JEE Main 2021 (Online) 22th July Evening Shift

Let L be the line of intersection of planes \(\overrightarrow r .(\widehat i - \widehat j + 2\widehat k) = 2\) and \(\overrightarrow r .(2\widehat i + \widehat j - \widehat k) = 2\). If \(P(\alpha ,\beta ,\gamma )\) is the foot of perpendic...

If the shortest distance between the straight lines \(3(x - 1) = 6(y - 2) = 2(z - 1)\) and \(4(x - 2) = 2(y - \lambda ) = (z - 3),\lambda \in R\) is \({1 \over {\sqrt {38} }}\), then the integral value of \(\lambda\) is equal to :

Let f : R \(\to\) R be defined as\(f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le 0} \cr } } \right.\). Then f is increasing function in the interval

The area (in sq. units) of the region bounded by the curves x2 + 2y \(-\) 1 = 0, y2 + 4x \(-\) 4 = 0 and y2 \(-\) 4x \(-\) 4 = 0, in the upper half plane is _______________.

If the constant term, in binomial expansion of \({\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}\) is 180, then r is equal to __________________.

The number of elements in the set {n \(\in\) {1, 2, 3, ......., 100} | (11)n > (10)n + (9)n} is ______________.

Let the circle S : 36x2 + 36y2 \(-\) 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x \(-\) 2y = 4 and 2x \(-\) y = 5 lies inside the circle S, then :

Let n denote the number of solutions of the equation z2 + 3\(\overline z\) = 0, where z is a complex number. Then the value of \(\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}}\) is equal to :

If \(\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R}\) where [x] is the greatest integer less than or eq...

Let y = y(x) be the solution of the differential equation \(\cos e{c^2}xdy + 2dx = (1 + y\cos 2x)\cos e{c^2}xdx\), with \(y\left( {{\pi \over 4}} \right) = 0\). Then, the value of \({(y(0) + 1)^2}\) is equal to :

Let y = y(x) be the solution of the differential equation \(\left( {(x + 2){e^{\left( {{{y + 1} \over {x + 2}}} \right)}} + (y + 1)} \right)dx = (x + 2)dy\), y(1) = 1. If the domain of y = y(x) is an open interval (\(\alpha\), \(\beta\)), t...

Let \({E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b\). Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have same e...

Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A \(\to\) A such that f(1) + f(2) = 3 \(-\) f(3) is equal to

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 \(-\) y2 = 3. If L is also a tangent to the parabola y2 = \(\alpha\)x, then \(\alpha\) is equal to :

If the domain of the function \(f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}\) is the interval (\(\alpha\), \(\beta\)], then \(\alpha\) + \(\beta\) is equal to :

Let f : R \(\to\) R be defined as $$f(x) = \left\{ {\matrix{
{{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr

} } ...

Let f : R \(\to\) R be a function defined as \(f(x) = \left\{ {\matrix{ {3\left( {1 - {{|x|} \over 2}} \right)} & {if} & {|x|\, \le 2} \cr 0 & {if} & {|x|\, > 2} \cr } } \right.\)Let g : R \(\to\) R be given by $$g(x) = f(x + 2)...

Which of the following Boolean expressions is not a tautology?

The values of \(\lambda\) and \(\mu\) such that the system of equations \(x + y + z = 6\), \(3x + 5y + 5z = 26\), \(x + 2y + \lambda z = \mu\) has no solution, are :

Let A = [aij] be a real matrix of order 3 \(\times\) 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :

Let \(A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]\). Then the number of 3 \(\times\) 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________.

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________.

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 \(\times\) 2 matrices. The probability that such formed matrix have all different entries and are non-singular, is :

Let [x] denote the greatest integer less than or equal to x. Then, the values of x\(\in\)R satisfying the equation \({[{e^x}]^2} + [{e^x} + 1] - 3 = 0\) lie in the interval :

Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 \(-\) S6 is equal to:

The sum of all the elements in the set {n\(\in\) {1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to _____________.

Consider the following frequency distribution :
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The number of solutions of sin7x + cos7x = 1, x\(\in\) [0, 4\(\pi\)] is equal to

Let a vector \({\overrightarrow a }\) be coplanar with vectors \(\overrightarrow b = 2\widehat i + \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\). If \({\overrightarrow a}\) is perpendicular to ...

Let three vectors \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be such that \(\overrightarrow a \times \overrightarrow b = \overrightarrow c\), $$\overrightarrow b \times \overrightarrow c = \overrightarrow...