JEE Main 2022 (Online) 24th June Evening Shift

If the shortest distance between the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }\) and \({{x - 2} \over 1} = {{y - 4} \over 4} = {{z - 5} \over 5}\) is \({1 \over {\sqrt 3 }}\), then the sum of all possible valu...

Let the points on the plane P be equidistant from the points (\(-\)4, 2, 1) and (2, \(-\)2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is :

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by \({{{x^2}} \over {xy - {x^2}{y^2} - 1}}\). If the curve passes through the point (1, 1), then e . y(e) is equal to

Let \(\lambda\)\(^ *\) be the largest value of \(\lambda\) for which the function \({f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48\) is increasing for all x \(\in\) R. Then $${f_{{\lambda ^ * }}}(1) + {f_{{\lambda ^ * }}}( -...

The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________.

The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is _________.

Let a circle C : (x \(-\) h)2 + (y \(-\) k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ___________.

Let S = {z \(\in\) C : |z \(-\) 3| \(\le\) 1 and z(4 + 3i) + \(\overline z\)(4 \(-\) 3i) \(\le\) 24}. If \(\alpha\) + i\(\beta\) is the point in S which is closest to 4i, then 25(\(\alpha\) + \(\beta\)) is equal to ___________.

The value of the integral \(\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}}\) is equal to

\(\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\, + \,\,{{{n^2}} \over {({n^2} + {n^2})(n + n)}}} \right)\) is ...

If \(y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} < {x^3} < {{3\pi } \over 2}\), then

Let the maximum area of the triangle that can be inscribed in the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2\), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to t...

Let the hyperbola \(H:{{{x^2}} \over {{a^2}}} - {y^2} = 1\) and the ellipse \(E:3{x^2} + 4{y^2} = 12\) be such that the length of latus rectum of H is equal to the length of latus rectum of E. If \({e_H}\) and \({e_E}\) are the eccentriciti...

Let \(x * y = {x^2} + {y^3}\) and \((x * 1) * 1 = x * (1 * 1)\).
Then a value of \(2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)\) is :

Let \(f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.\)
where [t] denotes greatest integer \(\le\) t. I...

Consider the following statements:
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

Let the system of linear equations
x + y + \(\alpha\)z = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x\(^ *\), y\(^ *\), z\(^ *\)). If (\(\alpha\), x\(^ *\)), (y\(^ *\), \(\alpha\)) and (x\(^ *\), \(-\)y\(^ *\)) are collinear...

Let \(S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}\) and let \({T_n} = \{ A \in S:{A^{n(n + 1)}} = I\}\). Then the number of elements in $$\bigcap\limits_{n = 1}^{100}...

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____________.

A random variable X has the following probability distribution :

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In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability \({3 \over 4}\) and the remaining 6 questions correctly with probability \({1 \over 4}\). If the ...

The sum of all the real roots of the equation \(({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0\) is

The number of distinct real roots of the equation x7 \(-\) 7x \(-\) 2 = 0 is

Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is

The sum of all the elements of the set \(\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\}\) is __________.

Let the area of the triangle with vertices A(1, \(\alpha\)), B(\(\alpha\), 0) and C(0, \(\alpha\)) be 4 sq. units. If the points (\(\alpha\), \(-\)\(\alpha\)), (\(-\)\(\alpha\), \(\alpha\)) and (\(\alpha\)2, \(\beta\)) are collinear, then $...

The number of solutions of the equation \(\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x\), \(x \in [ - 3\pi ,3\pi ]\) is :

Let \(\widehat a\) and \(\widehat b\) be two unit vectors such that \(|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2\). If \(\theta\) \(\in\) (0, \(\pi\)) is the angle between \(\widehat a\) and \(\widehat b\), then among...