2
, (x2 – 4)2
,....., (x50 – 4)2
is :

Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance
\({3 \over 5}\)
from the origin. Then which one of the
following points lies on any of these lines ?

The distance of the point having position vector \(- \widehat i + 2\widehat j + 6\widehat k\)
from the straight line passing through the point
(2, 3, – 4) and parallel to the vector, \(6\widehat i + 3\widehat j - 4\widehat k\) is :