JEE Main 2022 (Online) 27th July Morning Shift

If the plane \(P\) passes through the intersection of two mutually perpendicular planes \(2 x+k y-5 z=1\) and \(3 k x-k y+z=5, k<3\) and intercepts a unit length on positive \(x\)-axis, then the intercept made by the plane \(P\) on the $$y$...

Let the line \(\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}\) intersect the plane containing the lines \(\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}\) and \(4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}\) at the point $$P(\alpha, \beta, \gamm...

Let \(M\) and \(N\) be the number of points on the curve \(y^{5}-9 x y+2 x=0\), where the tangents to the curve are parallel to \(x\)-axis and \(y\)-axis, respectively. Then the value of \(M+N\) equals ___________.

The area of the smaller region enclosed by the curves \(y^{2}=8 x+4\) and \(x^{2}+y^{2}+4 \sqrt{3} x-4=0\) is equal to

The remainder when \((2021)^{2022}+(2022)^{2021}\) is divided by 7 is

If the circle \(x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}\) passes through the point \((6,1)\) and its centre lies on the line \(x-2 c y=8\), then the length of intercept made by the circle on \(x\)-axis is :

Let the minimum value \(v_{0}\) of \(v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}\) is attained at \({ }{z}=z_{0}\). Then \(\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}\) is equal to :

Let \(S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}\). Then \(\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))\) is equal to ______________.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined as
\(f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}\) where \([t]\) is the greatest integer less than or equal to \(t\). If $$\mathop {\lim }\limits_{x \t...

Let \(I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x\). Then

Let a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined as :
$$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$$
where \(\mathrm{b} \in \mathbb{R}\). If \(f\) is continuous at $$x=4...

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be two distinct solutions of the differential equation \(\frac{d y}{d x}=x+y\), with \(y_{1}(0)=0\) and \(y_{2}(0)=1\) respectively. Then, the number of points of intersection of \(y=y_{1}(x)\) and $$y=...

Let \(y=y(x)\) be the solution curve of the differential equation
\(\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy - 4\sqrt 2 x\sin \left( {{x^2} - {\pi \over 4}} \right)} \right)dx = 0\), $$0 < x < \sqrt ...

If the length of the latus rectum of the ellipse \(x^{2}+4 y^{2}+2 x+8 y-\lambda=0\) is 4 , and \(l\) is the length of its major axis, then \(\lambda+l\) is equal to ____________.

Let \(f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}\) be functions defined by \(f(a)=\alpha\), where \(\alpha\) is the maximum of the powers of those primes \(p\) such that \(p^{\alpha}\) divides \(a\), and \(g(a)=a+1\), for all $$a \in \ma...

Let \(f(x)=2 x^{2}-x-1\) and \(\mathrm{S}=\{n \in \mathbb{Z}:|f(n)| \leq 800\}\). Then, the value of \(\sum\limits_{n \in S} f(n)\) is equal to ___________.

Let a vertical tower \(A B\) of height \(2 h\) stands on a horizontal ground. Let from a point \(P%\) on the ground a man can see upto height \(h\) of the tower with an angle of elevation \(2 \alpha\). When from \(P\), he moves a distance $...

An ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) passes through the vertices of the hyperbola \(H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1\). Let the major and minor axes of the ellipse \(E\) coincide with the transverse and conjug...

For \(k \in \mathbb{R}\), let the solutions of the equation \(\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0<|x|<\frac{1}{\sqrt{2}}\) be \(\alpha\) and \(\beta\), wher...

\((p \wedge r) \Leftrightarrow(p \wedge(\sim q))\) is equivalent to \((\sim p)\) when \(r\) is

Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha A^{2}+\beta A=2 I\). Then \(\alpha+\beta\) is equal to

Let \(S\) be the set containing all \(3 \times 3\) matrices with entries from \(\{-1,0,1\}\). The total number of matrices \(A \in S\) such that the sum of all the diagonal elements of \(A^{\mathrm{T}} A\) is 6 is ____________.

Let \(P(a, b)\) be a point on the parabola \(y^{2}=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^{2}+y^{2}-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of \(a\) and \(B\) be the produc...

Let \(S\) be the sample space of all five digit numbers. It \(p\) is the probability that a randomly selected number from \(S\), is a multiple of 7 but not divisible by 5 , then \(9 p\) is equal to :

Suppose \(a_{1}, a_{2}, \ldots, a_{n}\), .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is \(5: 17\) and , \(110 < {a_{15}} < 120\), then the...

Let \(R_{1}\) and \(R_{2}\) be two relations defined on \(\mathbb{R}\) by
\(a \,R_{1} \,b \Leftrightarrow a b \geq 0\) and \(a \,R_{2} \,b \Leftrightarrow a \geq b\)
Then,

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _____________.

Let \(A(1,1), B(-4,3), C(-2,-5)\) be vertices of a triangle \(A B C, P\) be a point on side \(B C\), and \(\Delta_{1}\) and \(\Delta_{2}\) be the areas of triangles \(A P B\) and \(A B C\), respectively. If \(\Delta_{1}: \Delta_{2}=4: 7\), ...

Let \(\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}\) and \(\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}\) be two vectors, such that \(\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}\). Then the projection of \(\vec{b}-2 \vec{a}\) on $$\vec{b}...

\(\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}|\)...