JEE Main 2022 (Online) 27th June Evening Shift

Let the foot of the perpendicular from the point (1, 2, 4) on the line \({{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1} \over 3}\) be P. Then the distance of P from the plane \(3x + 4y + 12z + 23 = 0\) is :

The shortest distance between the lines \({{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}\) and \({{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}\), is :

If the area of the region \(\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}\) is A, then \({{256A} \over \pi }\) is equal to __________.

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of \({\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}\) is 939, then the sum of all the possible integral values of n is _________.

The set of values of k, for which the circle \(C:4{x^2} + 4{y^2} - 12x + 8y + k = 0\) lies inside the fourth quadrant and the point \(\left( {1, - {1 \over 3}} \right)\) lies on or inside the circle C, is :

Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 = 3x \(-\) 4y \(-\) 11 = 0 at Q. The line L2 touches C at the point Q. Then the distance o...

The number of points of intersection of \(|z - (4 + 3i)| = 2\) and \(|z| + |z - 4| = 6\), z \(\in\) C, is :

If m and n respectively are the number of local maximum and local minimum points of the function \(f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \over {2 + {e^t}}}dt}\), then the ordered pair (m, n) is equal to

Let f be a differentiable function in \(\left( {0,{\pi \over 2}} \right)\). If \(\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x}\), then \({1 \over {\sqrt 3 }}f'\left( {{1 \over {\sqrt 3 }}} \right)\) is equal to

The integral \(\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx}\), where [ . ] denotes the greatest integer function, is equal to

If the solution curve of the differential equation \((({\tan ^{ - 1}}y) - x)dy = (1 + {y^2})dx\) passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is

Let \(y = y(x)\) be the solution of the differential equation \((1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 < x < 1\), and \(y(0) = 0\). If $$\int_{{{ - 1} \over 2}}^{{1 \over 2}} {\sqrt {1 - {x^2}} y(x)dx = k...

If \(y(x) = {\left( {{x^x}} \right)^x},\,x > 0\), then \({{{d^2}x} \over {d{y^2}}} + 20\) at x = 1 is equal to ____________.

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S \(\to\) S as
\(f(n) = \left\{ {\matrix{ {2n} & , & {if\,n = 1,2,3,4,5} \cr {2n - 11} & , & {if\,n = 6,7,8,9,10} \cr } } \right.\).
Let g : S \(\to\) S be a function such that...

The value of \(\cot \left( {\sum\limits_{n = 1}^{50} {{{\tan }^{ - 1}}\left( {{1 \over {1 + n + {n^2}}}} \right)} } \right)\) is :

Let [t] denote the greatest integer \(\le\) t and {t} denote the fractional part of t. The integral value of \(\alpha\) for which the left hand limit of the function
$$f(x) = [1 + x] + {{{\alpha ^{2[x] + {\{x\}}}} + [x] - 1} \over {2[x] + \...

Which of the following statement is a tautology?

Let \(f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R\). Then the sum of the squares of all the values of a, for which \(2f'(10) - f'(5) + 100 = 0\), is

Let A and B be two 3 \(\times\) 3 matrices such that \(AB = I\) and \(|A| = {1 \over 8}\). Then \(|adj\,(B\,adj(2A))|\) is equal to

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is \(3x + y - 29 = 0\), is \({x^2} + a{y^2} + bxy + cx + dy + k = 0\), then \(a + b + c + d + k\) is equal to :

Let A be a matrix of order 2 \(\times\) 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is ___________.

If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x \(-\) 6y = 30, then the probability that y < 1 is :

Let S = {E1, E2, ........., E8} be a sample space of a random experiment such that \(P({E_n}) = {n \over {36}}\) for every n = 1, 2, ........, 8. Then the number of elements in the set $$\left\{ {A \subseteq S:P(A) \ge {4 \over 5}} \right\}...

Let \(\alpha\), \(\beta\) be the roots of the equation \({x^2} - 4\lambda x + 5 = 0\) and \(\alpha\), \(\gamma\) be the roots of the equation \({x^2} - \left( {3\sqrt 2 + 2\sqrt 3 } \right)x + 7 + 3\lambda \sqrt 3 = 0\), \(\lambda\) > 0. ...

Let \(S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....\). Then 4S is equal to

If a1, a2, a3 ...... and b1, b2, b3 ....... are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4 b4 is equal to -

The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y, where x < y, are 6 and \({9 \over 4}\) respectively. Then \({x^4} + {y^2}\) is equal to :

Let for some real numbers \(\alpha\) and \(\beta\), \(a = \alpha - i\beta\). If the system of equations \(4ix + (1 + i)y = 0\) and \(8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0\) has more than o...

\(\alpha = \sin 36^\circ\) is a root of which of the following equation?

Let \(\overrightarrow a\) and \(\overrightarrow b\) be the vectors along the diagonals of a parallelogram having area \(2\sqrt 2\). Let the angle between \(\overrightarrow a\) and \(\overrightarrow b\) be acute, $$|\overrightarrow a | ...