JEE Main 2021 (Online) 17th March Evening Shift

If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line \({{x + 1} \over 2} = {{y - 3} \over 1} = {{z + 2} \over { - 1}}\) and containing the line $${{x - 2} \over 3} = {{1 - y} \over 2} = {{z + 1...

Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx \(-\) nz = 0 and x \(-\) 2y + z = 0, equal to 9. If the locus of the point P is x2 + y2 + z2 = 9, then the value of l \(-\) n is equal ...

Consider the function f : R \(\to\) R defined by
\(f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr 0,\,\,x = 0 \hfill \cr} \right.\). Then f is :

Let f : [\(-\)1, 1] \(\to\) R be defined as f(x) = ax2 + bx + c for all x\(\in\)[\(-\)1, 1], where a, b, c\(\in\)R such that f(\(-\)1) = 2, f'(\(-\)1) = 1 for x\(\in\)(\(-\)1, 1) the maximum value of f ''(x) is \({{1 \over 2}}\). If f(x) ...

Let f : [\(-\)3, 1] \(\to\) R be given as \(f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.\)If the area bounded by y = f(x) and x-axis i...

Let the coefficients of third, fourth and fifth terms in the expansion of \({\left( {x + {a \over {{x^2}}}} \right)^n},x \ne 0\), be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to ___________.

The value of \(\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)}\) is equal to :

Two tangents are drawn from a point P to the circle x2 + y2 \(-\) 2x \(-\) 4y + 4 = 0, such that the angle between these tangents is \({\tan ^{ - 1}}\left( {{{12} \over 5}} \right)\), where \({\tan ^{ - 1}}\left( {{{12} \over 5}} \right)\) ...

Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, the...

Let S1, S2 and S3 be three sets defined asS1 = {z\(\in\)C : |z \(-\) 1| \(\le\) \(\sqrt 2\)}S2 = {z\(\in\)C : Re((1 \(-\) i)z) \(\ge\) 1}S3 = {z\(\in\)C : Im(z) \(\le\) 1}Then the set S1 \(\cap\) S2 \(\cap\) S3 :

Let f : R \(\to\) R be defined as f(x) = e\(-\)xsinx. If F : [0, 1] \(\to\) R is a differentiable function with that F(x) = \(\int_0^x {f(t)dt}\), then the value of \(\int_0^1 {(F'(x) + f(x)){e^x}dx}\) lies in the interval

Let \({I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx\), where n\(\in\)N. If (20)I10 = \(\alpha\)I9 + \(\beta\)I8, for natural numbers \(\alpha\) and \(\beta\), then \(\alpha\) \(-\) \(\beta\) equals to ___________.

If the integral
\(\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma\), where \(\alpha\), \(\beta\), \(\gamma\) are integers and [x] denotes the greatest integer less than or...

Let y = y(x) be the solution of the differential equation \(\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx,0 \le x \le {\pi \over 2},y(0) = 0\). Then, \(y\left( {{\pi \over 3}} \right)\) is equal to :

If the curve y = y(x) is the solution of the differential equation \(2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})dx = {2x^{9/4}}dx\), x > 0 which passes through the point \(\left( {1,1 - {4 \over 3}{{\log }_e}2} \right)\), then the value of y(...

The number of solutions of the equation \({\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}\), for x\(\in\)[\(-\)1, 1], and [x] denotes the greatest integer less than or equal to...

The value of the limit \(\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}\) is equal to :

The value of \(\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}\), where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

If the Boolean expression \((p \wedge q) \odot (p \otimes q)\) is a tautology, then \(\odot\) and \(\otimes\) are respectively given by :

If x, y, z are in arithmetic progression with common difference d, x \(\ne\) 3d, and the determinant of the matrix \(\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]\) is zero, then...

If 1, log10(4x \(-\) 2) and log10\(\left( {{4^x} + {{18} \over 5}} \right)\) are in arithmetic progression for a real number x, then the value of the determinant $$\left| {\matrix{
{2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} ...

Let \(A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]\) and \(B = \left[ {\matrix{ \alpha \cr \beta \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr } } \right]\) such that AB = B and a + d = 2021,...

Let L be a tangent line to the parabola y2 = 4x \(-\) 20 at (6, 2). If L is also a tangent to the ellipse \({{{x^2}} \over 2} + {{{y^2}} \over b} = 1\), then the value of b is equal to :

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be \({1 \over 2}\) and probability of occurrence of 0 at the odd place be \({1 \over 3}\). Then th...

Consider a set of 3n numbers having variance 4. In this set, the mean of first 2n numbers is 6 and the mean of the remaining n numbers is 3. A new set is constructed by adding 1 into each of first 2n numbers, and subtracting 1 from each of ...

Let tan\(\alpha\), tan\(\beta\) and tan\(\gamma\); \(\alpha\), \(\beta\), \(\gamma\) \(\ne\) \({{(2n - 1)\pi } \over 2}\), n\(\in\)N be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of $$\...

The number of solutions of the equation x + 2tanx = \({\pi \over 2}\) in the interval [0, 2\(\pi\)] is :

Let O be the origin. Let \(\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\) and \(\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k\), x, y\(\in\)R, x > 0, be such that $$\left| {\overrightarrow {PQ} } \righ...

Let \(\overrightarrow x\) be a vector in the plane containing vectors \(\overrightarrow a = 2\widehat i - \widehat j + \widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - \widehat k\). If the vector \(\overrightarrow x\) i...