JEE Main 2022 (Online) 24th June Morning Shift

Let a line having direction ratios, 1, \(-\)4, 2 intersect the lines \({{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z + 2} \over 1}\) and \({x \over 2} = {{y - 7} \over 3} = {z \over 1}\) at the points A and B. Then (AB)2 is equal to ____...

If the shortest distance between the lines \(\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)\) and $$\overrightarrow r = \left( { - \widehat j + 2\widehat k} \right) + \...

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

For the function \(f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1\), which one of the following is NOT correct?

If the tangent at the point (x1, y1) on the curve \(y = {x^3} + 3{x^2} + 5\) passes through the origin, then (x1, y1) does NOT lie on the curve :

The sum of absolute maximum and absolute minimum values of the function \(f(x) = |2{x^2} + 3x - 2| + \sin x\cos x\) in the interval [0, 1] is :

Let \(\lambda x - 2y = \mu\) be a tangent to the hyperbola \({a^2}{x^2} - {y^2} = {b^2}\). Then \({\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}\) is equal to :

Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then \({{{R_2}} \over {{R_1}}}\) is equal to ______________.

The remainder when 32022 is divided by 5 is :

Let \(A = \{ z \in C:1 \le |z - (1 + i)| \le 2\}\)
and \(B = \{ z \in A:|z - (1 - i)| = 1\}\). Then, B :

Let \(f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}\). Then the value of \(\left| {\int_0^{\pi /2} {f(\theta )d\theta } } \right|\) is _____________.

Let \(\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha\) and \(\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta\).
If $$\int\limits_{\beta - {...

If x = x(y) is the solution of the differential equation \(y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0\); then x(e) is equal to :

If two tangents drawn from a point (\(\alpha\), \(\beta\)) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10\(\alpha\) + 5)2 + (16\(\beta\)2 + 5...

The number of one-one functions f : {a, b, c, d} \(\to\) {0, 1, 2, ......, 10} such
that 2f(a) \(-\) f(b) + 3f(c) + f(d) = 0 is ___________.

The set of all values of k for which \({({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R\), is the interval :

The domain of the function \(f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}\) is :

The number of points where the function
\(f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1 < x < 1} \cr {|x + 1| + |x - 2|} & {if} & {x \ge 1} \cr } } \right.\)
[t] denote...

The number of choices for \(\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \}\), such that \((p\Delta q) \Rightarrow ((p\Delta \sim q) \vee (( \sim p)\Delta q))\) is a tautology, is :

The number of values of \(\alpha\) for which the system of equations :
x + y + z = \(\alpha\)
\(\alpha\)x + 2\(\alpha\)y + 3z = \(-\)1
x + 3\(\alpha\)y + 5z = 4
is inconsistent, is

Let S = {\(\sqrt{n}\) : 1 \(\le\) n \(\le\) 50 and n is odd}.
Let a \(\in\) S and \(A = \left[ {\matrix{ 1 & 0 & a \cr { - 1} & 1 & 0 \cr { - a} & 0 & 1 \cr } } \right]\).
If $$\sum\limits_{a\, \in \,S}^{} {\det (adj\,A) = ...

Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, \(-\)2 marks for each wrong answer and 0 mark if the question is not attempted. Then, t...

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come...

If a random variable X follows the Binomial distribution B(33, p) such that \(3P(X = 0) = P(X = 1)\), then the value of \({{P(X = 15)} \over {P(X = 18)}} - {{P(X = 16)} \over {P(X = 17)}}\) is equal to :

If the sum of the squares of the reciprocals of the roots \(\alpha\) and \(\beta\) of the equation 3x2 + \(\lambda\)x \(-\) 1 = 0 is 15, then 6(\(\alpha\)3 + \(\beta\)3)2 is equal to :

If \(\{ {a_i}\} _{i = 1}^n\), where n is an even integer, is an arithmetic progression with common difference 1, and \(\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120}\), then n is equal to :

Let \(A\left( {{3 \over {\sqrt a }},\sqrt a } \right),\,a > 0\), be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If \(D(3\cos \theta ,a\sin \theta )\) is a point in the fourth quadrant such...

Let \(S = \left\{ {\theta \in [ - \pi ,\pi ] - \left\{ { \pm \,\,{\pi \over 2}} \right\}:\sin \theta \tan \theta + \tan \theta = \sin 2\theta } \right\}\). If \(T = \sum\limits_{\theta \, \in \,S}^{} {\cos 2\theta }\), then T + n(S) is...

Let \(\widehat a\), \(\widehat b\) be unit vectors. If \(\overrightarrow c\) be a vector such that the angle between \(\widehat a\) and \(\overrightarrow c\) is \({\pi \over {12}}\), and $$\widehat b = \overrightarrow c + 2\left( {\over...