JEE Main 2022 (Online) 25th June Evening Shift

Let p be the plane passing through the intersection of the planes \(\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5\) and $$\overrightarrow r \,.\,\left( {2\widehat i - \widehat j + \widehat k} \right) = 3$...

Let l1 be the line in xy-plane with x and y intercepts \({1 \over 8}\) and \({1 \over {4\sqrt 2 }}\) respectively, and l2 be the line in zx-plane with x and z intercepts \(- {1 \over 8}\) and \(- {1 \over {6\sqrt 3 }}\) respectively. If d...

Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm2 / sec) at which the wet conical surface...

If the angle made by the tangent at the point (x0, y0) on the curve \(x = 12(t + \sin t\cos t)\), \(y = 12{(1 + \sin t)^2}\), \(0 < t < {\pi \over 2}\), with the positive x-axis is \({\pi \over 3}\), then y0 is equal to:

Let \(f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R\). If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ____________.

The area of the region enclosed between the parabolas y2 = 2x \(-\) 1 and y2 = 4x \(-\) 3 is

The coefficient of x101 in the expression \({(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}\), x > 0, is

If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of \({\left( {2{x^3} + {3 \over x}} \right)^{10}}\) is \({5^{10}} - \beta \,.\,{3^9}\), then \(\beta\) is equal to ____________.

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

Let z1 and z2 be two complex numbers such that \({\overline z _1} = i{\overline z _2}\) and \(\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi\). Then :

If \({b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N}\), then

The value of b > 3 for which \(12\int\limits_3^b {{1 \over {({x^2} - 1)({x^2} - 4)}}dx = {{\log }_e}\left( {{{49} \over {40}}} \right)}\), is equal to ___________.

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

The line y = x + 1 meets the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 2} = 1\) at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be \({5 \over 4}\). If the equation of the normal at the point \(\left( {{8 \over {\sqrt {5} }},{{12} \over {5}}} \right)\) on the hyperbola is ...

The value of \({\tan ^{ - 1}}\left( {{{\cos \left( {{{15\pi } \over 4}} \right) - 1} \over {\sin \left( {{\pi \over 4}} \right)}}} \right)\) is equal to :

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)\) is equal to

Let \(f(x) = \left[ {2{x^2} + 1} \right]\) and \(g(x) = \left\{ {\matrix{ {2x - 3,} & {x < 0} \cr {2x + 3,} & {x \ge 0} \cr } } \right.\), where [t] is the greatest integer \(\le\) t. Then, in the open interval (\(-\)1, 1), the ...

The negation of the Boolean expression ((\(\sim\) q) \(\wedge\) p) \(\Rightarrow\) ((\(\sim\) p) \(\vee\) q) is logically equivalent to :

The system of equations
\(- kx + 3y - 14z = 25\)
\(- 15x + 4y - kz = 3\)
\(- 4x + y + 3z = 4\)
is consistent for all k in the set

Let \(A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)\) and \(B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)\). Then the number of elements in the set {(n, m) : n, m \(\in\) {1, 2, .......

If the line \(y = 4 + kx,\,k > 0\), is the tangent to the parabola \(y = x - {x^2}\) at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

The total number of three-digit numbers, with one digit repeated exactly two times, is ______________.

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is \({1 \over n}\). If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :

Let \(A = \{ x \in R:|x + 1| < 2\}\) and \(B = \{ x \in R:|x - 1| \ge 2\}\). Then which one of the following statements is NOT true?

Let a, b \(\in\) R be such that the equation \(a{x^2} - 2bx + 15 = 0\) has a repeated root \(\alpha\). If \(\alpha\) and \(\beta\) are the roots of the equation \({x^2} - 2bx + 21 = 0\), then \({\alpha ^2} + {\beta ^2}\) is equal to :

The sum 1 + 2 . 3 + 3 . 32 + ......... + 10 . 39 is equal to :

If the mean deviation about the mean of the numbers 1, 2, 3, .........., n, where n is odd, is \({{5(n + 1)} \over n}\), then n is equal to ______________.

The value of 2sin (12\(^\circ\)) \(-\) sin (72\(^\circ\)) is :

Let \(\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k\), \(\lambda\) \(\in\) R. If \(\overrightarrow a\) is a vector such that \(\overrightarrow a \times \overrightarrow b = 13\widehat i - \widehat j - 4\widehat k\) and...