JEE Main 2022 (Online) 27th June Morning Shift

If two straight lines whose direction cosines are given by the relations \(l + m - n = 0\), \(3{l^2} + {m^2} + cnl = 0\) are parallel, then the positive value of c is :

Let the mirror image of the point (a, b, c) with respect to the plane 3x \(-\) 4y + 12z + 19 = 0 be (a \(-\) 6, \(\beta\), \(\gamma\)). If a + b + c = 5, then 7\(\beta\) \(-\) 9\(\gamma\) is equal to ______________.

Let
\({A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}\) and
\({A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}\). If 27 (Area A1) = 5 (Area A2), then k is equal to :

If the coefficient of x10 in the binomial expansion of \({\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}\) is \({5^k}\,.\,l\), where l, k \(\in\) N and l is co-prime to 5, then k is eq...

A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x \(-\) y + 4 = 0, then the area of R is ____________.

The area of the polygon, whose vertices are the non-real roots of the equation \(\overline z = i{z^2}\) is :

The value of the integral \(\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}\) is equal to :

Let \({{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}\), where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

If \({{dy} \over {dx}} + {{{2^{x - y}}({2^y} - 1)} \over {{2^x} - 1}} = 0\), x, y > 0, y(1) = 1, then y(2) is equal to :

If \({\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| < 2\), then :

Let the eccentricity of an ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \(a > b\), be \({1 \over 4}\). If this ellipse passes through the point \(\left( { - 4\sqrt {{2 \over 5}} ,3} \right)\), then \({a^2} + {b^2}\) is...

Let f : R \(\to\) R be a function defined by \(f(x) = {{2{e^{2x}}} \over {{e^{2x}} + e}}\). Then $$f\left( {{1 \over {100}}} \right) + f\left( {{2 \over {100}}} \right) + f\left( {{3 \over {100}}} \right) + \,\,\,.....\,\,\, + \,\,\,f\left(...

If \(\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}\), where C is a constant, then \({{{d^3}f} \over {d{x^3}}}\) at x = 1 is equal to :

\({\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)\) is equal to :

Let a be an integer such that \(\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}\) exists, where [t] is greatest integer \(\le\) t. Then a is equal to :

The boolean expression \(( \sim (p \wedge q)) \vee q\) is equivalent to :

Let the system of linear equations \(x + 2y + z = 2\), \(\alpha x + 3y - z = \alpha\), \(- \alpha x + y + 2z = - \alpha\) be inconsistent. Then \(\alpha\) is equal to :

The positive value of the determinant of the matrix A, whose
Adj(Adj(A)) = \(\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} \cr {28} & { - 14} & {14} \cr } } \right)\), is _____________.

A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola \(y = {\left( {x - {1 \over 4}} \right)^2} + \alpha\), where \(\alpha\) > 0. Then (4\(\alpha\) \(-\) 8)2 is equal to _______...

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _____________.

Five numbers \({x_1},{x_2},{x_3},{x_4},{x_5}\) are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order \(({x_1} < {x_2} < {x_3} < {x_4} < {x_5})\). The probability that \({x_2} = 7\) and $${x_4} ...

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(x = 4), then the sum of the mean and the variance of X is :

The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 \(-\) 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to :

The number of distinct real roots of x4 \(-\) 4x + 1 = 0 is :

If the sum of all the roots of the equation \({e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0\) is \({\log _e}p\), then p is equal to ____________.

\(x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } }\), where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc \(\ne\) 0, then :

If the sum of the first ten terms of the series
\({1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5 \over {2501}} + \,\,....\)
is \({m \over n}\), where m and n are co-prime numbers, then m + n is equal to _____________...

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If (\(\alpha\), \(\beta\)) is the centroid of \(\Delta\)ABC, then 15(\(\alpha\) + \(\beta\)) is ...

The value of \(\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)\) is equal to :

Let \(\overrightarrow a = \widehat i + \widehat j - \widehat k\) and \(\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k\). Then the number of vectors \(\overrightarrow b\) such that $$\overrightarrow b \times \overrightarrow ...