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Let \( A = \{ x \in R : |x + 1| < 2 \} \) and \( B = \{ x \in R : |x - 1| \ge 2 \} \). Then which one of the following statements is NOT true?

Let \( a, b \in R \) be such that the equation \( a x^{2} - 2 b x + 15 = 0 \) has a repeated root \( \alpha \). If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2} - 2 b x + 21 = 0 \), then \( \alpha^{2} + \beta^{2} \) is equal to:

Let \( z_{1} \) and \( z_{2} \) be two complex numbers such that \( \bar{z}_{1} = i \bar{z}_{2} \) and \( \arg\!\left(\frac{z_{1}}{z_{2}}\right) = \pi \). Then

The system of equation
\( -k x + 3y - 14z = 25 \)
\( -15x + 4y - k z = 3 \)
\( -4x + y + 3z = 4 \)
is consistent for all \( k \) in the set

\( \displaystyle \lim_{x \to \frac{\pi}{2}} \tan^{2} x \left( \left(2 \sin^{2} x + 3 \sin x + 4 \right)^{\frac{1}{2}} - \left(\sin^{2} x + 6 \sin x + 2 \right)^{\frac{1}{2}} \right) \) is equal to

The area of the region enclosed between the parabolas \( y^{2} = 2x - 1 \) and \( y^{2} = 4x - 3 \) is

The coefficient of \( x^{101} \) in the expression \( (5 + x)^{500} + x(5 + x)^{499} + x^{2}(5 + x)^{498} + \ldots + x^{500}, \, x > 0, \) is

The sum \( 1 + 2 \cdot 3 + 3 \cdot 3^{2} + \ldots + 10 \cdot 3^{9} \) is equal to:

Let \( P \) be the plane passing through the intersection of the planes \( \vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 5 \) and \( \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 3 \), and the point \( (2, 1, -2) \). Let the position vectors of the points \( X \) and \( Y \) be \( \hat{i} - 2\hat{j} + 4\hat{k} \) and \( 5\hat{i} - \hat{j} + 2\hat{k} \) respectively. Then the points

A circle touches both the y-axis and the line \( x + y = 0 \). Then the locus of its center is:

Water is being filled at the rate of \( 1 \, \text{cm}^{3} / \text{sec} \) in a right circular conical vessel (vertex downwards) of height \( 35 \, \text{cm} \) and diameter \( 14 \, \text{cm} \). When the height of the water level is \( 10 \, \text{cm} \), the rate (in \( \text{cm}^{2} / \text{sec} \)) at which the wet conical surface area of the vessel increases is

If \( b_n = \int_{0}^{\pi/2} \frac{\cos^2(nx)}{\sin x} \, dx , \, n \in \mathbb{N} , \) then

If \( y=y(x) \) is the solution of the differential equation \( 2x^{2}\frac{dy}{dx}-2xy+3y^{2}=0 \) such that \( y(e)=\frac{e}{3} \), then \( y(1) \) is equal to

$If\ the\ angle\ made\ by\ the\ tangent\ at\ the\ point\ (x_0, y_0)\ on\ the\ curve$ $x = 12\Big(t + \sin t \cos t\Big), y = 12\big(1 + \sin t\big)^2, 0 < t < \frac{\pi}{2},\ with\ the\ positive\ x\text{-axis}\ is\ \frac{\pi}{3},\ then\ y_0\ is\ equal\ to\ :$

The value of \( 2\sin(12^{\circ})-\sin(72^{\circ}) \) is:

A biased die is marked with numbers \( 2,4,8,16,32,32 \) on its faces and the probability of getting a face with mark \( n \) is \( \frac{1}{n} \). If the die is thrown thrice, then the probability, that the sum of the numbers obtained is \( 48 \), is:

The negation of the Boolean expression \( \big((\sim q)\land p\big)\Rightarrow \big((\sim p)\lor q\big) \) is logically equivalent to:

If the line \( y = 4 + kx, \; k > 0 \), is the tangent to the parabola \( y = x - x^2 \) at the point \( P \) and \( V \) is the vertex of the parabola, then the slope of the line through \( P \) and \( V \) is:

The value of \( \tan^{-1}\!\left( \frac{\cos\!\left( \frac{15\pi}{4} \right)}{\sin\!\left( \frac{\pi}{4} \right)} - 1 \right) \) is equal to:

The line \( y = x + 1 \) meets the ellipse \( \frac{x^2}{4} + \frac{y^2}{2} = 1 \) at two points \( P \) and \( Q \). If \( r \) is the radius of the circle with \( PQ \) as diameter then \( (3r)^2 \) is equal to:

Then the number of elements in the set \( \{(n,m): n,m \in \{1,2,\ldots,10\} \text{ and } nA^n + mB^m = I \} \) is ________.

, where $[t]$ is the greatest integer $\leq t$. Then, in the open interval $(-1, 1)$, the number of points where $fog$ is discontinuous is equal to __________.

The value of \( b > 3 \) for which \( 12 \int_{3}^{b} \frac{1}{(x^2 - 1)(x^2 - 4)} \, dx = \log_e \!\left( \frac{49}{40} \right) \) is equal to ________.

If the sum of the co-efficients of all the positive even powers of \( x \) in the binomial expansion of \( \left( 2x^3 + \frac{3}{x} \right)^{10} \) is \( 5^{10} - \beta \cdot 3^9 \), then \( \beta \) is equal to __________.

If the mean deviation about the mean of the numbers \( 1, 2, 3, \ldots, n \), where \( n \) is odd, is \( \frac{5(n+1)}{n} \), then \( n \) is equal to __________.

Let $\vec{b} = \hat{i} + \hat{j} + \lambda \hat{k}, \lambda \in \mathbb{R}$. If $\vec{a}$ is a vector such that $\vec{a} \times \vec{b} = 13\hat{i} - \hat{j} - 4\hat{k}$ and $\vec{a} \cdot \vec{b} + 21 = 0$, then $(\vec{b} - \vec{a}) \cdot (\hat{k} - \hat{j}) + (\vec{b} + \vec{a}) \cdot (\hat{i} - \hat{k})$ is equal to ____________.

The total number of three-digit numbers, with one digit repeated exactly two times, is __________.

Let \( f(x)=|(x-1)|(x^2-2x-3)|+x-3, \ x\in \mathbb{R} \). If \( m \) and \( M \) are respectively the number of points of local minimum and local maximum of \( f \) in the interval \( (0,4) \), then \( m+M \) is equal to __________.

Let the eccentricity of the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be \( \frac{5}{4} \). If the equation of the normal at the point \( \left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right) \) on the hyperbola is \( 8\sqrt{5}\,x+\beta y=\lambda \), then \( \lambda-\beta \) is equal to __________.

Let \( l_1 \) be the line in \( xy \)-plane with \( x \) and \( y \) intercepts \( \frac{1}{8} \) and \( \frac{1}{4\sqrt{2}} \) respectively, and \( l_2 \) be the line in \( zx \)-plane with \( x \) and \( z \) intercepts \( -\frac{1}{8} \) and \( -\frac{1}{6\sqrt{3}} \) respectively. If \( d \) is the shortest distance between the line \( l_1 \) and \( l_2 \), then \( d^{-2} \) is equal to __________.

JEE Main 2022 25th June Evening Shift

Physics Chemistry Mathematics

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The system of equation
\( -k x + 3y - 14z = 25 \)
\( -15x + 4y - k z = 3 \)
\( -4x + y + 3z = 4 \)
is consistent for all \( k \) in the set