JEE Main 2021 (Online) 24th February Evening Shift

Let \(\lambda\) be an integer. If the shortest distance between the lines x \(-\) \(\lambda\) = 2y \(-\) 1 = \(-\)2z and x = y + 2\(\lambda\) = z \(-\) \(\lambda\) is \({{\sqrt 7 } \over {2\sqrt 2 }}\), then the value of | \(\lambda\) | is ...

Let a, b\(\in\)R. If the mirror image of the point P(a, 6, 9) with respect to the line \({{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}\) is (20, b, \(-\)a\(-\)9), then | a + b |, is equal to :

The vector equation of the plane passing through the intersection of the planes \(\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1\) and \(\overrightarrow r .\left( {\widehat i - 2\widehat j} \right) = - 2\), an...

If the curve y = ax2 + bx + c, x\(\in\)R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :

For which of the following curves, the line \(x + \sqrt 3 y = 2\sqrt 3\) is the tangent at the point \(\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)\)?

Let \(f:R \to R\) be defined as\(f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr } } \right.\)Let A = {x $$ \i...

The area of the region : \(R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\}\) is :

If \(n \ge 2\) is a positive integer, then the sum of the series \({}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ... + {}^n{C_2}} \right)\) is :

For integers n and r, let \(\left( {\matrix{ n \cr r \cr } } \right) = \left\{ {\matrix{ {{}^n{C_r},} & {if\,n \ge r \ge 0} \cr {0,} & {otherwise} \cr } } \right.\) The maximum value of k for which the sum $$\sum\lim...

Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (\(-\)5, 0). If the locus of the point P is a circle of radius r, then 4r2 is equal to ________

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x \(-\) 2)2 + (y \(-\) 3)2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

Let \(i = \sqrt { - 1}\). If \({{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 + i)}^{24}}}} = k\), and \(n = [|k|]\) be the greatest integral part of | k |. ...

The value of the integral, \(\int\limits_1^3 {[{x^2} - 2x - 2]dx}\), where [x] denotes the greatest integer less than or equal to x, is :

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 \(-\) x) for all x\(\in\) (0, 2), f(0) = 1 and f(2) = e2. Then the value of \(\int\limits_0^2 {f(x)} dx\) is :

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) \(\ne\) 0 for all x \(\in\) R. If \(\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right|\) = 0, for all x$$ \i...

If a curve y = f(x) passes through the point (1, 2) and satisfies \(x {{dy} \over {dx}} + y = b{x^4}\), then for what value of b, \(\int\limits_1^2 {f(x)dx = {{62} \over 5}}\)?

If a + \(\alpha\) = 1, b + \(\beta\) = 2 and \(af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0\), then the value of the expression \({{f(x) + f\left( {{1 \over x}} \right)} \over {x + {1 \over x}}}\) is ________...

The angle of elevation of a jet plane from a point A on the ground is 60\(^\circ\). After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30\(^\circ\). If the jet plane is flying at a constant height, t...

A possible value of \(\tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right)\) is :

For the statements p and q, consider the following compound statements :(a) \(( \sim q \wedge (p \to q)) \to \sim p\)(b) \(((p \vee q) \wedge \sim p) \to q\)Then which of the following statements is correct?

The negation of the statement \(\sim p \wedge (p \vee q)\) is :

Let A and B be 3 \(\times\) 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 \(-\) B2A2) X = O, where X is a 3 \(\times\) 1 column matrix of unknown variables and O is...

For the system of linear equations:\(x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R\),consider the following statements :(A) The system has unique solution if \(k \ne 2,k \ne - 2\).(B) The system has unique solution if k = \(-\)2(C) The ...

If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x \(-\) 1, then the co-ordinates of P are :

The students S1, S2, ....., S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is ___________.

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be \(\left( {{{10} \over 3},{7 \over 3}} \right)\). If \(\alpha\), \(\beta\) are the roots of the equation $$a{x^2} + bx + 1 ...

The number of the real roots of the equation \({(x + 1)^2} + |x - 5| = {{27} \over 4}\) is ________.

The sum of first four terms of a geometric progression (G. P.) is \({{65} \over {12}}\) and the sum of their respective reciprocals is \({{65} \over {18}}\). If the product of first three terms of the G.P. is 1, and the third term is $$\alp...

If the variance of 10 natural numbers 1, 1, 1, ....., 1, k is less than 10, then the maximum possible value of k is ________.