JEE Main 2020 (Online) 9th January Morning Slot

The projection of the line segment joining the
points (1, –1, 3) and (2, –4, 11) on the line
joining the points (–1, 2, 3) and (3, –2, 10)
is ____________.

A spherical iron ball of 10 cm radius is
coated with a layer of ice of uniform
thickness the melts at a rate of 50 cm3/min.
When the thickness of ice is 5 cm, then the rate
(in cm/min.) at which of the thickness of ice
decreases, is :

The coefficient of x4 is the expansion of
(1 + x + x2)10 is _____.

A circle touches the y-axis at the point (0, 4)
and passes through the point (2, 0). Which of
the following lines is not a tangent to this circle?

Let z be complex number such that
\(\left| {{{z - i} \over {z + 2i}}} \right| = 1\) and |z| = \({5 \over 2}\). Then the value of |z + 3i| is :

If for all real triplets (a, b, c), ƒ(x) = a + bx + cx2;
then \(\int\limits_0^1 {f(x)dx}\) is equal to :

The value of
\(\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx\) is equal to :

If for x \(\ge\) 0, y = y(x) is the solution of the
differential equation
(x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0,
then y(3) is equal to _______.

If e1 and e2 are the eccentricities of the ellipse,
\({{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1\) and the hyperbola, \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then ...

If ƒ'(x) = tan–1(secx + tanx), \(- {\pi \over 2} < x < {\pi \over 2}\),
and
ƒ(0) = 0, then ƒ(1) is equal to :

The integral \(\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}}\) is equal to :
(where C is a constant of integration)

Let ƒ be any function continuous on [a, b] and
twice differentiable on (a, b). If for all x \(\in\) (a, b),
ƒ'(x) > 0 and ƒ''(x) < 0, then for any c \(\in\) (a, b),
\({{f(c) - f(a)} \over {f(b) - f(c)}}\) is greater than :

If $$f(x) = \left\{ {\matrix{
{{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr
{b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr
{{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \ov...

The number of distinct solutions of the equation
\({\log _{{1 \over 2}}}\left| {\sin x} \right| = 2 - {\log _{{1 \over 2}}}\left| {\cos x} \right|\) in the interval
[0, 2\(\pi\)], is ____.

Negation of the statement :
\(\sqrt 5\) is an integer or 5 is an irrational is :

If for some \(\alpha\) and \(\beta\) in R, the intersection of the
following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + \(\alpha\)z = 5
is a line in R3, then \(\alpha\) + \(\beta\) is equal to :

If the matrices A = \(\left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 \cr } } \right]\),
B = adjA and
C = 3A, then \({{\left| {adjB} \right|} \over {\left| C \right|}}\) is equal to :

If the number of five digit numbers with distinct
digits and 2 at the 10th place is 336 k, then k
is equal to :

In a box, there are 20 cards, out of which 10
are lebelled as A and the remaining 10 are
labelled as B. Cards are drawn at random, one
after the other and with replacement, till a
second A-card is obtained. The probability that
the second A...

The number of real roots of the equation,
e4x + e3x – 4e2x + ex + 1 = 0 is :

The product \({2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}\) ... to \(\infty\) is equal
to :

Let the observations xi (1 \(\le\) i \(\le\) 10) satisfy the
equations, \(\sum\limits_{i = 1}^{10} {\left( {{x_1} - 5} \right)}\) = 10 and \(\sum\limits_{i = 1}^{10} {{{\left( {{x_1} - 5} \right)}^2}}\) = 40.
If \(\mu\) and $$\lambda...

Let C be the centroid of the triangle with
vertices (3, –1), (1, 3) and (2, 4). Let P be the
point of intersection of the lines x + 3y – 1 = 0
and 3x – y + 1 = 0. Then the line passing through
the points C and P also passes through the
poin...

The value of
\({\cos ^3}\left( {{\pi \over 8}} \right)\)\({\cos}\left( {{3\pi \over 8}} \right)\)+\({\sin ^3}\left( {{\pi \over 8}} \right)\)\({\sin}\left( {{3\pi \over 8}} \right)\)
is :

If the vectors, \(\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k\),

\(\overrightarrow q = a\widehat i + \left( {a + 1} \right)\widehat j + a\widehat k\) and

$$\overrightarrow r = a\widehat i + a\wideha...