JEE Main 2021 (Online) 16th March Evening Shift

If the foot of the perpendicular from point (4, 3, 8) on the line \({L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}\), l \(\ne\) 0 is (3, 5, 7), then the shortest distance between the line L1 and line $${L_2}:{{x - 2} \over ...

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression $$3 + {{x - 11} \over {{{(y - 19)}^2}{{(z - 12)}^2}}} + {{y - 19} \over {{{(x - 11...

If the distance of the point (1, \(-\)2, 3) from the plane x + 2y \(-\) 3z + 10 = 0 measured parallel to the line, \({{x - 1} \over 3} = {{2 - y} \over m} = {{z + 3} \over 1}\) is \(\sqrt {{7 \over 2}}\), then the value of |m| is equal to ...

The maximum value of \(f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R\) is...

Let f be a real valued function, defined on R \(-\) {\(-\)1, 1} and given by f(x) = 3 loge \(\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}\).Then in which of the following intervals, function f(x) is increasing?

Let n be a positive integer. Let $$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^...

Let the lengths of intercepts on x-axis and y-axis made by the circle x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2\({\sqrt 2 }\) and 2\({\sqrt 5 }\), respectively. Then the shortest distance from origin to a tangent to this circle which is perp...

The least value of |z| where z is complex number which satisfies the inequality \(\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1}\), is equal to :

Consider the integral \(I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}\), where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that \(\int_0^1 {P(x)dx}\) = 1 and P(x) leaves remainder 5 when it is divided by (x \(-\) 2). Then the value of 9(b + c) is equal to :

Let C1 be the curve obtained by the solution of differential equation \(2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0\). Let the curve C2 be the solution of \({{2xy} \over {{x^2} - {y^2}}} = {{dy} \over {dx}}\). If both the curves pass through...

If y = y(x) is the solution of the differential equation \({{dy} \over {dx}}\) + (tan x) y = sin x, \(0 \le x \le {\pi \over 3}\), with y(0) = 0, then \(y\left( {{\pi \over 4}} \right)\) equal to :

If the points of intersections of the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1\) and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to :

For real numbers \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\), if \(\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \over {({x^4} + 3{x^2} + 1){{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}}dx}\)...

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy $${\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x$...

Let f : S \(\to\) S where S = (0, \(\infty\)) be a twice differentiable function such that f(x + 1) = xf(x). If g : S \(\to\) R be defined as g(x) = loge f(x), then the value of |g''(5) \(-\) g''(1)| is equal to :

Let \(\alpha\) \(\in\) R be such that the function $$f(x) = \left\{ {\matrix{
{{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr
{\alpha ,} & {x = 0} \cr

} } \...

Let f : R \(\to\) R and g : R \(\to\) R be defined as \(f(x) = \left\{ {\matrix{ {x + a,} & {x < 0} \cr {|x - 1|,} & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{x + 1,} & {x < 0} \cr
{{{(x - 1)}^2} + ...

Let \(A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]\) and \(B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]\) be two 2 \(\times\) 1 matrices with real entries such that A = XB, where $$X = {1 \ove...

Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is :

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number ...

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :

In \(\Delta\)ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of \(\Delta\)ABC is 30 cm2 and R and r are respectively the radii of circumcircle and incircle of \(\Delta\)ABC, then the value of 2R + r (in cm)...

Let \({1 \over {16}}\), a and b be in G.P. and \({1 \over a}\), \({1 \over b}\), 6 be in A.P., where a, b > 0. Then 72(a + b) is equal to ___________.

Sn(x) = loga1/2x + loga1/3x + loga1/6x + loga1/11x + loga1/18x + loga1/27x + ...... up to n-terms, where a > 1. If S24(x) = 1093 and S12(2x) = 265, then value of a is equal to ____________.

Let A = {2, 3, 4, 5, ....., 30} and '\(\simeq\)' be an equivalence relation on A \(\times\) A, defined by (a, b) \(\simeq\) (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with or...

Consider the statistics of two sets of observations as follows :



Size
Mean
Variance




Observation I
10
2
2


Observation II
n
3
1


If the variance of the combined set of thes...

Let A(\(-\)1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of \(\Delta\)ABC and \(\Delta\)PQC respectively, such that A1 = 3A2, the...

Let \(\overrightarrow c\) be a vector perpendicular to the vectors, \(\overrightarrow a\) = \(\widehat i\) + \(\widehat j\) \(-\) \(\widehat k\) and \(\overrightarrow b\) = \(\widehat i\) + 2\(\widehat j\) + \(\widehat k\). If $$\overrig...

Let \(\overrightarrow a\) = \(\widehat i\) + 2\(\widehat j\) \(-\) 3\(\widehat k\) and \(\overrightarrow b = 2\widehat i\) \(-\) 3\(\widehat j\) + 5\(\widehat k\). If \(\overrightarrow r\) \(\times\) \(\overrightarrow a\) = $$\overright...