JEE Main 2020 (Online) 3rd September Evening Slot

Let a plane P contain two lines
\(\overrightarrow r = \widehat i + \lambda \left( {\widehat i + \widehat j} \right)\), \(\lambda \in R\) and
\(\overrightarrow r = - \widehat j + \mu \left( {\widehat j - \widehat k} \right)\), $$\mu \in...

The plane which bisects the line joining, the
points (4, –2, 3) and (2, 4, –1) at right angles
also passes through the point :

If the surface area of a cube is increasing at a
rate of 3.6 cm2/sec, retaining its shape; then
the rate of change of its volume (in cm3/sec),
when the length of a side of the cube is
10 cm, is :

If the term independent of x in the expansion of
\({\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}\) is k, then 18 k is equal to :

If z1
, z2
are complex numbers such that
Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and
arg(z1 - z2) = \({\pi \over 6}\), then Im(z1
+ z2
) is equal to :

If the value of the integral
\(\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx\)
is \({k \over 6}\), then k is equal to :

Suppose f(x) is a polynomial of degree four,
having critical points at –1, 0, 1. If
T = {x \(\in\) R |
f(x) = f(0)}, then the sum of squares of all the
elements of T is :

If x3dy + xy dx = x2dy + 2y dx; y(2) = e and
x > 1, then y(4) is equal to :

Let e1
and e2
be the eccentricities of the
ellipse, \({{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1\)(b < 5) and the hyperbola,
\({{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1\) respectively satisfying e1e2
= 1. If \(\alpha\)
...

If \(\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx\) = A(x)\({\tan ^{ - 1}}\left( {\sqrt x } \right)\) + B(x) + C,
where C is a constant of integration, then the
ordered pair (A(x), B(x)) can be :

\(\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}\) (\(a\) \(\ne\) 0) is equa...

Let p, q, r be three statements such that the
truth value of (p \(\wedge\) q) \(\to\) (\(\sim\)q \(\vee\) r) is F. Then the
truth values of p, q, r are respectively :

Let A be a 3 \(\times\) 3 matrix such that
adj A = \(\left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & 0 & 2 \cr 1 & { - 2} & { - 1} \cr } } \right]\) and B = adj(adj A).
If |A| = \(\lambda\) and |(B-1)T| = \(\mu\) , then th...

Let S be the set of all integer solutions, (x, y, z),
of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 \(\le\) x2
+ y2
+ z2 \(\le\) 150. Then, the
number of elements in the set S is equal to
...

Let the latus ractum of the parabola y2
= 4x be
the common chord to the circles C1
and C2
each of them having radius 2\(\sqrt 5\). Then, the
distance between the centres of the circles C1
and C2
is :

If the tangent to the curve, y = ex
at a point
(c, ec) and the normal to the parabola, y2 = 4x
at the point (1, 2) intersect at the same point on
the x-axis, then the value of c is ________ .

The total number of 3-digit numbers, whose
sum of digits is 10, is __________.

The probability that a randomly chosen 5-digit
number is made from exactly two digits is :

The set of all real values of \(\lambda\) for which the
quadratic equations, (\(\lambda\)2
+ 1)x2
– 4\(\lambda\)x + 2 = 0
always have exactly one root in the interval
(0, 1) is :

If m arithmetic means (A.Ms) and three
geometric means (G.Ms) are inserted between
3 and 243 such that 4th A.M. is equal to 2nd
G.M., then m is equal to _________ .

If the sum of the series
20 + 19\({3 \over 5}\) + 19\({1 \over 5}\) + 18\({4 \over 5}\) + ...
upto nth term is 488
and the nth term is negative, then :

Let R1
and R2
be two relation defined as
follows :
R1
= {(a, b) \(\in\) R2
: a2
+ b2 \(\in\) Q} and
R2
= {(a, b) \(\in\) R2
: a2
+ b2 \(\notin\) Q},
where Q is the
set of all rational numbers. Then :

Let xi
(1 \(\le\) i \(\le\) 10) be ten observations of a
random variable X. If \(\sum\limits_{i = 1}^{10} {\left( {{x_i} - p} \right)} = 3\) and \(\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - p} \right)}^2}} = 9\) where 0 \(\ne\) p ...

If a \(\Delta\)ABC has vertices A(–1, 7), B(–7, 1) and
C(5, –5), then its orthocentre has coordinates :

Let a, b c \(\in\) R be such that a2
+ b2
+ c2
= 1. If \(a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)\),
where
\({\theta = {\pi \over 9}}\), then the angle be...