JEE Main 2020 (Online) 6th September Morning Slot

The shortest distance between the lines
\({{x - 1} \over 0} = {{y + 1} \over { - 1}} = {z \over 1}\) and x + y + z + 1 = 0, 2x – y + z
+ 3 = 0 is :

The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real
numbers greater than 1. Then the average speed of the car over the time interval [t1
, t2
] is
attained at the point :

The area (in sq. units) of the region A = {(x, y) : |x| + |y| \(\le\) 1, 2y2 \(\ge\) |x|}

If {p} denotes the fractional part of the number p, then
\(\left\{ {{{{3^{200}}} \over 8}} \right\}\), is equal to :

The region represented by {z = x + iy \(\in\) C : |z| – Re(z) \(\le\) 1} is also given by the inequality :
{z = x + iy \(\in\) C : |z| – Re(z) \(\le\) 1}

If I1 = \(\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx\) and
I2 = \(\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx\) such that I2
= \(\alpha\)I1
then \(\alpha\)
equals to :

\(\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)dt} } \over {\left( {x - 1} \right)\sin \left( {x - 1} \right)}}} \right)\)

The general solution of the differential equation
\(\sqrt {1 + {x^2} + {y^2} + {x^2}{y^2}}\) + xy\({{dy} \over {dx}}\) = 0 is :
(where C is a constant of integration)

Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent
to the ellipse,
\({{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\)
from any of its foci?

If f(x + y) = f(x)f(y) and \(\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2\) , x, y \(\in\) N, where N is the set of all natural number, then the
value of
\({{f\left( 4 \right)} \over {f\left( 2 \right)}}\) is :

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If
AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point
A such that MD2 + MC2 is minimum is ______.

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the
foot of the hill is found to be 45o. After walking a distance of 80 meters towards the top, up a slope
inclined at an angle of 30o to the h...

Let f : R \(\to\) R be defined as
$$f\left( x \right) = \left\{ {\matrix{
{{x^5}\sin \left( {{1 \over x}} \right) + 5{x^2},} & {x < 0} \cr
{0,} & {x = 0} \cr
{{x^5}\cos \left( {{1 \over x}} \right) + \lambda {x^2},} & {x > 0}...

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

Let m and M be respectively the minimum and maximum values of
$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \si...

The values of \(\lambda\) and \(\mu\) for which the system of linear equations
x + y + z = 2
x + 2y + 3z = 5
x + 3y + \(\lambda\)z = \(\mu\)
has infinitely many solutions are, respectively:

Let L1
be a tangent to the parabola y2 = 4(x + 1) and L2
be a tangent to the parabola
y2 = 8(x + 2) such that L1
and L2
intersect at right angles. Then L1
and L2
meet on the straight
line :

Two families with three members each and one family with four members are to be seated in a row.
In how many ways can they be seated so that the same family members are not separated?

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is :

If \(\alpha\) and \(\beta\) be two roots of the equation x2 – 64x + 256 = 0. Then the value of
\({\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}\) is :

Let a , b, c , d and p be any non zero distinct real numbers such that
(a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then :

Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more
than the total number of subsets of B, then the value of m.n is ______.

If \(\sum\limits_{i = 1}^n {\left( {{x_i} - a} \right)} = n\) and \(\sum\limits_{i = 1}^n {{{\left( {{x_i} - a} \right)}^2}} = na\)
(n, a > 1) then the standard deviation of n
observations x1
, x2
, ..., xn
is :

A ray of light coming from the point (2, \(2\sqrt 3\)) is incident at an angle 30o on the line x = 1 at the
point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB
passes through the point :

If \(\overrightarrow a\)
and \(\overrightarrow b\)
are unit vectors, then the greatest value of
\(\sqrt 3 \left| {\overrightarrow a + \overrightarrow b } \right| + \left| {\overrightarrow a - \overrightarrow b } \right|\) is_____.