JEE Main 2022 (Online) 25th June Morning Shift

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, \(-\)1, \(-\)1), parallel to the line PQ meets the plane S at R, then QR2 is equal to :

Let the lines
\({L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R\)
$${L_2}:\overrightarrow r = \left( {\widehat i + 3\widehat j + \widehat k} \right) + \mu \left( {\widehat i + \wi...

If \({1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}\), then the remainder when K is divided by 6 is :

Let Cr denote the binomial coefficient of xr in the expansion of \({(1 + x)^{10}}\). If for \(\alpha\), \(\beta\) \(\in\) R, \({C_1} + 3.2{C_2} + 5.3{C_3} +\) ....... upto 10 terms $$ = {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\le...

Let a circle C touch the lines \({L_1}:4x - 3y + {K_1} = 0\) and \({L_2} = 4x - 3y + {K_2} = 0\), \({K_1},{K_2} \in R\). If a line passing through the centre of the circle C intersects L1 at \(( - 1,2)\) and L2 at \((3, - 6)\), then the equ...

Let the abscissae of the two points P and Q be the roots of \(2{x^2} - rx + p = 0\) and the ordinates of P and Q be the roots of \({x^2} - sx - q = 0\). If the equation of the circle described on PQ as diameter is $$2({x^2} + {y^2}) - 11x -...

Let a circle C in complex plane pass through the points \({z_1} = 3 + 4i\), \({z_2} = 4 + 3i\) and \({z_3} = 5i\). If \(z( \ne {z_1})\) is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then...

The value of \(\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}\) is equal to:

Let \(g:(0,\infty ) \to R\) be a differentiable function such that $$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} +...

Let \(y = y(x)\) be the solution of the differential equation \((x + 1)y' - y = {e^{3x}}{(x + 1)^2}\), with \(y(0) = {1 \over 3}\). Then, the point \(x = - {4 \over 3}\) for the curve \(y = y(x)\) is :

If the solution curve \(y = y(x)\) of the differential equation \({y^2}dx + ({x^2} - xy + {y^2})dy = 0\), which passes through the point (1, 1) and intersects the line \(y = \sqrt 3 x\) at the point \((\alpha ,\sqrt 3 \alpha )\), then value...

Let f : R \(\to\) R be defined as \(f(x) = {x^3} + x - 5\). If g(x) is a function such that \(f(g(x)) = x,\forall 'x' \in R\), then g'(63) is equal to ________________.

Let f : N \(\to\) R be a function such that \(f(x + y) = 2f(x)f(y)\) for natural numbers x and y. If f(1) = 2, then the value of \(\alpha\) for which
\(\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)}\)
holds, is :

Let \(f:R \to R\) and \(g:R \to R\) be two functions defined by \(f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1\) and \(g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}\). Then, for which of the following range of \(\alpha\), the inequality $$f\left( ...

Let \(f:R \to R\) be a function defined by \(f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}\). If the function \(g(x) = f(f(f(x))) + f(f(x))\), then the greatest integer less than or equa...

Let f(x) be a polynomial function such that \(f(x) + f'(x) + f''(x) = {x^5} + 64\). Then, the value of \(\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}\) is equal to:

Consider the following two propositions:
\(P1: \sim (p \to \sim q)\)
\(P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)\)
If the proposition \(p \to (( \sim p) \vee q)\) is evaluated as FALSE, then :

Let A be a 3 \(\times\) 3 real matrix such that
$$A\left( {\matrix{
1 \cr
1 \cr
0 \cr

} } \right) = \left( {\matrix{
1 \cr
1 \cr
0 \cr

} } \right);A\left( {\matrix{
1 \cr
0 \cr
1 \cr

} } \r...

Let \(A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]\). If M and N are two matrices given by \(M = \sum\limits_{k = 1}^{10} {{A^{2k}}}\) and \(N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}}\) then MN2 is :

If \(y = {m_1}x + {c_1}\) and \(y = {m_2}x + {c_2}\), \({m_1} \ne {m_2}\) are two common tangents of circle \({x^2} + {y^2} = 2\) and parabola y2 = x, then the value of \(8|{m_1}{m_2}|\) is equal to :

Let \(x = 2t\), \(y = {{{t^2}} \over 3}\) be a conic. Let S be the focus and B be the point on the axis of the conic such that \(SA \bot BA\), where A is any point on the conic. If k is the ordinate of the centroid of the \(\Delta\)SAB, the...

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

Let A be a 3 \(\times\) 3 matrix having entries from the set {\(-\)1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.

Let E1 and E2 be two events such that the conditional probabilities \(P({E_1}|{E_2}) = {1 \over 2}\), \(P({E_2}|{E_1}) = {3 \over 4}\) and \(P({E_1} \cap {E_2}) = {1 \over 8}\). Then :

Let a, b and c be the length of sides of a triangle ABC such that \({{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}\). If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the va...

For a natural number n, let \({\alpha _n} = {19^n} - {12^n}\). Then, the value of \({{31{\alpha _9} - {\alpha _{10}}} \over {57{\alpha _8}}}\) is ___________.

The greatest integer less than or equal to the sum of first 100 terms of the sequence \({1 \over 3},{5 \over 9},{{19} \over {27}},{{65} \over {81}},\) ...... is equal to ___________.

The number of values of x in the interval \(\left( {{\pi \over 4},{{7\pi } \over 4}} \right)\) for which \(14\cos e{c^2}x - 2{\sin ^2}x = 21 - 4{\cos ^2}x\) holds, is ____________.

Let \(\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k\) \({a_i} > 0\), \(i = 1,2,3\) be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of \(\overrightarrow a\) on...

Let \(\theta\) be the angle between the vectors \(\overrightarrow a\) and \(\overrightarrow b\), where \(|\overrightarrow a | = 4,\) \(|\overrightarrow b | = 3\) and \(\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)\). Then $${...