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Let \( \alpha \) be a root of the equation \( 1 + x^2 + x^4 = 0 \). Then the value of \( \alpha^{1011} + \alpha^{2022} - \alpha^{3033} \) is equal to:

Let \( \arg(z) \) represent the principal argument of the complex number \( z \). Then, \( |z| = 3 \) and \( \arg(z-1) - \arg(z+1) = \pi/4 \) intersect.

If \( B = I - {^{5}C_{1}}(\operatorname{adj}A) + {^{5}C_{2}}(\operatorname{adj}A)^2 - \ldots - {^{5}C_{5}}(\operatorname{adj}A)^5 \), then the sum of all elements of the matrix \( B \) is:

The sum of the infinite series \( 1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \frac{51}{6^5} + \frac{70}{6^6} + \cdots \) is equal to:

The value of \( \displaystyle \lim_{x \to 1} \frac{(x^2-1)\sin^2(\pi x)}{x^4-2x^3+2x-1} \) is equal to:

Let \( f:\mathbb{R}\to \mathbb{R} \) be a function defined by \( f(x)=(x-3)^{n_1}(x-5)^{n_2}, \, n_1,n_2 \in \mathbb{N} \). Then, which of the following is NOT true?

Let \( f \) be a real valued continuous function on \( [0,1] \) and \( f(x)=x+\int_{0}^{1}(x-t)f(t)\,dt \). Then, which of the following points \( (x,y) \) lies on the curve \( y=f(x) \)?

If \( \int_{0}^{2}\left(\sqrt{2x}-\sqrt{2x-x^{2}}\right)dx=\int_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right)dy+\int_{1}^{2}\left(2-\frac{y^{2}}{2}\right)dy+I \) then \( I \) equals:

If \( y=y(x) \) is the solution of the differential equation \( (1+e^{2x})\frac{dy}{dx}+2(1+y^{2})e^{x}=0 \) and \( y(0)=0 \), then \( 6\left(y'(0)+y(\log_{e}\sqrt{3})\right)^{2} \) is equal to:

Let \( P: y^{2}=4ax,\; a>0 \) be a parabola with focus \( S \). Let the tangents to the parabola \( P \) make an angle of \( \frac{\pi}{4} \) with the line \( y=3x+5 \) touch the parabola \( P \) at \( A \) and \( B \). Then the value of \( a \) for which \( A \), \( B \) and \( S \) are collinear is:

Let a triangle \( ABC \) be inscribed in the circle \( x^{2}-\sqrt{2}(x+y)+y^{2}=0 \) such that \( \angle BAC=\frac{\pi}{2} \). If the length of side \( AB \) is \( \sqrt{2} \), then the area of the \( \triangle ABC \) is equal to:

Let \( \dfrac{x-2}{3}=\dfrac{y+1}{-2}=\dfrac{z+3}{-1} \) lie on the plane \( px-qy+z=5 \), for some \( p,q\in\mathbb{R} \). The shortest distance of the plane from the origin is:

The distance of the origin from the centroid of the triangle whose two sides have the equations \( x - 2y + 1 = 0 \) and \( 2x - y - 1 = 0 \) and whose orthocenter is \( \left(\frac{7}{3}, \frac{7}{3}\right) \) is:

Let \( Q \) be the mirror image of the point \( P(1,2,1) \) with respect to the plane \( x+2y+2z=16 \). Let \( T \) be a plane passing through the point \( Q \) and contains the line \( \vec{r}=-\hat{k}+ \lambda(\hat{i}+\hat{j}+2\hat{k}), \ \lambda \in \mathbb{R} \). Then which of the following points lies on \( T \)?

Let \( A, B, C \) be three points whose position vectors respectively are
\( \vec{a}=\hat{i}+4\hat{j}+3\hat{k} \)
\( \vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k}, \ \alpha \in \mathbb{R} \)
\( \vec{c}=3\hat{i}-2\hat{j}+5\hat{k} \)
If \( \alpha \) is the smallest positive integer for which \( \vec{a}, \vec{b}, \vec{c} \) are non-collinear, then the length of the median, in \( \Delta ABC \), through \( A \) is:

The probability that a relation \( R \) from \( \{x,y\} \) to \( \{x,y\} \) is both symmetric and transitive, is equal to:

The number of values of \( a \in \mathbb{N} \) such that the variance of \( 3,7,12,a,43-a \) is a natural number is:

From the base of a pole of height \( 20 \, \text{meter} \), the angle of elevation of the top of a tower is \( 60^\circ \). The pole subtends an angle \( 30^\circ \) at the top of the tower. Then the height of the tower is:

Negation of the Boolean statement \( (p \vee q) \Rightarrow ((\sim r) \vee p) \) is equivalent to:

Let \( n \geq 5 \) be an integer. If \( 9^{n} - 8n - 1 = 64\alpha \) and \( 6^{n} - 5n - 1 = 25\beta \), then \( \alpha - \beta \) is equal to:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} + (\vec{b} \times \vec{c}) = \vec{0} \) and \( \vec{b} \cdot \vec{c} = 5 \). Then, the value of \( 3(\vec{c} \cdot \vec{a}) \) is equal to ______.

Let \( y = y(x),\; x>1 \), be the solution of the differential equation \( (x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1} \) with \( y(2) = \frac{1+e^{4}}{2e^{4}} \). If \( y(3) = \frac{e^{\alpha}+1}{\beta e^{\alpha}} \), then the value of \( \alpha + \beta \) is equal to ______.

Let \( 3, 6, 9, 12, \ldots \) upto \( 78 \) terms and \( 5, 9, 13, 17, \ldots \) upto \( 59 \) terms be two series. Then, the sum of the terms common to both the series is equal to ______.

The number of solutions of the equation \( \sin x = \cos^{2} x \) in the interval \( (0,10) \) is ______.

For real numbers \( a, b \) (\( a > b > 0 \)), let
\( \text{Area}\left\{(x,y): x^2 + y^2 \le a^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \ge 1 \right\} = 30\pi \) and \( \text{Area}\left\{(x,y): x^2 + y^2 \ge b^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \right\} = 18\pi \).
Then the value of \( (a-b)^2 \) is equal to ______.

Let \( f \) and \( g \) be twice differential even functions on \( (-2,2) \) such that \( f\!\left(\frac{1}{4}\right)=0 \), \( f\!\left(\frac{1}{2}\right)=0 \), \( f(1)=1 \) and \( g\!\left(\frac{3}{4}\right)=0 \), \( g(1)=2 \). Then, the minimum number of solutions of \( f(x)g''(x)+f'(x)g'(x)=0 \) in \( (-2,2) \) is equal to ______.

Let the coefficients of \( x^{-1} \) and \( x^{-3} \) in the expansion of \( \left(2x^{1/5}-\frac{1}{x^{1/5}}\right)^{15} \), \( x>0 \), be \( m \) and \( n \) respectively. If \( r \) is a positive integer such that \( mn^{2}=\,^{15}\!C_{r}\cdot 2^{r} \), then the value of \( r \) is equal to ______.

The total number of four digits numbers such that each of first three digits is divisible by the last digit, is equal to ______.

where \( \alpha \) is a non-zero real number an \( N=\displaystyle\sum_{k=1}^{49}M^{2k} \). If \( (I-M^{2})N=-2I \), then the positive integral values of \( \alpha \) is ______.

Let \( f(x) \) and \( g(x) \) be two real polynomials of degree \( 2 \) and \( 1 \) respectively. If \( f(g(x)) = 8x^2 - 2x \), and \( g(f(x)) = 4x^2 + 6x + 1 \), then the value of \( f(2) + g(2) \) is ________.

JEE Main 2022 29th June Evening Shift

Physics Chemistry Mathematics

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Let \( f:\mathbb{R}\to \mathbb{R} \) be a function defined by \( f(x)=(x-3)^{n_1}(x-5)^{n_2}, \, n_1,n_2 \in \mathbb{N} \). Then, which of the following is NOT true?