JEE Main 2020 (Online) 2nd September Evening Slot

A plane passing through the point (3, 1, 1)
contains two lines whose direction ratios are 1,
–2, 2 and 2, 3, –1 respectively. If this plane also
passes through the point (\(\alpha\), –3, 5), then
\(\alpha\) is
equal to:

Let f : (–1,
\(\infty\))
\(\to\) R be defined by f(0) = 1 and
f(x) = \({1 \over x}{\log _e}\left( {1 + x} \right)\), x \(\ne\) 0. Then the function f :

The equation of the normal to the curve
y = (1+x)2y + cos
2(sin–1x) at x = 0 is :

Consider a region R = {(x, y) \(\in\) R : x2 \(\le\) y \(\le\) 2x}.
if a line y = \(\alpha\) divides the area of region R into
two equal parts, then which of the following is
true?

For a positive integer n,
\({\left( {1 + {1 \over x}} \right)^n}\) is expanded
in increasing powers of x. If three consecutive
coefficients in this expansion are in the ratio,
2 : 5 : 12, then n is equal to________.

The imaginary part of
\({\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}\) can be :

Let [t] denote the greatest integer less than or
equal to t. Then the value of \(\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx}\) is ______.

If a curve y = f(x), passing through the point
(1, 2), is the solution of the differential equation,
2x2dy= (2xy + y2)dx, then \(f\left( {{1 \over 2}} \right)\) is equal to :

If y = \(\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}}\),
then \({{dy} \over {dx}}\) at x = 0 is _______.

Let f : R \(\to\) R be a function which satisfies
f(x + y) = f(x) + f(y) \(\forall\) x, y \(\in\) R. If f(1) = 2 and
g(n) = \(\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)}\), n \(\in\) N then the value of n, for
w...

For some \(\theta \in \left( {0,{\pi \over 2}} \right)\), if the eccentricity of the
hyperbola, x2–y2sec2\(\theta\) = 10 is
\(\sqrt 5\) times the
eccentricity of the ellipse, x2sec2\(\theta\) + y2 = 5, then
the length of the latus rect...

\(\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}\) is equal to :

Which of the following is a tautology ?

Let a, b, c \(\in\) R be all non-zero and satisfy
a3 + b3 + c3 = 2. If the matrix
A = \(\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)\)
satisfies ATA = I, then a value of abc can be :

Let A = {X = (x, y, z)T: PX = 0 and
x2 + y2 + z2 = 1} where
\(P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right]\),
then the set A :

The area (in sq. units) of an equilateral triangle
inscribed in the parabola y2 = 8x, with one of
its vertices on the vertex of this parabola, is :

Let n > 2 be an integer. Suppose that there are
n Metro stations in a city located along a
circular path. Each pair of stations is connected
by a straight track only. Further, each pair of
nearest stations is connected by blue line,
whereas...

Let EC denote the complement of an event E.
Let E1
, E2
and E3
be any pairwise independent
events with P(E1) > 0
and P(E1 \(\cap\) E2 \(\cap\) E3) = 0.
Then P(\(E_2^C \cap E_3^C/{E_1}\)) is equal to :

Let f(x) be a quadratic polynomial such that
f(–1) + f(2) = 0. If one of the roots of f(x) = 0
is 3, then its other root lies in :

If the sum of first 11 terms of an A.P.,
a1, a2, a3, ....
is 0 (a \(\ne\) 0), then the sum of the A.P.,
a1
, a3
, a5
,....., a23 is ka1
, where k is equal to :

Let S be the sum of the first 9 terms of the
series :
{x + k\(a\)} + {x2 + (k + 2)\(a\)} + {x3 + (k + 4)\(a\)}
+ {x4 + (k + 6)\(a\)} + .... where a \(\ne\) 0 and x \(\ne\) 1.
If S = $${{{x^{10}} - x + 45a\left( {x - 1} \right)} \over {...

If the variance of the terms in an increasing A.P., b1
, b2
, b3
,....,b11 is 90, then the common
difference of this A.P. is_______.

The set of all possible values of
\(\theta\) in the interval
(0, \(\pi\)) for which the points (1, 2) and (sin
\(\theta\), cos \(\theta\)) lie on the same side of the line x + y =
1 is :

If the equation cos4 \(\theta\) + sin4 \(\theta\) +
\(\lambda\) = 0 has real
solutions for
\(\theta\), then
\(\lambda\) lies in the interval :

Let the position vectors of points 'A' and 'B' be
\(\widehat i + \widehat j + \widehat k\) and \(2\widehat i + \widehat j + 3\widehat k\), respectively. A point
'P' divides the line segment AB internally in the
ratio
\(\lambda\) : 1 (
$$\l...