JEE Main 2022 (Online) 26th June Morning Shift

If the two lines \({l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2\) and \({l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}\) are perpendicular, then an angle between the lines l2 and $${l_3}:{{1 - x} \over 3} =...

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x \(-\) 3y + 5z = 8. If the mirror image of the point \(\left( {2, - {1 \over 2},2} \right)\) in the rotated plane is B(a, b, ...

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x \(-\) x2 + 2| \(-\) x in the interval [\(-\)1, 2] is :

Let S be the set of all the natural numbers, for which the line \({x \over a} + {y \over b} = 2\) is a tangent to the curve \({\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2\) at the point (a, b), ab \(\ne\) 0. Then ...

Let \(f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10\), \(x \in [ - 1,1]\). If [a, b] is the range of the function f, then 4a \(-\) b is equal to :

The area bounded by the curve y = |x2 \(-\) 9| and the line y = 3 is :

The remainder when (2021)2023 is divided by 7 is :

Let C be a circle passing through the points A(2, \(-\)1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle \({(x - 5)^2} + {(y - 1)^2} = {{13} \over 2}\), then r2 is equal to ...

Let \(A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}\) and \(B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}\). Then A \(\cap\) B is :

Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then \(\int\limits_{ - 6}^0 {f(x)dx}\) is equal to __________.

The value of the integral \({{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{\sin x} \over {1 + {{\cos }^2}x}}dx}\) is equal to __________.

Let the solution curve y = y(x) of the differential equation \((4 + {x^2})dy - 2x({x^2} + 3y + 4)dx = 0\) pass through the origin. Then y(2) is equal to _____________.

Let \(S = (0,2\pi ) - \left\{ {{\pi \over 2},{{3\pi } \over 4},{{3\pi } \over 2},{{7\pi } \over 4}} \right\}\). Let \(y = y(x)\), x \(\in\) S, be the solution curve of the differential equation $${{dy} \over {dx}} = {1 \over {1 + \sin 2x}}...

Let \(f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\}\). If \({f^{n + 1}}(x) = f({f^n}(x))\) for all n \(\in\) N, then \({f^6}(6) + {f^7}(7)\) is equal to :

\(\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}\) is equal to :

Let f, g : R \(\to\) R be two real valued functions defined as \(f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x < 0} \cr {{e^x}} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{x^2} + {k_1}x} & , & {x < 0} ...

Let \(\Delta\), \(\nabla\) \(\in\) {\(\wedge\), \(\vee\)} be such that p \(\nabla\) q \(\Rightarrow\) ((p \(\Delta\) q) \(\nabla\) r) is a tautology. Then (p \(\nabla\) q) \(\Delta\) r is logically equivalent to :

Let A be a 3 \(\times\) 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :

The ordered pair (a, b), for which the system of linear equations
3x \(-\) 2y + z = b
5x \(-\) 8y + 9z = 3
2x + y + az = \(-\)1
has no solution, is :

Let the normal at the point on the parabola y2 = 6x pass through the point (5, \(-\)8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

Let the common tangents to the curves \(4({x^2} + {y^2}) = 9\) and \({y^2} = 4x\) intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l ...

There are ten boys B1, B2, ......., B10 and five girls G1, G2, ........, G5 in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B1 and B2 together should not be the members of a group, is...

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is :

The sum of the cubes of all the roots of the equation \({x^4} - 3{x^3} - 2{x^2} + 3x + 1 = 0\) is _________.

Let A = {n \(\in\) N : H.C.F. (n, 45) = 1} and
Let B = {2k : k \(\in\) {1, 2, ......., 100}}. Then the sum of all the elements of A \(\cap\) B is ____________.

Let \(A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } }\) and \(B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } }\). Then A + B is equal to _____________.

The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of \(\Delta\)PQR is :

If \({\sin ^2}(10^\circ )\sin (20^\circ )\sin (40^\circ )\sin (50^\circ )\sin (70^\circ ) = \alpha - {1 \over {16}}\sin (10^\circ )\), then \(16 + {\alpha ^{ - 1}}\) is equal to __________.

If \(\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2\) and \(\overrightarrow c \,.\,\overrightarrow a = 3\), then the value of $$\left[ {\overrightarrow a \times \left( {\overrightarrow b \ti...