JEE Main 2020 (Online) 8th January Evening Slot

The mirror image of the point (1, 2, 3) in a plane
is \(\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)\). Which of the following
points lies on this plane ?

Let ƒ(x) be a polynomial of degree 3 such that
ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point
at x = –1 and ƒ'(x) has a critical point at x = 1.
Then ƒ(x) has a local minima at x = _______.

The length of the perpendicular from the origin,
on the normal to the curve, x2 + 2xy – 3y2 = 0
at the point (2,2) is

The area (in sq. units) of the region
{(x,y) \(\in\) R2 : x2 \(\le\) y \(\le\) 3 – 2x}, is :

If \(\alpha\) and \(\beta\) be the coefficients of x4 and x2
respectively in the expansion of
\({\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}\), then

If a line, y = mx + c is a tangent to the circle,
(x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + y2 = 1 at the point \(\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)\), then :

\(\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}\) is equal to

If \(I = \int\limits_1^2 {{{dx} \over {\sqrt {2{x^3} - 9{x^2} + 12x + 4} }}}\), then :

The differential equation of the family of
curves, x2 = 4b(y + b), b \(\in\) R, is :

Let ƒ : (1, 3) \(\to\) R be a function defined by
\(f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}\) , where [x] denotes the greatest
integer \(\le\) x. Then the range of ƒ is

If a hyperbola passes through the point
P(10, 16) and it has vertices at (± 6, 0), then the
equation of the normal to it at P is :

Let S be the set of all functions ƒ : [0,1] \(\to\) R,
which are continuous on [0,1] and differentiable
on (0,1). Then for every ƒ in S, there exists a
c \(\in\) (0,1), depending on ƒ, such that

Which of the following statements is a tautology?

The system of linear equations
\(\lambda\)x + 2y + 2z = 5
2\(\lambda\)x + 3y + 5z = 8
4x + \(\lambda\)y + 6z = 10 has

If \(A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right)\) and \(I = \left( {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right)\) then 10A–1 is
equal to :

Let a line y = mx (m > 0) intersect the parabola,
y2 = x at a point P, other than the origin. Let
the tangent to it at P meet the x-axis at the point
Q. If area (\(\Delta\)OPQ) = 4 sq. units, then m is equal
to __________.

The number of 4 letter words (with or without
meaning) that can be formed from the eleven
letters of the word 'EXAMINATION' is
_______.

Let A and B be two events such that the
probability that exactly one of them occurs is \({2 \over 5}\) and the probability that A or B occurs is \({1 \over 2}\) ,
then the probability of both of them occur
together is :

Let \(\alpha = {{ - 1 + i\sqrt 3 } \over 2}\). If \(a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}}\) and \(b = \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}}\), then a and b are the roots of the quadratic equation...

Let S be the set of all real roots of the equation,
3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :

The sum, \(\sum\limits_{n = 1}^7 {{{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 4}}\) is equal to
________.

If the 10th term of an A.P. is \({1 \over {20}}\) and its 20th term
is \({1 \over {10}}\), then the sum of its first 200 terms is

The mean and variance of 20 observations are
found to be 10 and 4, respectively. On
rechecking, it was found that an observation 9
was incorrect and the correct observation was
11. Then the correct variance is

If \({{\sqrt 2 \sin \alpha } \over {\sqrt {1 + \cos 2\alpha } }} = {1 \over 7}\) and \(\sqrt {{{1 - \cos 2\beta } \over 2}} = {1 \over {\sqrt {10} }}\)
\(\alpha ,\beta \in \left( {0,{\pi \over 2}} \right)\) then tan(\(\alpha\) + 2$$\bet...

Let \(\overrightarrow a = \widehat i - 2\widehat j + \widehat k\) and \(\overrightarrow b = \widehat i - \widehat j + \widehat k\) be two
vectors. If \(\overrightarrow c\) is a vector such that $$\overrightarrow b \times \overrightarrow...