JEE Main 2021 (Online) 20th July Morning Shift

Let P be a plane passing through the points (1, 0, 1), (1, \(-\)2, 1) and (0, 1, \(-\)2). Let a vector \(\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k\) be such that \(\overrightarrow a\) is parallel to the ...

Let \(A = [{a_{ij}}]\) be a 3 \(\times\) 3 matrix, where \({a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.\)Let a function f...

Let 'a' be a real number such that the function f(x) = ax2 + 6x \(-\) 15, x \(\in\) R is increasing in \(\left( { - \infty ,{3 \over 4}} \right)\) and decreasing in \(\left( {{3 \over 4},\infty } \right)\). Then the function g(x) = ax2 $$-$...

Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 and x = \(\sqrt 5\) is \(\alpha\)\(\sqrt 5\) + \(\beta\) + \(\gamma\) cos\(-\)1$$\left( ...

The coefficient of x256 in the expansion of (1 \(-\) x)101 (x2 + x + 1)100 is :

The number of rational terms in the binomial expansion of \({\left( {{4^{{1 \over 4}}} + {5^{{1 \over 6}}}} \right)^{120}}\) is _______________.

If z and \(\omega\) are two complex numbers such that \(\left| {z\omega } \right| = 1\) and \(\arg (z) - \arg (\omega ) = {{3\pi } \over 2}\), then \(\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)\) is :...

Let a be a positive real number such that \(\int_0^a {{e^{x - [x]}}} dx = 10e - 9\) where [ x ] is the greatest integer less than or equal to x. Then a is equal to:

The value of the integral \(\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx}\) is equal to:

Let y = y(x) be the solution of the differential equation \(x\tan \left( {{y \over x}} \right)dy = \left( {y\tan \left( {{y \over x}} \right) - x} \right)dx\), \(- 1 \le x \le 1\), \(y\left( {{1 \over 2}} \right) = {\pi \over 6}\). Then t...

Let y = y(x) be the solution of the differential equation \({e^x}\sqrt {1 - {y^2}} dx + \left( {{y \over x}} \right)dy = 0\), y(1) = \(-\)1. Then the value of (y(3))2 is equal to :

Let [ x ] denote the greatest integer \(\le\) x, where x \(\in\) R. If the domain of the real valued function \(f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}}\) is (\(-\) \(\infty\), a) \(]\cup\) [b, c) $$\cup...

The number of real roots of the equation \({\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4}\) is :

Let a function f : R \(\to\) R be defined as \(f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right.\) where [ x ] is the greatest...

If the value of \(\mathop {\lim }\limits_{x \to 0} {(2 - \cos x\sqrt {\cos 2x} )^{\left( {{{x + 2} \over {{x^2}}}} \right)}}\) is equal to ea, then a is equal to __________.

The Boolean expression \((p \wedge \sim q) \Rightarrow (q \vee \sim p)\) is equivalent to :

Let \(A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]\), a\(\in\)R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of deter...

Let \(A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)\) and B = 7A20 \(-\) 20A7 + 2I, where I is an identity matrix of order 3 \(\times\) 3. If B = [bij], then b13is equal to ___________...

Let a, b, c, d in arithmetic progression with common difference \(\lambda\). If \(\left| {\matrix{ {x + a - c} & {x + b} & {x + a} \cr {x - 1} & {x + c} & {x + b} \cr {x - b + d} & {x + d} & {x + c} \cr } } \right| = 2\), t...

Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

Let y = mx + c, m > 0 be the focal chord of y2 = \(-\) 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4\(\sqrt 2\) (m + c) is equal to _____________.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ...

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :

The probability of selecting integers a\(\in\)[\(-\) 5, 30] such that x2 + 2(a + 4)x \(-\) 5a + 64 > 0, for all x\(\in\)R, is :

If in a triangle ABC, AB = 5 units, \(\angle B = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)\) and radius of circumcircle of \(\Delta\)ABC is 5 units, then the area (in sq. units) of \(\Delta\)ABC is :

If \(\alpha\) and \(\beta\) are the distinct roots of the equation \({x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0\), then the value of \({\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1)\) is equal to :

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are :

Let \(\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k\) and \(\overrightarrow b = \widehat i + \widehat j\). If \(\overrightarrow c\) is a vector such that $$\overrightarrow a .\,\overrightarrow c = \left| {\overrightarrow c ...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\overrightarrow a\) + $$\overrightarrow...

If the shortest distance between the lines \(\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\widehat j + 2\widehat k)\), \(\lambda\) \(\in\) R, \(\alpha\) > 0 and $$\overrightarrow {{r_2}} ...