JEE Main 2021 (Online) 26th August Morning Shift

A plane P contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(- 2x + y + z + 8 = 0\). Then which of the following points lies on P?

Let the line L be the projection of the line \({{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 4} \over 2}\) in the plane x \(-\) 2y \(-\) z = 3. If d is the distance of the point (0, 0, 6) from L, then d2 is equal to _______________.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then $...

The area of the region \(S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\}\) is ____________.

If \({{}^{20}{C_r}}\) is the co-efficient of xr in the expansion of (1 + x)20, then the value of \(\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}}\) is equal to :

If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (\(-\)30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is :

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to _____________.

The equation \(\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}\) represents a circle with :

Let \(z = {{1 - i\sqrt 3 } \over 2}\), \(i = \sqrt { - 1}\). Then the value of $$21 + {\left( {z + {1 \over z}} \right)^3} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^3} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^3} + .... + {\left( {...

The value of \(\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx}\) is :

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}}\) is :

Let y = y(x) be a solution curve of the differential equation \((y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0\), \(x \in \left( {0,{\pi \over 2}} \right)\). If \(\mathop {\lim }\limits_{x \to 0 + } xy(x) = 1\), then the value of $$y\left(...

Let \(f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)\), 0 < x < 1. Then :

If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then \({{{d^2}y} \over {d{x^2}}}\) at x = 0 is equal to ___________.

On the ellipse \({{{x^2}} \over 8} + {{{y^2}} \over 4} = 1\) let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its ecc...

Let a, b \(\in\) R, b \(\in\) 0, Define a function \(f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {for\,x \le 0} \cr {{{\tan 2x - \sin 2x} \over {b{x^3}}},} & {for\,x > 0} \cr } } \right.\). If f is continuous at x ...

If the truth value of the Boolean expression \(\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)\) is false, then the truth values of the statements p,...

Let \(\theta \in \left( {0,{\pi \over 2}} \right)\). If the system of linear equations\((1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0\)\({\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0\)$${\cos ^2...

If \(A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)\), \(B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)\), $$i = \s...

If \({}^1{P_1} + 2.{}^2{P_2} + 3.{}^3{P_3} + .... + 15.{}^{15}{P_{15}} = {}^q{P_r} - s,0 \le s \le 1\), then \({}^{q + s}{C_{r - s}}\) is equal to ______________.

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______________.

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = \({5 \over 9}\), is :

The sum of all integral values of k (k \(\ne\) 0) for which the equation \({2 \over {x - 1}} - {1 \over {x - 2}} = {2 \over k}\) in x has no real roots, is ____________.

The sum of the series \({1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}\) when x = 2 is :

If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. if \(\alpha\) and \(\sqrt \beta\) are the mean and standard deviatio...

Let ABC be a triangle with A(\(-\)3, 1) and \(\angle\)ACB = \(\theta\), 0 < \(\theta\) < \({\pi \over 2}\). If the equation of the median through B is 2x + y \(-\) 3 = 0 and the equation of angle bisector of C is 7x \(-\) 4y \(-\) 1 = 0, t...

The sum of solutions of the equation \({{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right) - \left\{ {{\pi \over 4}, - {\pi \over 4}} \right\}\) is :

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k\) and \(\overrightarrow b = \widehat j - \widehat k\). If \(\overrightarrow c\) is a vector such that \(\overrightarrow a \times \overrightarrow c = \overrightarrow b\) an...