JEE Main 2021 (Online) 26th August Evening Shift

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes \(\overrightarrow r \,.\,\left( {\widehat i + \widehat j + 4\widehat k} \right) = 16\) and $$\overrightarrow r \,.\,\left( { - \widehat i + \w...

Let Q be the foot of the perpendicular from the point P(7, \(-\)2, 13) on the plane containing the lines \({{x + 1} \over 6} = {{y - 1} \over 7} = {{z - 3} \over 8}\) and \({{x - 1} \over 3} = {{y - 2} \over 5} = {{z - 3} \over 7}\). Then (...

The local maximum value of the function \(f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}\), x > 0, is

Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 \(-\) 3x2 \(-\) 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal t...

Let \(\left( {\matrix{ n \cr k \cr } } \right)\) denotes \({}^n{C_k}\) and $$\left[ {\matrix{
n \cr
k \cr

} } \right] = \left\{ {\matrix{
{\left( {\matrix{
n \cr
k \cr

} } \right),} & {if\,0 \le k \le ...

A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y \(-\) 5 = 0 at two points P and Q such that PQ is a diameter of C1. Then the diameter of C is :

If \({\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)\), then p and q are roots of the equation :

The least positive integer n such that \({{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1}\) is a positive integer, is ___________.

If the value of the integral \(\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }\), where \(\alpha\), \(\beta\) \(\in\) R, 5\(\alpha\) + 6\(\beta\) = 0, and [x] denotes the greatest integer less than or equ...

The value of \(\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx\) is

Let y(x) be the solution of the differential equation 2x2 dy + (ey \(-\) 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = \(\sqrt 5\) inches and \(\angle\)PCB = tan-1(2). The acute angle through which the pencil must be rotated about C s...

The point \(P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)\) lies on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) having eccentricity \({{\sqrt 5 } \over 2}\). If the tangent and normal at P to the hyperbola intersect it...

The locus of the mid points of the chords of the hyperbola x2 \(-\) y2 = 4, which touch the parabola y2 = 8x, is :

The domain of the function \({{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)\) is :

If \(\sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p}\), then the value of tan p is :

Let [t] denote the greatest integer less than or equal to t. Let f(x) = x \(-\) [x], g(x) = 1 \(-\) x + [x], and h(x) = min{f(x), g(x)}, x \(\in\) [\(-\)2, 2]. Then h is :

\(\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)\) is equal to :

Consider the two statements :(S1) : (p \(\to\) q) \(\vee\) (\(\sim\) q \(\to\) p) is a tautology .(S2) : (p \(\wedge\) \(\sim\) q) \(\wedge\) (\(\sim\) p \(\wedge\) q) is a fallacy.Then :

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right)\). Then A2025 \(-\) A2020 is equal to :

Let A be a 3 \(\times\) 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.

A fair die is tossed until six is obtained on it. Let x be the number of required tosses, then the conditional probability P(x \(\ge\) 5 | x > 2) is :

Two fair dice are thrown. The numbers on them are taken as \(\lambda\) and \(\mu\), and a system of linear equationsx + y + z = 5x + 2y + 3z = \(\mu\) x + 3y + \(\lambda\)z = 1is constructed. If p is the probability that the system has a un...

Let \(\lambda\) \(\ne\) 0 be in R. If \(\alpha\) and \(\beta\) are the roots of the equation x2 \(-\) x + 2\(\lambda\) = 0, and \(\alpha\) and \(\gamma\) are the roots of equation 3x2 \(-\) 10x + 27\(\lambda\) = 0, then $${{\beta \gamma } \...

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is _____________.

Let a1, a2, ......., a10 be an AP with common difference \(-\) 3 and b1, b2, ........., b10 be a GP with common ratio 2. Let ck = ak + bk, k = 1, 2, ......, 10. If c2 = 12 and c3 = 13, then \(\sum\limits_{k = 1}^{10} {{c_k}}\) is equal to ...

Let the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x \(-\) y is ______________.

The value of $$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} ...

A hall has a square floor of dimension 10 m \(\times\) 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is \({\cos ^{ - 1}}{1 \over 5}\), then the height of the hall (in meters) is :

If the projection of the vector \(\widehat i + 2\widehat j + \widehat k\) on the sum of the two vectors \(2\widehat i + 4\widehat j - 5\widehat k\) and \(- \lambda \widehat i + 2\widehat j + 3\widehat k\) is 1, then \(\lambda\) is equal to...