JEE Main 2021 (Online) 27th July Morning Shift

Let the plane passing through the point (\(-\)1, 0, \(-\)2) and perpendicular to each of the planes 2x + y \(-\) z = 2 and x \(-\) y \(-\) z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to :

Let a plane P pass through the point (3, 7, \(-\)7) and contain the line, \({{x - 2} \over { - 3}} = {{y - 3} \over 2} = {{z + 2} \over 1}\). If distance of the plane P from the origin is d, then d2 is equal to ______________.

If the area of the bounded region \(R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}\) is , \(\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma\), then the value of $${(\alpha + \beta...

If the coefficients of x7 in \({\left( {{x^2} + {1 \over {bx}}} \right)^{11}}\) and x\(-\)7 in \({\left( {{x} - {1 \over {bx^2}}} \right)^{11}}\), b \(\ne\) 0, are equal, then the value of b is equal to :

Two tangents are drawn from the point P(\(-\)1, 1) to the circle x2 + y2 \(-\) 2x \(-\) 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal...

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :

Let \(A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\}\), \(B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\}\) and \(C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\}\).Then the minimum value of |r...

Let C be the set of all complex numbers. Let\({S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\}\)\({S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\}\) and \({S_3} = \{ z \in C||z - \overline z | \ge 8\}\).Then the number of elements in $...

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}}\) is equal to :

The value of the definite integral\(\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}\) is equal to :

Let the domain of the function\(f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)\) be (a, b). Then the value of the integral $$\int\limits_a^b {{{{{\sin }^3}x} \over {({{\sin }^3}x + {{\sin }^3}(a +...

Let \(F:[3,5] \to R\) be a twice differentiable function on (3, 5) such that \(F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt}\). If \(F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}\), then \(\alpha\) + $$\b...

Let y = y(x) be solution of the differential equation \({\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y\), with y(0) = 0.If \(y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2\), then the value of \(\alpha\) is equal t...

If \(y = y(x),y \in \left[ {0,{\pi \over 2}} \right)\) is the solution of the differential equation \(\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0\), with y(0) = 0, then \(5y'\left( {{\pi \over 2}} \right)\) is equal to _____...

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity \({1 \over 3}\) and the distance of the nearer focus from...

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S \(\to\) S such that f(m . n) = f(m) . f(n) for every m, n \(\in\) S and m . n \(\in\) S is equal to _____________.

Let \(f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R\) be defined as $$f(x) = \left\{ {\matrix{
{{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr
b & , & {x = 0} \cr
{{e^{\cot 4x/\c...

Let f : R \(\to\) R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of \(\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}\) is equal to :

Let \(f:[0,3] \to R\) be defined by \(f(x) = \min \{ x - [x],1 + [x] - x\}\) where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x \(\in\) [0, 3] where f i discontinuous, and Q denote the set cont...

The compound statement \((P \vee Q) \wedge ( \sim P) \Rightarrow Q\) is equivalent to :

Let \(A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]\). If A\(-\)1 = \(\alpha\)I + \(\beta\)A, \(\alpha\), \(\beta\) \(\in\) R, I is a 2 \(\times\) 2 identity matrix then 4(\(\alpha\) \(-\) \(\beta\)) is equal to :

For real numbers \(\alpha\) and \(\beta\), consider the following system of linear equations :x + y \(-\) z = 2, x + 2y + \(\alpha\)z = 1, 2x \(-\) y + z = \(\beta\). If the system has infinite solutions, then \(\alpha\) + \(\beta\) is equa...

Let \(f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|,x \in [0,\pi ]\). Then...

The probability that a randomly selected 2-digit number belongs to the set {n \(\in\) N : (2n \(-\) 2) is a multiple of 3} is equal to :

Let \(\alpha\), \(\beta\) be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then \(\alpha\)8 + \(\beta\)8 is equal to

If \({\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)\) are in an arithmetic progression, then the value of x is equal to _____________.

If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and \({{37} \over 4}\) respectively, then (a \(-\) b)2 is equal to :

If \(\sin \theta + \cos \theta = {1 \over 2}\), then 16(sin(2\(\theta\)) + cos(4\(\theta\)) + sin(6\(\theta\))) is equal to :

Let \(\overrightarrow a = \widehat i + \widehat j + 2\widehat k\) and \(\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k\). Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\le...

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b\) and \(\overrightarrow c = \widehat j - \widehat k\) be three vectors such that \(\overrightarrow a \times \overrightarrow b = \overrightarrow c\) and $...