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\( i = \sqrt{-1} \), and \( Q = A^T B A \), then the inverse of the matrix \( A Q^{2021} A^T \) is equal to:

If a line along a chord of the circle \( 4x^{2}+4y^{2}+120x+675=0 \), passes through the point \( (-30,0) \), and is tangent to the parabola \( y^{2}=30x \), then the length of this chord is:

The sum of series \( \frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots+\frac{2^{100}}{x^{2^{100}}+1} \) when \( x=2 \) is:

The equation \( \arg \left( \frac{z-1}{z+1} \right) = \frac{\pi}{4} \) represents a circle with:

A plane \( P \) contains the line \( x+2y+3z+1=0 = x-y-z-6 \), and is perpendicular to the plane \( -2x+y+z+8=0 \). Then which of the following points lies on \( P \)?

Let \( ABC \) be a triangle with \( A(-3,1) \) and \( \angle ACB=\theta \), \( 0<\theta<\frac{\pi}{2} \). If the equation of the median through \( B \) is \( 2x+y-3=0 \) and the equation of angle bisector of \( C \) is \( 7x-4y-1=0 \), then \( \tan\theta \) is equal to:

On the ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \) let \( P \) be a point in the second quadrant such that the tangent at \( P \) to the ellipse is perpendicular to the line \( x + 2y = 0 \). Let \( S \) and \( S' \) be the foci of the ellipse and \( e \) be its eccentricity. If \( A \) is the area of the triangle \( SPS' \) then, the value of \( (5 - e^2)A \) is:

Out of all the patients in a hospital \( 89\% \) are found to be suffering from heart ailment and \( 98\% \) are suffering from lungs infection. If \( K\% \) of them are suffering from both ailments then \( K \) can not belong to the set:

The mean and standard deviation of \( 20 \) observations were calculated as \( 10 \) and \( 2.5 \) respectively. It was found that by mistake one data value was taken as \( 25 \) instead of \( 35 \). If \( \alpha \) and \( \sqrt{\beta} \) are the mean and standard deviation respectively for correct data, then \( (\alpha, \beta) \) is:

Let \( y = y(x) \) be a solution curve of the differential equation \( (y+1)\tan^2 x \, dx + \tan x \, dy + y \, dx = 0, \, x \in \left(0, \frac{\pi}{2}\right) \). If \( \displaystyle \lim_{x \to 0^+} x y(x) = 1 \), then the value of \( y\!\left(\frac{\pi}{4}\right) \) is:

The value of $\int_{-1/\sqrt{2}}^{1/\sqrt{2}} \left( \left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2 \right)^{1/2} dx$ is:

Let \( \theta \in \left(0, \frac{\pi}{2}\right) \). If the system of linear equation.
\( (1 + \cos^2 \theta)x + \sin^2 \theta \, y + 4 \sin 3\theta \, z = 0 \)
\( \cos^2 \theta \, x + (1 + \sin^2 \theta) y + 4 \sin 3\theta \, z = 0 \)
\( \cos^2 \theta \, x + \sin^2 \theta \, y + (1 + 4 \sin 3\theta) z = 0 \)
Has a non-trivial solution, then the value of \( \theta \) is:

The sum of solutions of the equation \( \dfrac{\cos x}{1+\sin x} = |\tan 2x| \), \( x \in \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) - \left\{ \dfrac{\pi}{4}, -\dfrac{\pi}{4} \right\} \) is:

The value of \( \displaystyle \lim_{n \to \infty} \dfrac{1}{n} \displaystyle \sum_{r=0}^{2n-1} \dfrac{n^{2}}{n^{2}+4 r^{2}} \) is:

If the sum of an infinite GP \( a, ar, ar^{2}, ar^{3}, \ldots \) is \( 15 \) and the sum of the squares of its each term is \( 150 \), then the sum of \( ar^{2}, ar^{4}, ar^{6}, \ldots \) is:

If the truth value of the Boolean expression $((p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)) \rightarrow (p \wedge q)$ is false, then the truth values of the statements $p, q, r$ respectively can be:

Let $f(x) = \cos \left( 2 \tan^{-1} \sin \left( \cot^{-1} \sqrt{\frac{1-x}{x}} \right) \right), 0 < x < 1$. Then:

Let \( A \) and \( B \) be independent events such that \( P(A)=p \), \( P(B)=2p \). The largest value of \( p \), for which \( P \) (exactly one of \( A, B \) occurs) \( = \frac{5}{9} \), is:

If \( ^{20}C_{r} \) is the co-efficient of \( x^{r} \) in the expansion of \( (1+x)^{20} \), then the value of \( \displaystyle \sum_{r=0}^{20} r^{2}\,^{20}C_{r} \) is equal to:

Let \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \) and \( \vec{b}=\hat{j}-\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{a}\times \vec{c}=-\vec{b} \) and \( \vec{a}\cdot \vec{c}=3 \), then \( \vec{a}\cdot (\vec{b}\times \vec{c}) \) is equal to:

The locus of a point, which moves such that the sum of squares of its distances from the points \( (0,0) \), \( (1,0) \), \( (0,1) \), \( (1,1) \) is \( 18 \) units, is a circle of diameter \( d \). Then \( d^{2} \) is equal to ________.

If \( {}^{1}P_{1}+2\cdot {}^{2}P_{2}+3\cdot {}^{3}P_{3}+\ldots+15\cdot {}^{15}P_{15}={}^qP_{r-s},\; 0\le s\le 1 \), then \( {}^{\,q+s}C_{\,r-s} \) is equal to ________.

Let \( z=\frac{1-i\sqrt{3}}{2} \), \( i=\sqrt{-1} \). Then the value of \( 21+\left(z+\frac{1}{z}\right)^{3}+\left(z^{2}+\frac{1}{z^{2}}\right)^{3}+\left(z^{3}+\frac{1}{z^{3}}\right)^{3}+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^{3} \) is ________.

Let the line \( L \) be the projection of the line \( \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{2} \) in the plane \( x - 2y - z = 3 \). If \( d \) is the distance of the point \( (0, 0, 6) \) from \( L \), then \( d^2 \) is equal to ______.

The sum of all integral values of \( k \) (\( k \neq 0 \)) for which the equation \( \frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k} \) in \( x \) has no real roots, is _______.

If \( y = y(x) \) is an implicit function of \( x \) such that \( \log_{e}(x + y) = 4xy \), then \( \frac{d^{2}y}{dx^{2}} \) at \( x = 0 \) is equal to ________.

Let \( a, b \in \mathbb{R},\ b \neq 0 \). Define a function

If \( f \) is continuous at \( x = 0 \), then \( 10 - ab \) is equal to ________.

The area of the region \( S = \{(x, y) : 3x^{2} \le 4y \le 6x + 24\} \) is ________.

A wire of length \( 36 \, \text{m} \) is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum and the circumference of the circle is \( k \) (\( \text{meter} \)), then \( \left(\frac{4}{\pi} + 1\right)k \) is equal to ________.

The number of three-digit even numbers, formed by the digits \( 0 \), \( 1 \), \( 3 \),\( 4 \), \( 6 \), \( 7 \) if the repetition of digits is not allowed, is _________.

JEE Main 2021 26th August Morning Shift

Physics Chemistry Mathematics

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If \( {}^{1}P_{1}+2\cdot {}^{2}P_{2}+3\cdot {}^{3}P_{3}+\ldots+15\cdot {}^{15}P_{15}={}^qP_{r-s},\; 0\le s\le 1 \), then \( {}^{\,q+s}C_{\,r-s} \) is equal to ________.