JEE Main 2020 (Online) 8th January Morning Slot

The shortest distance between the lines
\({{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}\) and
\({{x + 3} \over { - 3}} = {{y + 7} \over 2} = {{z - 6} \over 4}\) is :

If c is a point at which Rolle's theorem holds
for the function,
f(x) = \({\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)\) in the
interval [3, 4], where a \(\in\) R, then ƒ''(c) is equal
to

Let the normal at a point P on the curve
y2 – 3x2 + y + 10 = 0 intersect the y-axis at \(\left( {0,{3 \over 2}} \right)\)
. If m is the slope of the tangent at P to
the curve, then |m| is equal to

Let ƒ(x) = xcos–1(–sin|x|), \(x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]\), then
which of the following is true?

For a > 0, let the curves C1 : y2 = ax and
C2 : x2 = ay intersect at origin O and a point P.
Let the line x = b (0 < b < a) intersect the chord
OP and the x-axis at points Q and R,
respectively. If the line x = b bisects the area
bounded by...

If the equation, x2 + bx + 45 = 0 (b \(\in\) R) has
conjugate complex roots and they satisfy
|z +1| = 2\(\sqrt {10}\) , then :

Let y = y(x) be a solution of the differential
equation,
\(\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0\), |x| < 1.
If \(y\left( {{1 \over 2}} \right) = {{\sqrt 3 } \over 2}\), then $$y\left( { - {1 \over {\sqrt 2 }}} \right...

Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1.
If \({{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right)\) and \(y\left( {\sqrt 3 } \right) = {\pi \over 6}\),
then y($${ -...

Let the line y = mx and the ellipse 2x2 + y2 = 1
intersect at a ponit P in the first quadrant. If the
normal to this ellipse at P meets the co-ordinate axes at \(\left( { - {1 \over {3\sqrt 2 }},0} \right)\) and (0, \(\beta\)), then $$\bet...

The inverse function of
f(x) = \({{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}\), x \(\in\) (-1, 1), is :

If \(\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c\)
where c is a constant of integration, then $$\lambda f\left( {{\pi \over 3}...

\(\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}\) is equal to

Which one of the following is a tautology?

For which of the following ordered pairs (\(\mu\), \(\delta\)),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = \(\mu\)
4x + 4y + 4z = \(\delta\)
is inconsistent ?

The number of all 3 × 3 matrices A, with
enteries from the set {–1, 0, 1} such that the sum
of the diagonal elements of AAT is 3, is

The locus of a point which divides the line
segment joining the point (0, –1) and a point on
the parabola, x2 = 4y, internally in the ratio
1 : 2, is :

If a, b and c are the greatest value of 19Cp, 20Cq
and 21Cr respectively, then :

An urn contains 5 red marbles, 4 black marbles
and 3 white marbles. Then the number of ways
in which 4 marbles can be drawn so that at the
most three of them are red is ___________.

Let A and B be two independent events such
that P(A) = \({1 \over 3}\) and P(B) = \({1 \over 6}\). Then, which of
the following is TRUE?

The least positive value of 'a' for which the
equation 2x2 + (a – 10)x + \({{33} \over 2}\)
= 2a has real
roots is

Let ƒ : R \(\to\) R be such that for all
x \(\in\) R (21+x + 21–x), ƒ(x) and (3x + 3–x) are in
A.P., then the minimum value of ƒ(x) is

The sum \(\sum\limits_{k = 1}^{20} {\left( {1 + 2 + 3 + ... + k} \right)}\) is :

The mean and the standard deviation (s.d.) of
10 observations are 20 and 2 resepectively.
Each of these 10 observations is multiplied by
p and then reduced by q, where p \(\ne\) 0 and
q \(\ne\) 0. If the new mean and new s.d. become
hal...

Let two points be A(1, –1) and B(0, 2). If a point
P(x', y') be such that the area of \(\Delta\)PAB = 5 sq.
units and it lies on the line, 3x + y – 4\(\lambda\) = 0,
then a value of \(\lambda\) is :

Let the volume of a parallelopiped whose
coterminous edges are given by
\(\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k\), \(\overrightarrow v = \widehat i + \widehat j + 3\widehat k\) and
$$\overrightarrow w = 2\wideh...