JEE Main 2021 (Online) 24th February Morning Shift

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the
planes 3x + y - 2z = 5 and 2x - 5y - z = 7, is :

The distance of the point (1, 1, 9) from the point of intersection of the line
\({{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2}\)
and the plane x + y + z = 17 is :

If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the
ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

The minimum value of \(\alpha\) for which the equation \({4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha\)
has at least one
solution in \(\left( {0,{\pi \over 2}} \right)\) is .......

The function
f(x) = \({{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x\) :

The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola
y2 = 9x, is :

The value of
-15C1 + 2.15C2 – 3.15C3 + ... - 15.15C15 + 14C1 + 14C3 + 14C5 + ...+ 14C11 is :

If one of the diameters of the circle x2 + y2 - 2x - 6y + 6 = 0 is a chord of another circle 'C',
whose center is at (2, 1), then its radius is ________.

If the least and the largest real values of a, for which the equation z + \(\alpha\)|z – 1| + 2i = 0
(z \(\in\) C and i = \(\sqrt { - 1}\)) has a solution, are p and q respectively; then 4(p2 + q2) is equal to __________.

If \(\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22\), (a > 2) and [x] denotes the greatest integer \(\le\) x, then\(\int\limits_{ - a}^a {\left( {x + \left[ x \right]} \right)} dx\) is equal to ...

\(\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}\) is equal to :

The population P = P(t) at time 't' of a certain species follows the differential equation
\({{dP} \over {dt}}\)
= 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) =
\({{x - {1 \over 2}} \over {x - 1}}\).
Then the composition function f(g(x)) is :

Two vertical poles are 150 m apart and the height of one is three times that of the other. If
from the middle point of the line joining their feet, an observer finds the angles of elevation of
their tops to be complementary, then the height...

If \(\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c\), where c is a constant of integration, then
the ordered pair (a, b) is equal to :

\(\mathop {\lim }\limits_{n \to \infty } \tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {1 + r + {r^2}}}} \right)} } \right\}\) is equal to ______.

If f : R \(\to\) R is a function defined by f(x)= [x - 1] \(\cos \left( {{{2x - 1} \over 2}} \right)\pi\), where [.] denotes the greatest
integer function, then f is :

The statement among the following that is a tautology is :

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

Let M be any 3 \(\times\) 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is ________.

Let P = \(\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]\), where \(\alpha\) \(\in\) R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k \(\in\) R. If ...

The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a
moving point of the parabola, is another parabola whose directrix is :

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at
least 2 Indians and double the number of foreigners as Indians. Then the number of ways,
the committee can be formed, is :

Let Bi (i = 1, 2, 3) be three independent events in a sample space. The probability that only B1 occur is \(\alpha\), only B2 occurs is \(\beta\) and only B3 occurs is \(\gamma\). Let p be the probability that none of the events Bi occur...

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd
number 2 times is equal to the probability of getting an even number 3 times, then the
probability of getting an odd number for odd number of tim...

Let p and q be two positive numbers such that p + q = 2 and p4+q4 = 272. Then p and q are
roots of the equation :

Let  A = {n \(\in\) N: n is a 3-digit number}
       B = {9k + 2: k \(\in\) N}
and C = {9k + \(l\): k \(\in\) N} for some \(l ( 0 < l < 9)\)
If the sum of all the elements of the set A \(\cap\) (B \(\cup\) C) is 274 \(\times\) 4...

A man is walking on a straight line. The arithmetic mean
of the reciprocals of the intercepts of this line on the
coordinate axes is \({1 \over 4}\). Three stones A, B and C are placed at the points
(1, 1), (2, 2) and (4, 4) respectively. T...

If
\({e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}\)
satisfies the equation t2 - 9t + 8 = 0, then the value of
$${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)...

Let three vectors \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be such that \(\overrightarrow c\) is coplanar with \(\overrightarrow a\) and \(\overrightarrow b\),
\(\overrightarrow a .\overrightarrow c\) ...