JEE Main 2022 (Online) 29th June Evening Shift

Let \({{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}\) lie on the plane \(px - qy + z = 5\), for some p, q \(\in\) R. The shortest distance of the plane from the origin is :

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line $$\overrightarrow r = - \widehat k + \lambda \left( {\widehat i + \widehat j ...

Let f : R \(\to\) R be a function defined by f(x) = (x \(-\) 3)n1 (x \(-\) 5)n2, n1, n2 \(\in\) N. Then, which of the following is NOT true?

For real numbers a, b (a > b > 0), let
Area \(\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi\)
and
Area $$\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over ...

Let n \(\ge\) 5 be an integer. If 9n \(-\) 8n \(-\) 1 = 64\(\alpha\) and 6n \(-\) 5n \(-\) 1 = 25\(\beta\), then \(\alpha\) \(-\) \(\beta\) is equal to

Let the coefficients of x\(-\)1 and x\(-\)3 in the expansion of \({\left( {2{x^{{1 \over 5}}} - {1 \over {{x^{{1 \over 5}}}}}} \right)^{15}},x > 0\), be m and n respectively. If r is a positive integer such that $$m{n^2} = {}^{15}{C_r}\,.\,...

Let a triangle ABC be inscribed in the circle \({x^2} - \sqrt 2 (x + y) + {y^2} = 0\) such that \(\angle BAC = {\pi \over 2}\). If the length of side AB is \(\sqrt 2\), then the area of the \(\Delta\)ABC is equal to :

Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z \(-\) 1) \(-\) arg(z + 1) = \({\pi \over 4}\) intersect :

Let f be a real valued continuous function on [0, 1] and \(f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}\).
Then, which of the following points (x, y) lies on the curve y = f(x) ?

If \(\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left( {2 - {{{y^2}} \over 2}} \right)dy + I} } }\), then I equa...

If y = y(x) is the solution of the differential equation \(\left( {1 + {e^{2x}}} \right){{dy} \over {dx}} + 2\left( {1 + {y^2}} \right){e^x} = 0\) and y (0) = 0, then $$6\left( {y'(0) + {{\left( {y\left( {{{\log }_e}\sqrt 3 } \right)} \righ...

Let y = y(x), x > 1, be the solution of the differential equation \((x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}\), with \(y(2) = {{1 + {e^4}} \over {2{e^4}}}\). If \(y(3) = {{{e^\alpha } + 1} \over {\beta {e^\alpha }}}\), then the va...

Let f and g be twice differentiable even functions on (\(-\)2, 2) such that \(f\left( {{1 \over 4}} \right) = 0\), \(f\left( {{1 \over 2}} \right) = 0\), \(f(1) = 1\) and \(g\left( {{3 \over 4}} \right) = 0\), \(g(1) = 2\). Then, the minimu...

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If \(f(g(x)) = 8{x^2} - 2x\) and \(g(f(x)) = 4{x^2} + 6x + 1\), then the value of \(f(2) + g(2)\) is _________.

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60\(^\circ\). The pole subtends an angle 30\(^\circ\) at the top of the tower. Then the height of the tower is :

The value of \(\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}\) is equal to:

Negation of the Boolean statement (p \(\vee\) q) \(\Rightarrow\) ((\(\sim\) r) \(\vee\) p) is equivalent to :

Let \(A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)\). If \(B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, - {}^5{C_5}{(adj\,A)^5}\), then the sum of all elements of the matrix B is

Let \(M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]\), where \(\alpha\) is a non-zero real number an \(N = \sum\limits_{k = 1}^{49} {{M^{2k}}}\). If \((I - {M^2})N = - 2I\), then the positive integral v...

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of \({\pi \over 4}\) with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is :

The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to ____________.

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to :

Let \(\alpha\) be a root of the equation 1 + x2 + x4 = 0. Then, the value of \(\alpha\)1011 + \(\alpha\)2022 \(-\) \(\alpha\)3033 is equal to :

The sum of the infinite series \(1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \over {{6^6}}} + \,\,.....\) is equal to :

Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.

The number of values of a \(\in\) N such that the variance of 3, 7, 12, a, 43 \(-\) a is a natural number is :

The distance of the origin from the centroid of the triangle whose two sides have the equations \(x - 2y + 1 = 0\) and \(2x - y - 1 = 0\) and whose orthocenter is \(\left( {{7 \over 3},{7 \over 3}} \right)\) is :

The number of solutions of the equation \(\sin x = {\cos ^2}x\) in the interval (0, 10) is _________.

Let A, B, C be three points whose position vectors respectively are
\(\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k\)
\(\overrightarrow b = 2\widehat i + \alpha \widehat j + 4\widehat k,\,\alpha \in R\)
$$\overrightarrow c ...

Let  \(\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k\),   \(\overrightarrow b = \widehat i + \widehat j + \widehat k\)   and   \(\overrightarrow c\)   be a vector such that   $$\overrightarrow a + \left( {\overrightarrow b ...