JEE Main 2020 (Online) 7th January Evening Slot

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (\(\alpha\), 7, 1)
is
\(\left( {{5 \over 3},{7 \over 3},{{17} \over 3}} \right)\), then \(\alpha\) is equal to______.

The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3
- 4x2
+ 8x + 11,
when x \(\in\) [0, 1] is:

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points.
If \(\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4\), then which one of the following is not true?

The area (in sq. units) of the region
{(x, y) \(\in\) R2 | 4x2 \(\le\) y \(\le\) 8x + 12} is :

The coefficient of x7
in the expression
(1 + x)10 + x(1 + x)9
+ x2(1 + x)8
+ ......+ x10 is:

Let the tangents drawn from the origin to the circle, x2
+ y2
- 8x - 4y + 16 = 0 touch it at the
points A and B. The (AB)2
is equal to :

If \({{3 + i\sin \theta } \over {4 - i\cos \theta }}\), \(\theta\) \(\in\) [0, 2\(\theta\)], is a real number, then an argument of sin\(\theta\) + icos\(\theta\) is :

The value of \(\alpha\) for which
\(4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5\), is:

If \(\theta\)1
and \(\theta\)2
be respectively the smallest and the largest values of \(\theta\) in (0, 2\(\pi\)) - {\(\pi\)} which satisfy
the equation,
2cot2\(\theta\) - \({5 \over {\sin \theta }}\) + 4 = 0, then
$$\int\limits_{{\...

Let y = y(x) be the solution curve of the differential equation,
\(\left( {{y^2} - x} \right){{dy} \over {dx}} = 1\), satisfying y(0) =
1. This curve intersects the x-axis at a point whose abscissa is :

Let y = y(x) be a function of x satisfying
\(y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}}\) where k is a constant and
\(y\left( {{1 \over 2}} \right) = - {1 \over 4}\). Then \({{dy} \over {dx}}\) at x = \({1 \over 2}\), is equal to :

If 3x + 4y = 12\(\sqrt 2\) is a tangent to the ellipse
\({{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1\) for some \(a\) \(\in\) R, then the distance
between the foci of the ellipse is :

If the function ƒ defined on \(\left( { - {1 \over 3},{1 \over 3}} \right)\) by
f(x) = $$\left\{ {\matrix{
{{1 \over x}{{\log }_e}\left( {{{1 + 3x} \over {1 - 2x}}} \right),} & {when\,x \ne 0} \cr
{k,} & {when\,x = 0} \cr

} } \r...

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A \(\subseteq\) B and B \(\subseteq\) D,
then A \(\subseteq\) C" is :

If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + \(\lambda\)z = \(\mu\)
has more than two solutions, then \(\mu\) - \(\lambda\)2
is equal to ______.

Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3.
If the determinant of B is 81, then the determinant of A is:

The number of ordered pairs (r, k) for which 6.35Cr
= (k2 - 3). 36Cr + 1, where k is an integer, is :

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is \({{1 \over 4}}\)
. If the probability that at most two machines will be out of service on the same day is $${\left( {{3 \over 4}...

Let \(\alpha\) and \(\beta\) be the roots of the equation x2
- x - 1 = 0. If pk
= \({\left( \alpha \right)^k} + {\left( \beta \right)^k}\)
, k \(\ge\) 1, then which one
of the following statements is not true?

Let \({a_1}\)
, \({a_2}\)
, \({a_3}\)
,....... be a G.P. such that \({a_1}\)
< 0, \({a_1}\)
+ \({a_2}\)
= 4 and \({a_3}\)
+ \({a_4}\)
= 16. If \(\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda\), then \(\lambda\) is
equal to:

If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :

Let X = {n \(\in\) N : 1 \(\le\) n \(\le\) 50}. If
A = {n \(\in\) X: n is a multiple of 2} and
B = {n \(\in\) X: n is a multiple of 7}, then the number of elements in the smallest subset of X
containing both A and B is ________.

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively,
then x.y is equal to _______.

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line
x = y is :

Let \(\overrightarrow a\)
, \(\overrightarrow b\)
and \(\overrightarrow c\)
be three unit vectors such that
\(\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0\). If $$\lambda = \overrightarrow a .\vec b + \vec b.\...