JEE Main 2021 (Online) 27th July Evening Shift

For real numbers \(\alpha\) and \(\beta\) \(\ne\) 0, if the point of intersection of the straight lines\({{x - \alpha } \over 1} = {{y - 1} \over 2} = {{z - 1} \over 3}\) and $${{x - 4} \over \beta } = {{y - 6} \over 3} = {{z - 7} \over 3}$...

The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3, \(-\)4, \(-\)5) and R(2, \(-\)3, 1) and the plane 2x + y + z = 7, is equal to ______________.

The area of the region bounded by y \(-\) x = 2 and x2 = y is equal to :

A possible value of 'x', for which the ninth term in the expansion of \({\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}}\) in the increasing powers of...

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept \(6\sqrt 5\) on the x-axis. Then the radius of the circle C is equal to :

Let C be the set of all complex numbers. LetS1 = {z\(\in\)C : |z \(-\) 2| \(\le\) 1} and S2 = {z\(\in\)C : z(1 + i) + \(\overline z\)(1 \(-\) i) \(\ge\) 4}.Then, the maximum value of \({\left| {z - {5 \over 2}} \right|^2}\) for z\(\in\)S1 ...

If the real part of the complex number \(z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)\) is zero, then the value of sin23\(\theta\) + cos2\(\theta\) is equal to _______________.

Let f : (a, b) \(\to\) R be twice differentiable function such that \(f(x) = \int_a^x {g(t)dt}\) for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

If \(\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} }\), then \(\alpha\) + \(\beta\) is equal to ____________.

Let y = y(x) be the solution of the differential equation (x \(-\) x3)dy = (y + yx2 \(-\) 3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :

Let y = y(x) be the solution of the differential equation dy = e\(\alpha\)x + y dx; \(\alpha\) \(\in\) N. If y(loge2) = loge2 and y(0) = loge\(\left( {{1 \over 2}} \right)\), then the value of \(\alpha\) is equal to _____________.

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, \(-\)4), one focus at (4, \(-\)4) and one vertex at (5, \(-\)4). If mx \(-\) y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m2 is...

Let f : R \(\to\) R be defined as \(f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1\). Then, the value of \(\sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}}\) is equal to :

The value of \(\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right)\) is equal to :

Let \(f:[0,\infty ) \to [0,3]\) be a function defined by \(f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.\)Then which of the following is true?

Which of the following is the negation of the statement "for all M > 0, there exists x\(\in\)S such that x \(\ge\) M" ?

Let A and B be two 3 \(\times\) 3 real matrices such that (A2 \(-\) B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determinant of the matrix A3 + B3 is equal to :

If \(A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]\) and M = A + A2 + A3 + ....... + A20, then the sum of all the elements of the matrix M is equal to _____________.

Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number (10)10 . (11)11 . (13)13 is equal to __________.

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability.
the smallest value of n, so that the probability of guessing at least 'n' correct answers is less than ...

Let \(\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}\) and \(\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}\). If \(8{x^2} + bx + c = 0\) is a quadratic equation whose roots are...

The number of real roots of the equation e4x \(-\) e3x \(-\) 4e2x \(-\) ex + 1 = 0 is equal to ______________.

Let N be the set of natural numbers and a relation R on N be defined by \(R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\}\). Then the relation R is :

Let A = {n \(\in\) N | n2 \(\le\) n + 10,000}, B = {3k + 1 | k\(\in\) N} an dC = {2k | k\(\in\)N}, then the sum of all the elements of the set A \(\cap\)(B \(-\) C) is equal to _____________.

Let the mean and variance of the frequency distribution\(\matrix{ {x:} & {{x_1} = 2} & {{x_2} = 6} & {{x_3} = 8} & {{x_4} = 9} \cr {f:} & 4 & 4 & \alpha & \beta \cr }\)be 6 and 6.8 respectively. If x3 is changed from 8 to 7, ...

The point P (a, b) undergoes the following three transformations successively :(a) reflection about the line y = x.(b) translation through 2 units along the positive direction of x-axis.(c) rotation through angle \({\pi \over 4}\) about th...

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :

If \(\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)\) are in arithmetic progression and \(\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)\) are also in arithmetic progress...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three vectors such that \(\overrightarrow a\) = \(\overrightarrow b\) \(\times\) (\(\overrightarrow b\) \(\times\) \(\overrightarrow c\)). If magnitudes of...

Let \(\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k\),   \(\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k\) and \(\overrightarrow c = -\alpha \widehat i - 2\widehat j + \widehat k\), where ...