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Let \( f(x) = \frac{x - 1}{x + 1} ,\; x \in \mathbb{R} - \{ 0, -1, 1 \} \). If \( f^{\,n+1}(x) = f\!\left(f^{\,n}(x)\right) \) for all \( n \in \mathbb{N} \), then \( f^{6}(6) + f^{7}(7) \) is equal to:

Let \( A = \left\{ z \in \mathbb{C} : \left| \frac{z + 1}{z - 1} \right| < 1 \right\} \) and \( B = \left\{ z \in \mathbb{C} : \arg \!\left( \frac{z - 1}{z + 1} \right) = \frac{2\pi}{3} \right\} \). Then \( A \cap B \) is:

Let \( A \) be a \( 3 \times 3 \) invertible matrix. If \( \left| \operatorname{adj}(24A) \right| = \left| \operatorname{adj}\!\left( 3 \operatorname{adj}(2A) \right) \right| \), then \( |A|^{2} \) is equal to:

The ordered pair \( (a, b) \), for which the system of linear equations.
\( 3x - 2y + z = b \)
\( 5x - 8y + 9z = 3 \)
\( 2x + y + az = -1 \)
Has no solution, is:

The remainder when \( (2021)^{2023} \) is divided by \( 7 \) is:

\( \displaystyle \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)} \) is equal to:

Let $f, g : \mathbb{R} \rightarrow \mathbb{R}$ be two real valued functions defined as

where $k_1$ and $k_2$ are real constants. If $(gof)$ is differentiable at $x = 0$, then $(gof)(-4) + (gof)(4)$ is equal to:

The sum of the absolute minimum and the absolute maximum values of the function \( f(x)=\left|3x-x^{2}+2\right|-x \) in the interval \( [-1,2] \) is:

Let \( S \) be the set of all the natural numbers, for which the line \( \dfrac{x}{a}+\dfrac{y}{b}=2 \) is a tangent to the curve \( \left(\dfrac{x}{a}\right)^{n}+\left(\dfrac{y}{b}\right)^{n}=2 \) at the point \( (a,b), ab\neq 0 \). Then:

The area bounded by the curve \( y=|x^{2}-9| \) and the line \( y=3 \) is:

Let \( R \) be the point \( (3,7) \) and let \( P \) and \( Q \) be two points on the line \( x+y=5 \) such that \( PQR \) is an equilateral triangle. Then the area of \( \triangle PQR \) :

Let \( C \) be a circle passing through the points \( A(2,-1) \) and \( B(3,4) \). The line segment \( AB \) is not a diameter of \( C \). If \( r \) is the radius of \( C \) and its centre lies on the circle \( (x-5)^{2}+(y-1)^{2}=\dfrac{13}{2} \), then \( r^{2} \) is equal to:

Let the normal at the point \( P \) on the parabola \( y^2 = 6x \) pass through the point \( (5, -8) \). If the tangent at \( P \) to the parabola intersects its directrix at the point \( Q \), then the ordinate of the point \( Q \) is:

If the two lines \( l_1 : \dfrac{x-2}{3} = \dfrac{y+1}{-2}, \; z = 2 \) and \( l_2 : \dfrac{x-1}{1} = \dfrac{2y+3}{\alpha} = \dfrac{z+5}{2} \) are perpendicular, then an angle between the lines \( l_2 \) and \( l_3 : \dfrac{1-x}{3} = \dfrac{2y-1}{-4} = \dfrac{z}{4} \) is:

Let the plane \( 2x + 3y + z + 20 = 0 \) be rotated through a right angle about its line of intersection with the plane \( x - 3y + 5z = 8 \). If the mirror image of the point \( \left( 2, -\dfrac{1}{2}, 2 \right) \) in the rotated plane is \( B(a,b,c) \), then:

If \( \vec{a}\cdot\vec{b} = 1 \), \( \vec{b}\cdot\vec{c} = 2 \) and \( \vec{c}\cdot\vec{a} = 3 \), then the value of \( \left[ \vec{a}\times(\vec{b}\times\vec{c}), \; \vec{b}\times(\vec{c}\times\vec{a}), \; \vec{c}\times(\vec{b}\times\vec{a}) \right] \) is:

Let a biased coin be tossed \( 5 \) times. If the probability of getting \( 4 \) heads is equal to the probability of getting \( 5 \) heads, then the probability of getting at most two heads is:

The mean of the numbers \( a, b, 8, 5, 10 \) is \( 6 \) and their variance is \( 6.8 \). If \( M \) is the mean deviation of the numbers about the mean, then \( 25M \) is equal to:

Let \( f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^{2} - 2x + 10, \, x \in [-1,1] \). If \([a,b]\) is the range of the function \( f \), then \( 4a - b \) is equal to:

Let \( \Delta, \nabla \in \{\wedge, \vee\} \) be such that \( p \nabla q \Rightarrow ((p \Delta q)\nabla r) \) is a tautology. Then \( (p \nabla q)\Delta r \) is logically equivalent to:

The sum of the cubes of all the roots of the equation \( x^{4} - 3x^{3} - 2x^{2} + 3x + 1 = 0 \) is __________ .

There are ten boys \( B_{1}, B_{2}, \ldots, B_{10} \) and five girls \( G_{1}, G_{2}, \ldots, G_{5} \) in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both \( B_{1} \) and \( B_{2} \) together should not be the members of a group, is __________ .

Let the common tangents to the curves \( 4(x^{2}+y^{2}) = 9 \) and \( y^{2} = 4x \) intersect at the point \( Q \). Let an ellipse, centered at the origin \( O \), has lengths of semi-minor and semi-major axes equal to \( OQ \) and \( 6 \), respectively. If \( e \) and \( l \) respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \( \frac{l}{e^{2}} \) is equal to______________.

Let \( f(x) = \max\{\, |x+1|, |x+2|, \ldots, |x+5| \,\} \). Then \( \int_{-6}^{0} f(x)\,dx \) is equal to______________.

Let the solution curve \( y = y(x) \) of the differential equation \( (4 + x^2)\,\,dy - 2x(x^2 + 3y + 4)\,dx = 0 \) pass through the origin. Then \( y(2) \) is equal to ________________ .

If \( \sin^2(10^\circ)\sin(20^\circ)\sin(40^\circ)\sin(50^\circ)\sin(70^\circ) = \alpha - \frac{1}{16}\sin(10^\circ) \), then \( 16 + \alpha^{-1} \) is equal to ____________ .

Let \( A = \{ n \in \mathbb{N} : H.C.F.(n, 45) = 1 \} \) and Let \( B = \{ 2k : k \in \{ 1, 2, \ldots , 100 \} \} \). Then the sum of all the elements of \( A \cap B \) is ____________ .

The value of the integral \( \frac{48}{\pi^{4}} \int_{0}^{{\pi}{}} \left(\frac{3\pi x^{2}}{2}-x^{3}\right)^{}\frac{\sin x}{1+\cos^{2}x}\,dx \) is equal to ____________ .

Let \( A = \displaystyle \sum_{i=1}^{10}\ \displaystyle \sum_{j=1}^{10} \min\{i, j\} \) and \( B =\displaystyle \sum_{i=1}^{10}\ \displaystyle \sum_{j=1}^{10} \max\{i, j\} \). Then \( A + B \) is equal to ____________ .

Let $S = (0, 2\pi) - \left\{ \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4} \right\}$. Let $y = y(x), x \in S$, be the solution curve of the differential equation

$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}.$

If the sum of abscissas of all the points of intersection of the curve $y = y(x)$ with the curve ($y = \sqrt{2} \sin x$) is $\frac{k\pi}{12}$, then $k$ is equal to ___________.

JEE Main 2022 26th June Morning Shift

Physics Chemistry Mathematics

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The remainder when \( (2021)^{2023} \) is divided by \( 7 \) is: