JEE Main 2021 (Online) 20th July Evening Shift

The lines x = ay \(-\) 1 = z \(-\) 2 and x = 3y \(-\) 2 = bz \(-\) 2, (ab \(\ne\) 0) are coplanar, if :

Consider the line L given by the equation \({{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}\). Let Q be the mirror image of the point (2, 3, \(-\)1) with respect to L. Let a plane P be such that it passes through Q, and the line L...

The sum of all the local minimum values of the twice differentiable function f : R \(\to\) R defined by \(f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)\) is :

For the natural numbers m, n, if \({(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}\) and \({a_1} = {a_2} = 10\), then the value of (m + n) is equal to :

Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (\(-\)4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y \(-\) 4 = 0. If $${{{r_1}} \over {{r_2}}} =...

If [x] denotes the greatest integer less than or equal to x, then the value of the integral \(\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx}\) is equal to :

If the real part of the complex number \({(1 - \cos \theta + 2i\sin \theta )^{ - 1}}\) is \({1 \over 5}\) for \(\theta \in (0,\pi )\), then the value of the integral \(\int_0^\theta {\sin x} dx\) is equal to:

Let \(g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx\), where \(f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R\). Then which one of the following is correct?

Let y = y(x) satisfies the equation \({{dy} \over {dx}} - |A| = 0\), for all x > 0, where \(A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr } } \right]\). If \(y(\pi ) = \pi + 2\), th...

Let a curve y = y(x) be given by the solution of the differential equation \(\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {{e^{2x}} - 1} dy\). If it intersects y-axis at y = \(-\)1, and the intersection point of t...

Let \(f:R - \left\{ {{\alpha \over 6}} \right\} \to R\) be defined by \(f(x) = {{5x + 3} \over {6x - \alpha }}\). Then the value of \(\alpha\) for which (fof)(x) = x, for all \(x \in R - \left\{ {{\alpha \over 6}} \right\}\), is :

Let in a right angled triangle, the smallest angle be \(\theta\). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin\(\theta\) is equal to :

The value of \(\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right)\) is equal to :

If \(f:R \to R\) is given by \(f(x) = x + 1\), then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \...

Let a function g : [ 0, 4 ] \(\to\) R be defined as \(g(x) = \left\{ {\matrix{ {\mathop {\max }\limits_{0 \le t \le x} \{ {t^3} - 6{t^2} + 9t - 3),} & {0 \le x \le 3} \cr {4 - x,} & {3 < x \le 4} \cr } } \right.\), then the numb...

If \(\mathop {\lim }\limits_{x \to 0} {{\alpha x{e^x} - \beta {{\log }_e}(1 + x) + \gamma {x^2}{e^{ - x}}} \over {x{{\sin }^2}x}} = 10,\alpha ,\beta ,\gamma \in R\), then the value of \(\alpha\) + \(\beta\) + \(\gamma\) is _____________.

The number of solutions of the equation \({\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0\), x > 0, is :

Consider the following three statements :(A) If 3 + 3 = 7 then 4 + 3 = 8(B) If 5 + 3 = 8 then earth is flat.(C) If both (A) and (B) are true then 5 + 6 = 17.Then, which of the following statements is correct?

The value of k \(\in\)R, for which the following system of linear equations3x \(-\) y + 4z = 3,x + 2y \(-\) 3z = \(-\)26x + 5y + kz = \(-\)3,has infinitely many solutions, is :

Let \(A = \{ {a_{ij}}\}\) be a 3 \(\times\) 3 matrix, where \({a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.\) then $$\...

Let P be a variable point on the parabola \(y = 4{x^2} + 1\). Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :

If the point on the curve y2 = 6x, nearest to the point \(\left( {3,{3 \over 2}} \right)\) is (\(\alpha\), \(\beta\)), then 2(\(\alpha\) + \(\beta\)) is equal to _____________.

Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 \(-\) k), the probability that exactly one of B and C occurs is (1 \(-\) 2k), the probability that exactly one of C and A occurs is (1 \(-\) k...

If sum of the first 21 terms of the series \({\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......\), where x > 0 is 504, then x is equal to

For k \(\in\) N, let \({1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {{{{A_k}} \over {\alpha + k}}}\), where \(\alpha > 0\). Then the value of $$100{\left( {{{{A_{14}} + {A_{15}}} \over {...

Let \(\left\{ {{a_n}} \right\}_{n = 1}^\infty\) be a sequence such that a1 = 1, a2 = 1 and \({a_{n + 2}} = 2{a_{n + 1}} + {a_n}\) for all n \(\ge\) 1. Then the value of \(47\sum\limits_{n = 1}^\infty {{{{a_n}} \over {{2^{3n}}}}}\) is equ...

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and \({{20} \over 3}\), respectively, then the value of | a \(-\) b | is equal to :

Consider a triangle having vertices A(\(-\)2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point \(\left( {0,{\alpha \over 2}} \right)\), then the value o...

In a triangle ABC, if \(\left| {\overrightarrow {BC} } \right| = 3\), \(\left| {\overrightarrow {CA} } \right| = 5\) and \(\left| {\overrightarrow {BA} } \right| = 7\), then the projection of the vector \(\overrightarrow {BA}\) on $$\overr...

For p > 0, a vector \({\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j\) is obtained by rotating the vector \({\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j\) by an angle \(\theta\) about origin in counter clockwise direc...