JEE Main 2022 (Online) 25th July Evening Shift

A plane \(E\) is perpendicular to the two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((P Q)^{2}\) is equal...

The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is :

The sum of the maximum and minimum values of the function \(f(x)=|5 x-7|+\left[x^{2}+2 x\right]\) in the interval \(\left[\frac{5}{4}, 2\right]\), where \([t]\) is the greatest integer \(\leq t\), is ______________.

Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve \(4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0\) at the point (\(-\)2, 3) be A. Then 8A is equal to ______________.

The remainder when \((11)^{1011}+(1011)^{11}\) is divided by 9 is

If the circles \({x^2} + {y^2} + 6x + 8y + 16 = 0\) and \({x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 - \sqrt 6 } \right)y = k + 6\sqrt 3 + 8\sqrt 6\), \(k > 0\), touch internally at the point \(P(\alpha ,\beta )\), then ...

For \(z \in \mathbb{C}\) if the minimum value of \((|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)\) is \(5 \sqrt{2}\), then a value Question: of \(p\) is _____________.

$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - ...

Let \([t]\) denote the greatest integer less than or equal to \(t\). Then the value of the integral \(\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x\) is equal to

Let \(f\) be a twice differentiable function on \(\mathbb{R}\). If \(f^{\prime}(0)=4\) and \(f(x) + \int\limits_0^x {(x - t)f'(t)dt = \left( {{e^{2x}} + {e^{ - 2x}}} \right)\cos 2x + {2 \over a}x}\), then \((2 a+1)^{5}\, a^{2}\) is equal t...

Let \({a_n} = \int\limits_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx}\) for every n \(\in\) N. Then the sum of all the elements of the set {n \(\in\) N : an \(\in\) (2, 30...

Let a smooth curve \(y=f(x)\) be such that the slope of the tangent at any point \((x, y)\) on it is directly proportional to \(\left(\frac{-y}{x}\right)\). If the curve passes through the points \((1,2)\) and \((8,1)\), then $$\left|y\left...

Let \(y=y(x)\) be the solution of the differential equation
\(\frac{d y}{d x}=\frac{4 y^{3}+2 y x^{2}}{3 x y^{2}+x^{3}}, y(1)=1\).
If for some \(n \in \mathbb{N}, y(2) \in[n-1, n)\), then \(n\) is equal to _____________.

If the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) meets the line \(\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1\) on the \(x\)-axis and the line \(\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1\) on the \(y\)-axis, then the eccentricity of the ellipse...

The number of bijective functions \(f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots .100\}\), such that \(f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots . . f(99)\), is ____________.

Let \(f(x)\) be a quadratic polynomial with leading coefficient 1 such that \(f(0)=p, p \neq 0\), and \(f(1)=\frac{1}{3}\). If the equations \(f(x)=0\) and \(f \circ f \circ f \circ f(x)=0\) have a common real root, then \(f(-3)\) is equal ...

Let the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{7}=1\) and the hyperbola \(\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}\) coincide. Then the length of the latus rectum of the hyperbola is :

Let \(x = \sin (2{\tan ^{ - 1}}\alpha )\) and \(y = \sin \left( {{1 \over 2}{{\tan }^{ - 1}}{4 \over 3}} \right)\). If \(S = \{ a \in R:{y^2} = 1 - x\}\), then \(\sum\limits_{\alpha \in S}^{} {16{\alpha ^3}}\) is equal to _______________...

\(\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}\) is equal to

Consider the following statements:
P : Ramu is intelligent.
Q : Ramu is rich.
R : Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:

The number of real values of \(\lambda\), such that the system of linear equations
2x \(-\) 3y + 5z = 9
x + 3y \(-\) z = \(-\)18
3x \(-\) y + (\(\lambda\)2 \(-\) | \(\lambda\) |)z = 16
has no solutions, is

Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{array}\right] $$ th...

The tangents at the points \(A(1,3)\) and \(B(1,-1)\) on the parabola \(y^{2}-2 x-2 y=1\) meet at the point \(P\). Then the area (in unit \({ }^{2}\) ) of the triangle \(P A B\) is :

If \(A\) and \(B\) are two events such that \(P(A)=\frac{1}{3}, P(B)=\frac{1}{5}\) and \(P(A \cup B)=\frac{1}{2}\), then \(P\left(A \mid B^{\prime}\right)+P\left(B \mid A^{\prime}\right)\) is equal to :

The sum \(\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}}\) is equal to

Let \(A=\{1,2,3,4,5,6,7\}\). Define \(B=\{T \subseteq A\) : either \(1 \notin T\) or \(2 \in T\}\) and \(C=\{T \subseteq A: T\) the sum of all the elements of \(T\) is a prime number \(\}\). Then the number of elements in the set $$B \cup C...

If the mean deviation about median for the numbers 3, 5, 7, 2k, 12, 16, 21, 24, arranged in the ascending order, is 6 then the median is :

Let the point \(P(\alpha, \beta)\) be at a unit distance from each of the two lines \(L_{1}: 3 x-4 y+12=0\), and \(L_{2}: 8 x+6 y+11=0\). If \(P\) lies below \(L_{1}\) and above \({ }{L_{2}}\), then \(100(\alpha+\beta)\) is equal to :

\(2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)\) is equal to :

Let \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}\) and let \(\vec{b}\) be a vector such that \(\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}\) and \(\vec{a} \cdot \vec{b}=3\). Then the projection of \(\vec{b}\) on the vector \(\vec{a}-\vec{b}\) is :