JEE Main 2021 (Online) 26th February Evening Shift

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P(\(\alpha\), \(\beta\), \(\gamma\)) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21(\(\alpha\) + \(\beta\) + $$\gam...

If the mirror image of the point (1, 3, 5) with respect to the plane 4x \(-\) 5y + 2z = 8 is (\(\alpha\), \(\beta\), \(\gamma\)), then 5(\(\alpha\) + \(\beta\) + \(\gamma\)) equals :

Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, \(-\)3) and (4, \(-\)2\(\sqrt 2\)), and given that a \(-\) 2\(\sqrt 2\) b = 3, then (a2 + b2 + ab) is equal to _________...

Let slope of the tangent line to a curve at any point P(x, y) be given by \({{x{y^2} + y} \over x}\). If the curve intersects the line x + 2y = 4 at x = \(-\)2, then the value of y, for which the point (3, y) lies on the curve, is :

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = \({\pi \over 2}\) in the first quad...

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal to :

Let A(1, 4) and B(1, \(-\)5) be two points. Let P be a point on the circle (x \(-\) 1)2 + (y \(-\) 1)2 = 1 such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on :

Let z be those complex numbers which satisfy| z + 5 | \(\le\) 4 and z(1 + i) + \(\overline z\)(1 \(-\) i) \(\ge\) \(-\)10, i = \(\sqrt { - 1}\).If the maximum value of | z + 1 |2 is \(\alpha\) + \(\beta\)\(\sqrt 2\), then the value o...

Let \(f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}\) be a differentiable function for all x\(\in\)R. Then f(x) equals :

For x > 0, if \(f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}\), then \(f(e) + f\left( {{1 \over e}} \right)\) is equal to :

If \({I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx}\), for m, \(n \ge 1\), and \(\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over {{{(1 + x)}^{m + 1}}}}} dx = \alpha {I_{m,n}}\alpha \in R\), then \(\alpha\) equals _____...

Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the square of the slope of the line L is __________.

Let \(A = \{ 1,2,3,....,10\}\) and \(f:A \to A\) be defined as\(f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.\)Then the number of possible functions \(g:A \to A\) such that $$gof ...

If 0 < a, b < 1, and tan\(-\)1a + tan\(-\)1b = \({\pi \over 4}\), then the value of \((a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....\) ...

Let f : R \(\to\) R be defined as \(f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.\) If f(...

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then \(\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}\) equals :

Let \(f(x) = {\sin ^{ - 1}}x\) and \(g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}\). If \(g(2) = \mathop {\lim }\limits_{x \to 2} g(x)\), then the domain of the function fog is :

Let F1(A, B, C) = (A \(\wedge\) \(\sim\) B) \(\vee\) [\(\sim\)C \(\wedge\) (A \(\vee\) B)] \(\vee\) \(\sim\) A and F2(A, B) = (A \(\vee\) B) \(\vee\) (B \(\to\) \(\sim\)A) be two logical expressions. Then :

Consider the following system of equations :x + 2y \(-\) 3z = a2x + 6y \(-\) 11z = bx \(-\) 2y + 7z = c,where a, b and c are real constants. Then the system of equations :

If the matrix \(A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]\) satisfies the equation $${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 &...

A natural number has prime factorization given by n = 2x3y5z, where y and z are such that y + z = 5 and y\(-\)1 + z\(-\)1 = \({5 \over 6}\), y > z. Then the number of odd divisions of n, including 1, is :

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is :

Let \(\alpha\) and \(\beta\) be two real numbers such that \(\alpha\) + \(\beta\) = 1 and \(\alpha\)\(\beta\) = \(-\)1. Let pn = (\(\alpha\))n + (\(\beta\))n, pn\(-\)1 = 11 and pn+1 = 29 for some integer n \(\ge\) 1. Then, the value of p$...

If the arithmetic mean and geometric mean of the pth and qth terms of the sequence \(-\)16, 8, \(-\)4, 2, ...... satisfy the equation 4x2 \(-\) 9x + 5 = 0, then p + q is equal to __________.

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.

The sum of the series \(\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}}\) is equal to :

Let X1, X2, ......., X18 be eighteen observations such that \(\sum\limits_{i = 1}^{18} {({X_i} - } \alpha ) = 36\) and \(\sum\limits_{i = 1}^{18} {({X_i} - } \beta {)^2} = 90\), where \(\alpha\) and \(\beta\) are distinct real numbers. If t...

If vectors \(\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k\) and \(\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k\) are collinear, then a possible unit vector parallel to the vector $$x\widehat i + y\wi...