JEE Main 2022 (Online) 28th June Evening Shift

Let the plane ax + by + cz = d pass through (2, 3, \(-\)5) and is perpendicular to the planes 2x + y \(-\) 5z = 10 and 3x + 5y \(-\) 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20...

Let the image of the point P(1, 2, 3) in the line \(L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}\) be Q. Let R (\(\alpha\), \(\beta\), \(\gamma\)) be a point that divides internally the line segment PQ in the ratio 1 : 3. The...

The area of the bounded region enclosed by the curve \(y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|\) and the x-axis is :

The term independent of x in the expansion of \((1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0\) is :

If one of the diameters of the circle \({x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0\) is a chord of the circle \({(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}\), then the value of r2 is equal to ____________.

Sum of squares of modulus of all the complex numbers z satisfying \(\overline z = i{z^2} + {z^2} - z\) is equal to ___________.

Let f : R \(\to\) R be a differentiable function such that \(f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0\) and \(f'\left( {{\pi \over 2}} \right) = 1\) and let $$g(x) = \int_x^{\pi /4} {(f'(t)\sec t +...

Let f : R \(\to\) R be a continuous function satisfying f(x) + f(x + k) = n, for all x \(\in\) R where k > 0 and n is a positive integer. If \({I_1} = \int\limits_0^{4nk} {f(x)dx}\) and \({I_2} = \int\limits_{ - k}^{3k} {f(x)dx}\), then ...

Let x = x(y) be the solution of the differential equation \(2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0\) such that x(1) = 0. Then, x(e) is equal to :

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 \(\tan x(\cos x - y)\). If the curve passes through the point \(\left( {{\pi \over 4},0} \right)\), then the value of \(\int\limits_0^{\pi /2} {y\,dx}\) is equal to ...

Let S = {1, 2, 3, 4}. Then the number of elements in the set { f : S \(\times\) S \(\to\) S : f is onto and f (a, b) = f (b, a) \(\ge\) a \(\forall\) (a, b) \(\in\) S \(\times\) S } is ______________.

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). Let e' and l' respectively be the eccentricity and length of the latus...

Let f, g : R \(\to\) R be functions defined by
\(f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{e^x} - x} & , & {x < 0} \cr
{{{(x - 1)}^2} - ...

The value of \(\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}\) is equal to :

If \(\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2\), then the value of (a \(-\) b) is equal to ___________.

The maximum number of compound propositions, out of p\(\vee\)r\(\vee\)s, p\(\vee\)r\(\vee\)\(\sim\)s, p\(\vee\)\(\sim\)q\(\vee\)s, \(\sim\)p\(\vee\)\(\sim\)r\(\vee\)s, \(\sim\)p\(\vee\)\(\sim\)r\(\vee\)\(\sim\)s, \(\sim\)p\(\vee\)q\(\vee\)$...

If the system of linear equations \(2x - 3y = \gamma + 5\), \(\alpha x + 5y = \beta + 1\), where \(\alpha\), \(\beta\), \(\gamma\) \(\in\) R has infinitely many solutions then the value of | 9\(\alpha\) + 3\(\beta\) + 5\(\gamma\) | is equ...

Let \(A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)\) where \(i = \sqrt { - 1}\). Then, the number of elements in the set { n \(\in\) {1, 2, ......, 100} : An = A } is ____________.

If vertex of a parabola is (2, \(-\)1) and the equation of its directrix is 4x \(-\) 3y = 21, then the length of its latus rectum is :

The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives at least 4 and at most 7 candies, C3 receives at least 2 and at most 6 candies, is equal to :

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) \(-\) f(c) = f(d) is :

Let f(x) be a quadratic polynomial such that f(\(-\)2) + f(3) = 0. If one of the roots of f(x) = 0 is \(-\)1, then the sum of the roots of f(x) = 0 is equal to :

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

Let for n = 1, 2, ......, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is \({1 \over {{{(n + 1)}^2}}}\). Then the value of $${1 \over {26}} + \sum\limits_{n = 1}^{50} {\left( {{S_n} +...

Let R1 = {(a, b) \(\in\) N \(\times\) N : |a \(-\) b| \(\le\) 13} and
R2 = {(a, b) \(\in\) N \(\times\) N : |a \(-\) b| \(\ne\) 13}. Then on N :

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of stud...

Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : \(-\)4x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 ...

If cot\(\alpha\) = 1 and sec\(\beta\) = \(- {5 \over 3}\), where \(\pi < \alpha < {{3\pi } \over 2}\) and \({\pi \over 2} < \beta < \pi\), then the value of \(\tan (\alpha + \beta )\) and the quadrant in which \(\alpha\) + \(\beta\) ...

Let \(\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k\) and \(\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are repre...

Let \(\overrightarrow a\) be a vector which is perpendicular to the vector \(3\widehat i + {1 \over 2}\widehat j + 2\widehat k\). If $$\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\wid...