JEE Main 2022 (Online) 29th June Morning Shift

If the mirror image of the point (2, 4, 7) in the plane 3x \(-\) y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

Let d be the distance between the foot of perpendiculars of the points P(1, 2, \(-\)1) and Q(2, \(-\)1, 3) on the plane \(-\)x + y + z = 1. Then d2 is equal to ___________.

Let \({P_1}:\overrightarrow r \,.\,\left( {2\widehat i + \widehat j - 3\widehat k} \right) = 4\) be a plane. Let P2 be another plane which passes through the points (2, \(-\)3, 2), (2, \(-\)2, \(-\)3) and (1, \(-\)4, 2). If the direction ra...

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square ...

The area enclosed by y2 = 8x and y = \(\sqrt2\) x that lies outside the triangle formed by y = \(\sqrt2\) x, x = 1, y = 2\(\sqrt2\), is equal to:

If the constant term in the expansion of
\({\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}\) is 2k.l, where l is an odd integer, then the value of k is equal to:

Let the tangent to the circle C1 : x2 + y2 = 2 at the point M(\(-\)1, 1) intersect the circle C2 : (x \(-\) 3)2 + (y \(-\) 2)2 = 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of...

Let \(\alpha\) and \(\beta\) be the roots of the equation x2 + (2i \(-\) 1) = 0. Then, the value of |\(\alpha\)8 + \(\beta\)8| is equal to :

Let \(S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}\). Let \(|z - 4i|\) attains minimum and maximum values, respectively, at z1 \(\in\) S and z2 \(\in\) S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5...

Let \(f:R \to R\) be a function defined by :
$$f(x) = \left\{ {\matrix{
{\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr
{{x^2} + 2x - 6} & ; & {2 < x < 3} \cr
{[x - 3] + 9} & ; & {3 \le x \le 5} \cr
{2x + 1} & ; & {...

\(\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx}\),
where [t] denotes greatest integer less than or equal to t, is equal to:

Let the solution curve of the differential equation
\(x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16{x^2}}\), \(y(1) = 3\) be \(y = y(x)\). Then y(2) is equal to:

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}\) with $$y\left( {{\pi \over 4}} \right)...

Let c, k \(\in\) R. If \(f(x) = (c + 1){x^2} + (1 - {c^2})x + 2k\) and \(f(x + y) = f(x) + f(y) - xy\), for all x, y \(\in\) R, then the value of \(|2(f(1) + f(2) + f(3) + \,\,......\,\, + \,\,f(20))|\) is equal to ____________.

Let \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\), a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \(4(2\sqrt 2 + \sqrt {14} )\). If the eccentricity H is $${{\sqrt {11} }...

The domain of the function \({\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)\) is :

\(50\tan \left( {3{{\tan }^{ - 1}}\left( {{1 \over 2}} \right) + 2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right) + 4\sqrt 2 \tan \left( {{1 \over 2}{{\tan }^{ - 1}}(2\sqrt 2 )} \right)\) is equal to ____________.

Let \(\Delta\) \(\in\) {\(\wedge\), \(\vee\), \(\Rightarrow\), \(\Leftrightarrow\)} be such that (p \(\wedge\) q) \(\Delta\) ((p \(\vee\) q) \(\Rightarrow\) q) is a tautology. Then \(\Delta\) is equal to :

If the system of linear equations
2x + y \(-\) z = 7
x \(-\) 3y + 2z = 1
x + 4y + \(\delta\)z = k, where \(\delta\), k \(\in\) R has infinitely many solutions, then \(\delta\) + k is equal to:

Let \(A = [{a_{ij}}]\) be a square matrix of order 3 such that \({a_{ij}} = {2^{j - i}}\), for all i, j = 1, 2, 3. Then, the matrix A2 + A3 + ...... + A10 is equal to :

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of \({\pi \over 2}\) at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1...

Let b1b2b3b4 be a 4-element permutation with bi \(\in\) {1, 2, 3, ........, 100} for 1 \(\le\) i \(\le\) 4 and bi \(\ne\) bj for i \(\ne\) j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then ...

The probability that a randomly chosen 2 \(\times\) 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

Let \(\{ {a_n}\} _{n = 0}^\infty\) be a sequence such that \({a_0} = {a_1} = 0\) and \({a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1\) for all n \(\ge\) 0. Then, \(\sum\limits_{n = 2}^\infty {{{{a_n}} \over {{7^n}}}}\) is equal to:

Let a set A = A1 \(\cup\) A2 \(\cup\) ..... \(\cup\) Ak, where Ai \(\cap\) Aj = \(\phi\) for i \(\ne\) j, 1 \(\le\) j, j \(\le\) k. Define the relation R from A to A by R = {(x, y) : y \(\in\) Ai if and only if x \(\in\) Ai, 1 \(\le\) i $$\...

Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be \({24 \over 5}\) and \({194 \over 25}\) respectively. If the mean and variance of the first 4 observation are \({7 \over 2}\) and a respectively, then (4a + x5) is equal ...

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle \({\pi \over 4}\) at the origin, is equal to :

The number of elements in the set \(S = \{ \theta \in [ - 4\pi ,4\pi ]:3{\cos ^2}2\theta + 6\cos 2\theta - 10{\cos ^2}\theta + 5 = 0\}\) is __________.

The number of solutions of the equation \(2\theta - {\cos ^2}\theta + \sqrt 2 = 0\) in R is equal to ___________.

Let \(\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k\), \(\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k\) and \(\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k\) where $$\alpha ,\,\beta \i...