JEE Main 2021 (Online) 25th February Evening Shift

A line 'l' passing through origin is perpendicular to the lines\({l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + 2t)\widehat j + (4 + 2t)\widehat k\)$${l_2}:\overrightarrow r = (3 + 2s)\widehat i + (3 + 2s)\widehat j + (2 + s)\wideh...

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, \(-\)1, 1), then the projection of \(\overrightarrow {OP}\) on this plane is of length :

If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________.

If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is __________.

The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is __________.

If \(\alpha\), \(\beta\) \(\in\) R are such that 1 \(-\) 2i (here i2 = \(-\)1) is a root of z2 + \(\alpha\)z + \(\beta\) = 0, then (\(\alpha\) \(-\) \(\beta\)) is equal to :

The value of \(\int\limits_{ - 2}^2 {|3{x^2} - 3x - 6|dx}\) is ___________.

\(\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ........ + {n \over {{{(2n + 1)}^2}}}} \right]\) is equal to :

If \({I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx}\), then :

If the curve, y = y(x) represented by the solution of the differential equation (2xy2 \(-\) y)dx + xdy = 0, passes through the intersection of the lines, 2x \(-\) 3y = 1 and 3x + 2y = 8, then |y(1)| is equal to _________.

If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A \(\times\) B. Then :

A function f(x) is given by \(f(x) = {{{5^x}} \over {{5^x} + 5}}\), then the sum of the series $$f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} ...

A hyperbola passes through the foci of the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1\) and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentrici...

The integral \(\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx\), x > 0, is equal to : (where c is a constant of integration)

cosec\(\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]\) is equal to :

If \(\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}\) exists and is equal to b, then the value of a \(-\) 2b is __________.

A function f is defined on [\(-\)3, 3] as\(f(x) = \left\{ {\matrix{ {\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le 2} \cr {[|x|],} & {2 < |x| \le 3} \cr } } \right.\) where [x] denotes the greatest integer \(\le\) x. The number...

The contrapositive of the statement "If you will work, you will earn money" is :

The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x \(-\) y + 4z = 8

If for the matrix, \(A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]\), \(A{A^T} = {I_2}\), then the value of \({\alpha ^4} + {\beta ^4}\) is :

Let A be a 3 \(\times\) 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 \(\to\) 2R2 + 5R3 on 2A, then det(B) is equal to :

The shortest distance between the line x \(-\) y = 1 and the curve x2 = 2y is :

A line is a common tangent to the circle (x \(-\) 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to _________.

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A per...

Let \(\alpha\) and \(\beta\) be the roots of x2 \(-\) 6x \(-\) 2 = 0. If an = \(\alpha\)n \(-\) \(\beta\)n for n \(\ge\) 1, then the value of \({{{a_{10}} - 2{a_8}} \over {3{a_9}}}\) is :

The minimum value of \(f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}\), where a, \(x \in R\) and a > 0, is equal to :

If 0 < x, y < \(\pi\) and cosx + cosy \(-\) cos(x + y) = \({3 \over 2}\), then sinx + cosy is equal to :

Let \(\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k\) and \(\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k\). If the area of the parallelogram whose adjacent sides are represented by the vectors $$\ove...