JEE Main 2021 (Online) 31st August Evening Shift

The distance of the point (\(-\)1, 2, \(-\)2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x \(-\) 2y + z = 0 is :

Suppose, the line \({{x - 2} \over \alpha } = {{y - 2} \over { - 5}} = {{z + 2} \over 2}\) lies on the plane \(x + 3y - 2z + \beta = 0\). Then \((\alpha + \beta )\) is equal to _______.

Let f(x) be a cubic polynomial with f(1) = \(-\)10, f(\(-\)1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = \(-\)1. Then f(3) is equal to ____________.

If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = \({3 \over 2}\) and the curve y = 1 + 4x \(-\) x2, then 12 m is equal to _____________.

If the coefficient of a7b8 in the expansion of (a + 2b + 4ab)10 is K.216, then K is equal to _____________.

Let B be the centre of the circle x2 + y2 \(-\) 2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). Then 8.\(\left( {{{area\,\Delta APQ} \over {area\,\Delta BPQ}}} \right)\) is equal to ____...

If z is a complex number such that \({{z - i} \over {z - 1}}\) is purely imaginary, then the minimum value of | z \(-\) (3 + 3i) | is :

If [x] is the greatest integer \(\le\) x, then \({\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx\) is equal to :

If \({{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}\), y(0) = 0, then for y = 1, the value of x lies in the interval :

If \(y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\phi '\left( {{{{y^2}} \over {{x^2}}}} \right)}}} \right]\), x > 0, \(\phi\) > 0, and y(1) = \(-\)1, then $$\phi \left( {{{...

The locus of mid-points of the line segments joining (\(-\)3, \(-\)5) and the points on the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 9} = 1\) is :

An angle of intersection of the curves, \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) and x2 + y2 = ab, a > b, is :

Let f : N \(\to\) N be a function such that f(m + n) = f(m) + f(n) for every m, n\(\in\)N. If f(6) = 18, then f(2) . f(3) is equal to :

If \(\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }\)
\(\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C\), when C is constant...

The domain of the function\(f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)\) is :

If \(\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}\) and \(\beta = \mathop {\lim }\limits_{x \to 0 } {(\cos x)^{\cot x}}\) are the roots of the equation, ...

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

Negation of the statement (p \(\vee\) r) \(\Rightarrow\) (q \(\vee\) r) is :

If \(\alpha\) + \(\beta\) + \(\gamma\) = 2\(\pi\), then the system of equations x + (cos \(\gamma\))y + (cos \(\beta\))z = 0(cos \(\gamma\))x + y + (cos \(\alpha\))z = 0(cos \(\beta\))x + (cos \(\alpha\))y + z = 0has :

The number of elements in the set \(\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}\), where I is 2 \(\times\) 2 identity matrix, is :

A tangent line L is drawn at the point (2, \(-\)4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equal to ___________.

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :

The sum of the roots of the equation \(x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0\), is :

Let a1, a2, a3, ..... be an A.P. If \({{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}\), p \(\ne\) 10, then \({{{a_{11}}} \over {{a_{10}}}}\) is equal to :

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________.

If \(S = {7 \over 5} + {9 \over {{5^2}}} + {{13} \over {{5^3}}} + {{19} \over {{5^4}}} + ....\), then 160 S is equal to ________.

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

Let A be the set of all points (\(\alpha\), \(\beta\)) such that the area of triangle formed by the points (5, 6), (3, 2) and (\(\alpha\), \(\beta\)) is 12 square units. Then the least possible length of a line segment joining the origin to...

The number of solutions of the equation \({32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4}\) is :

Let \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) three vectors mutually perpendicular to each other and have same magnitude. If a vector \({ \overrightarrow r }\) satisfies.
$$\overrightarrow a \times \{ (\overrightarro...