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For real numbers \( \alpha \) and \( \beta \neq 0 \), if the point of intersection of the straight lines \(\displaystyle \frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3} \) and \(\displaystyle\frac{x-4}{\beta}=\frac{y-6}{3}=\frac{z-7}{3} \), lies on the plane \( x+2y-z=8 \), then \( \alpha-\beta \) is equal to:

Let \( N \) be the set natural numbers and a relation \( R \) on \( N \) be defined by \( R=\{(x,y)\in N\times N: x^{3}-3x^{2}y-xy^{2}+3y^{3}=0\} \). Then the relation \( R \) is:

Let \( A \) and \( B \) be two \( 3\times 3 \) real matrices such that \( (A^{2}-B^{2}) \) is invertible matrix. If \( A^{5}=B^{5} \) and \( A^{3}B^{2}=A^{2}B^{3} \), then the value of the determinant of the matrix \( A^{3}+B^{3} \) is equal to:

Let \( y=y(x) \) be the solution of the differential equation \( (x-x^{3})\,dy=(y+y x^{2}-3x^{4})\,dx,\ x>2 \). If \( y(3)=3 \), then \( y(4) \) is equal to:

Let \( f:\mathbb{R}\to\mathbb{R} \) be defined as \( f(x+y)+f(x-y)=2\,f(x)\,f(y) \), \( f\!\left(\frac{1}{2}\right)=-1 \). Then, the value of \( \displaystyle \sum_{k=1}^{20}\frac{1}{\sin (k)\;\sin(k+f(k))} \) is equal to:

The value of \( \displaystyle \lim_{x\to 0}\left(\frac{x}{\sqrt[8]{\,1-\sin x\,}-\sqrt[8]{\,1+\sin x\,}}\right) \) is equal to:

The area of the region bounded by \( y - x = 2 \) and \( x^{2} = y \) is equal to:

Let \( \displaystyle \alpha = \max_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} \text{ and } \beta = \min_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\} \). If \( 8x^{2} + bx + c = 0 \) is a quadratic equation whose roots are \( \alpha^{1/5} \) and \( \beta^{1/5} \), then the value \( c - b \) is equal to:

Consider a circle \( C \) which touches the y-axis at \( (0, 6) \) and cuts off an intercept \( 6\sqrt{5} \) on the x-axis. Then the radius of the circle \( C \) is equal to:

The point \( P(a, b) \) undergoes the following three transformations successively:
(a) Reflection about the line \( y = x \)
(b) Translation through \( 2 \) units along the positive direction of x-axis
(c) Rotation through angle \( \dfrac{\pi}{4} \) about the origin in the anti-clockwise direction
If the co-ordinates of the final position of the point \( P \) are \( \left( -\dfrac{1}{\sqrt{2}}, \dfrac{7}{\sqrt{2}} \right) \), then the value of \( 2a + b \) is equal to:

If \( \tan\left( \dfrac{\pi}{9} \right), x, \tan\left( \dfrac{7\pi}{18} \right) \) are in arithmetic progression and \( \tan\left( \dfrac{\pi}{9} \right), y, \tan\left( \dfrac{5\pi}{18} \right) \) are also in arithmetic progression, then \( |x - 2y| \) is equal to:

A student appeared in an examination consisting of \( 8 \) true–false type questions. The student guesses the answers with equal probability. The smallest value of \( n \), so that the probability of guessing at least ‘\( n \)’ correct answers is less than \( \dfrac{1}{2} \), is:

Let \( f:(a,b)\to \mathbb{R} \) be twice differentiable function such that \( f(x)=\int\limits_{a}^{x} g(t)\,dt \) for a differentiable function \( g(x) \). If \( f(x)=0 \) has exactly five distinct roots in \( (a,b) \) then \( g(x)\,g'(x)=0 \) has at least:

Which of the following is the negation of the statement “for all \( M>0 \), there exists \( x\in S \) such that \( x\ge M \)”?

Let the mean and variance of the frequency distribution

be \( 6 \) and \( 6.8 \) respectively. If \( x_{3} \) is changed from \( 8 \) to \( 7 \), then the mean for the new data will be:

Let \( f:[0,\infty)\to [0,3] \) be a function defined by Then which of the following is true?

A possible value of '\(x\)' for which the ninth term in the expansion of in the increasing powers of \( 3^{\left(-\dfrac{1}{8}\right)\log_{3}\!\left(5^{x-1}+1\right)} \) is equal to \( 180 \), is:

Two sides of a parallelogram are along the line \( 4x+5y=0 \) and \( 7x+2y=0 \). If the equation of one of the diagonals of the parallelogram is \( 11x+7y=9 \), then other diagonal passes through the point:

Let \( C \) be the set of all complex numbers. Let
\( S_1=\{z\in \mathbb{C}: |z-2|\leq 1\} \) and
\( S_2=\{z\in \mathbb{C}: z(1+i)+\bar{z}(1-i)\geq 4\} \)
Then, the maximum value of \( \left|z-\dfrac{5}{2}\right|^2 \) for \( z\in S_1\cap S_2 \) is equal to :

Let \( \vec{a},\vec{b} \) and \( \vec{c} \) be three vectors such that \( \vec{a}=\vec{b}\times(\vec{b}\times\vec{c}) \). If magnitudes of the vectors \( \vec{a},\vec{b} \) and \( \vec{c} \) are \( \sqrt{2},1 \) and \( 2 \) respectively and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \theta \) \( (0<\theta<\dfrac{\pi}{2}) \), then the value of \( 1+\tan\theta \) is equal to :

Let \( n \) a non-negative integer. Then the number of divisors of the form “\( 4n+1 \)” of the number \( 10^{(10)}\cdot (11)^{11}\cdot (13)^{13} \) is equal to ____________.

Let \( \vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k},\ \vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k} \) and \( \vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k} \), where \( \alpha \) and \( \beta \) are integers. If \( \vec{a}\cdot\vec{b}=-1 \) and \( \vec{b}\cdot\vec{c}=10 \), then \( (\vec{a}\times\vec{b})\cdot\vec{c} \) is equal to __________.

The number of real roots of the equation \( e^{4x}-e^{3x}-4e^{2x}-e^{x}+1=0 \) is equal to ____________.

Let \( A=\{n\in \mathbb{N}\mid n^{2}\leq n+10{,}000\}, B=\{3k+1\mid k\in \mathbb{N}\} \) and \( C=\{2k\mid k\in \mathbb{N}\} \), then the sum of all the elements of the set \( A\cap (B-C) \) is equal to ____________.

If the real part of the complex number \( z = \dfrac{3 + 2 i \cos \theta}{1 - 3 i \cos \theta}, \ \theta \in \left(0, \dfrac{\pi}{2}\right) \) is zero, then the value of \( \sin^{2} 3\theta + \cos^{2} \theta \) is equal to ______.

If \(\displaystyle A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\) and \( M=A+A^{2}+A^{3}+\ldots+A^{20} \), then the sum of all the elements of the matrix \( M \) is equal to ____________.

Let \( E \) be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at \( (3,-4) \), one focus at \( (4,-4) \) and one vertex at \( (5,-4) \). If \( mx-y=4, \ m>0 \) is a tangent to the ellipse \( E \), then the value of \( 5m^{2} \) is equal to ____________.

The distance of the point \( P(3,4,4) \) from the point of intersection of the line joining the points \( Q(3,-4,-5) \) and \( R(2,-3,1) \) and the plane \( 2x+y+z=7 \), is equal to ____________.

Let \( y=y(x) \) be the solution of the differential equal to \( dy=e^{\alpha x + y} \, dx:\ \alpha \in \mathbb{N} \). If \( y(\log_{e} 2)=\log_{e} 2 \) and \( y(0)=\log_{e}\!\left(\dfrac{1}{2}\right) \), then the value of \( \alpha \) is equal to ____________.

If \( \displaystyle \int_{0}^{{\pi}{}} (\sin^{3}x )\, e^{-\sin^{2}x}\,dx = \alpha - \dfrac{\beta}{e}\displaystyle \int_{0}^{1} \sqrt{t}\,e^{t}\,dt \), then \( \alpha + \beta \) is equal to ____________.

JEE Main 2021 27th July Evening Shift

Physics Chemistry Mathematics

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The number of real roots of the equation \( e^{4x}-e^{3x}-4e^{2x}-e^{x}+1=0 \) is equal to ____________.