JEE Main 2022 (Online) 30th June Morning Shift

The distance of the point (3, 2, \(-\)1) from the plane \(3x - y + 4z + 1 = 0\) along the line \({{2 - x} \over 2} = {{y - 3} \over 2} = {{z + 1} \over 1}\) is equal to :

Consider a triangle ABC whose vertices are A(0, \(\alpha\), \(\alpha\)), B(\(\alpha\), 0, \(\alpha\)) and C(\(\alpha\), \(\alpha\), 0), \(\alpha\) > 0. Let D be a point moving on the line x + z \(-\) 3 = 0 = y and G be the centroid of $$\De...

If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + \(\alpha\), 0) and (0, 50 + \(\alpha\)), \(\alpha\) > 0, then (x, y) also lies on the line :

Let \(f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R\). Then f :

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infecte...

If for some \(\alpha\) > 0, the area of the region \(\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}\) is equal to \({3 \over 2}\), then the area of the region \(\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\}\) is equal to ____________.

For two positive real numbers a and b such that \({1 \over {{a^2}}} + {1 \over {{b^3}}} = 4\), then minimum value of the constant term in the expansion of \({\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}\) is :

Consider three circles:
\({C_1}:{x^2} + {y^2} = {r^2}\)
\({C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}\)
\({C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}\)
If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one sid...

The real part of the complex number \({{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}\) is equal to :

\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}}\) is equal to :

Let \(f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx}\). If \(f(1) + f'(1) = \alpha e - {1 \over 6}\), then the value of 150\(\alpha\) is equal to ___________.

Let \({{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R\), represents a circle with center (\(\alpha\), \(\beta\)). Then, \(\alpha\) + 2\(\beta\) is equal to :

Let the eccentricity of the ellipse \({x^2} + {a^2}{y^2} = 25{a^2}\) be b times the eccentricity of the hyperbola \({x^2} - {a^2}{y^2} = 5\), where a is the minimum distance between the curves y = ex and y = logex. Then $${a^2} + {1 \over {...

Let m and M respectively be the minimum and the maximum values of \(f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{\pi \over 8}} \right]\). Then m + M is equal to :

Let \(\alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right)\) and $$\beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \...

Suppose \(\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}\) exists and is equal to L, where
$$F(x) = \left| {\matrix{
{a + \sin {x \over 2}} & { - b\cos x} & 0 \cr
{ - b\cos x} & 0 & {a + \sin {x \over 2}} \cr
0 & {a + ...

The conditional statement
\(((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q))\) is :

Let \(A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]\) and \(B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right],\,\alpha \in C\). Then the absolut...

Let A and B be two square matrices of order 2. If \(det\,(A) = 2\), \(det\,(B) = 3\) and \(\det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}{5^c}\) for some a, b, c, \(\in\) N, then a + b + c is equal to :

Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of \(\Delta\)POQ is equal to ___________.

The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.

If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then \({{P(X = 2)} \over {P(X = 3)}}\) is equal to :

The probability distribution of X is :

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Let \({S_1} = \left\{ {x \in R - \{ 1,2\} :{{(x + 2)({x^2} + 3x + 5)} \over { - 2 + 3x - {x^2}}} \ge 0} \right\}\) and \({S_2} = \left\{ {x \in R:{3^{2x}} - {3^{x + 1}} - {3^{x + 2}} + 27 \le 0} \right\}\). Then, \({S_1} \cup {S_2}\) is equ...

Let S be the set of all integral values of \(\alpha\) for which the sum of squares of two real roots of the quadratic equation \(3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0\) is minimum. Then S :

The value of \(1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}\) is equal to:

Let for \(f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1\) and \(f'(1) = 0\). If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) i...

Let \(\alpha\)1, \(\alpha\)2 (\(\alpha\)1 < \(\alpha\)2) be the values of \(\alpha\) fo the points (\(\alpha\), \(-\)3), (2, 0) and (1, \(\alpha\)) to be collinear. Then the equation of the line, passing through (\(\alpha\)1, \(\alpha\)2) a...

Let \({S_1} = \{ x \in [0,12\pi ]:{\sin ^5}x + {\cos ^5}x = 1\}\)
and \({S_2} = \{ x \in [0,8\pi ]:{\sin ^7}x + {\cos ^7}x = 1\}\)
Then \(n({S_1}) - n({S_2})\) is equal to ______________.

Let a vector \(\overrightarrow c\) be coplanar with the vectors \(\overrightarrow a = - \widehat i + \widehat j + \widehat k\) and \(\overrightarrow b = 2\widehat i + \widehat j - \widehat k\). If the vector \(\overrightarrow c\) also ...