JEE Main 2022 (Online) 26th June Evening Shift

If the plane \(2x + y - 5z = 0\) is rotated about its line of intersection with the plane \(3x - y + 4z - 7 = 0\) by an angle of \({\pi \over 2}\), then the plane after the rotation passes through the point :

If the lines \(\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)\) and $$\overrightarrow r = \left( {\alpha \widehat i - \widehat j} \right) + \mu \left( {2\wideh...

Let \(\overrightarrow a = \widehat i + \widehat j + 2\widehat k\), \(\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\) be three given vectors. Let $$\overrightar...

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :

The area of the region bounded by y2 = 8x and y2 = 16(3 \(-\) x) is equal to:

If \(\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}\) m and n are coprime, then m + n is equal to ___________...

If \({z^2} + z + 1 = 0\), \(z \in C\), then \(\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|\) is equal to _________.

The integral \({{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}}\) is equal to ____________.

If \(y = y(x)\) is the solution of the differential equation \(x{{dy} \over {dx}} + 2y = x\,{e^x}\), \(y(1) = 0\) then the local maximum value of the function \(z(x) = {x^2}y(x) - {e^x},\,x \in R\) is :

If the solution of the differential equation \({{dy} \over {dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \right){e^{2x}}\) satisfies \(y(0) = 0\), then the value of y(2) is _______________.

Let f : R \(\to\) R satisfy \(f(x + y) = {2^x}f(y) + {4^y}f(x)\), \(\forall\)x, y \(\in\) R. If f(2) = 3, then \(14.\,{{f'(4)} \over {f'(2)}}\) is equal to ____________.

If m is the slope of a common tangent to the curves \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\) and \({x^2} + {y^2} = 12\), then \(12{m^2}\) is equal to :

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse \({x^2} + 2{y^2} = 4\) is an ellipse with eccentricity :

Let f : R \(\to\) R be defined as f (x) = x \(-\) 1 and g : R \(-\) {1, \(-\)1} \(\to\) R be defined as \(g(x) = {{{x^2}} \over {{x^2} - 1}}\).
Then the function fog is :

The normal to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1\) at the point \(\left( {8,3\sqrt 3 } \right)\) on it passes through the point :

Let a line L1 be tangent to the hyperbola \({{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1\) and let L2 be the line passing through the origin and perpendicular to L1. If the locus of the point of intersection of L1 and L2 is $${({x^2} + {y^2...

If \(\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c}\), \(g(1) = 0\), then \(g\left( {{1 \over 2}} \right)\) is equal to :

If the inverse trigonometric functions take principal values then $${\cos ^{ - 1}}\left( {{3 \over {10}}\cos \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right)} \right) + {2 \over 5}\sin \left( {{{\tan }^{ - 1}}\left( {{4 \over 3}} \right...

\(\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}\) is equal to :

Let f(x) = min {1, 1 + x sin x}, 0 \(\le\) x \(\le\) 2\(\pi\). If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

Let r \(\in\) {p, q, \(\sim\)p, \(\sim\)q} be such that the logical statement
r \(\vee\) (\(\sim\)p) \(\Rightarrow\) (p \(\wedge\) q) \(\vee\) r
is a tautology. Then r is equal to :

If the system of equations
\(\alpha\)x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = \(\beta\)
has infinitely many solutions, then the ordered pair (\(\alpha\), \(\beta\)) is equal to :

Let \(X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}\) and \(Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}\), \(\alpha\), $$\beta$...

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ___________.

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to _______________.

Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then \({\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}}\) is equal to _________.

If \(A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}\) and \(B = \sum\limits_{n = 1}^\infty {{{{{( - 1)}^n}} \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}\), then \({A \over B}\) is equal to :

If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to ___________.

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct v...

\(16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )\) is equal to :