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If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \( i^{\text{th}} \) roll than the number obtained in the \( (i-1)^{\text{th}} \) roll, \( i = 2, 3 \) is equal to

The integral \(\displaystyle \int_{1/4}^{3/4} \, \cos\!\left( 2\cot^{-1}\!\sqrt{\frac{1-x}{1+x}} \right) \, dx \) is equal to

The value of the integral \(\displaystyle\int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) dx \) is

and \( A \) be a \( 2 \times 2 \) matrix such that \( AB^{-1} = A^{-1} \). If \( BCB^{-1} = A \) and \( C^{4} + \alpha C^{2} + \beta I = 0 \), then \( 2\beta - \alpha \) is equal to

Let the range of the function \( f(x) = \dfrac{1}{2 + \sin 3x + \cos 3x}, \; x \in \mathbb{R} \) be \( [a, b] \). If \( \alpha \) and \( \beta \) are respectively the A.M. and the G.M. of \( a \) and \( b \), then \( \dfrac{\alpha}{\beta} \) is equal to

Let \( z \) be a complex number such that the real part of \( \dfrac{z-2i}{z+2i} \) is zero. Then, the maximum value of \( \lvert z - (6 + 8i) \rvert \) is equal to

Consider the line \( L \) passing through the points \( (1, 2, 3) \) and \( (2, 3, 5) \). The distance of the point \(\displaystyle \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) \) from the line \( L \) along the line \(\displaystyle \frac{3x-11}{2} = \frac{3y-11}{1} = \frac{3z-19}{2} \) is equal to

Let \(\displaystyle \int_{0}^{x}\sqrt{1-\left(y'(t)\right)^{2}}\,dt=\int_{0}^{x}y(t)\,dt,\ 0\le x\le 3,\ y\ge 0,\ y(0)=0 \). Then at \( x=2 \), \( y''+y+1 \) is equal to

Let the foci of a hyperbola \( H \) coincide with the foci of the ellipse \(\displaystyle E:\ \frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1 \) and the eccentricity of the hyperbola \( H \) be the reciprocal of the eccentricity of the ellipse \( E \). If the length of the transverse axis of \( H \) is \( \alpha \) and the length of its conjugate axis is \( \beta \), then \( 3\alpha^{2}+2\beta^{2} \) is equal to

The area (in square units) of the region enclosed by the ellipse \( x^{2}+3y^{2}=18 \) in the first quadrant below the line \( y=x \) is

Let \( a, ar, ar^{2},\ldots \) be an infinite G.P. If \( \sum\limits_{n=0}^{\infty}ar^{n}=57 \) and \( \sum\limits_{n=0}^{\infty}a^{3}r^{3n}=9747 \), then \( a+18r \) is equal to

The sum of the coefficient of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of \(\displaystyle \left(x^{2/3}+\frac{1}{2}x^{-2/5}\right)^{9} \) is

Between the following two statements:
Statements I : Let \( \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). Then the vector \( \vec{r} \) satisfying \( \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \) and \( \vec{a} \cdot \vec{r} = 0 \) is of magnitude \( \sqrt{10} \).
Statement II : In a triangle \( ABC \), \( \cos 2A + \cos 2B + \cos 2C \ge -\frac{3}{2} \).

\( \displaystyle \lim_{x\to\frac{\pi}{2}} \left( \dfrac{\displaystyle\int_{x^{3}}^{\left({x}/{2}\right)^{3}} \left(\sin\!\left(2t^{1/3}\right)+\cos\!\left(t^{1/3}\right)\right)\\,dt}{\left(x-\frac{\pi}{2}\right)^{2}} \right) \) is equal to

Two vertices of a triangle \( ABC \) are \( A(3,-1) \) and \( B(-2,3) \), and its orthocenter is \( P(1,1) \). If the coordinates of the point \( C \) are \( (\alpha,\beta) \) and the centre of the circle circumscribing the triangle \( PAB \) is \( (h,k) \), then the value of \( (\alpha+\beta)+2(h+k) \) equals

Let \( \alpha,\beta;\alpha>\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x-\sqrt{3}=0 \). Let \( P_{n}=\alpha^{n}-\beta^{n},\, n\in\mathbb{N} \). Then \( \left(11\sqrt{3}-10\sqrt{2}\right)P_{10}+\left(11\sqrt{2}+10\right)P_{11}-11P_{12} \) is equal to

If \( \log_{e} y = 3\sin^{-1}x \), then \( (1-x^{2})y''-xy' \) at \( x=\dfrac{1}{2} \) is equal to

Let \( \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \ \vec{b} = -\hat{i} + \hat{k}, \ \vec{c} = \beta \hat{j} - \hat{k} \), where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha , \beta) \), for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_{1}, \beta_{1}) \) and \( (\alpha_{2}, \beta_{2}) \). Then \( \alpha_{1}^{2} + \beta_{1}^{2} - \alpha_{2}\beta_{2} \) is equal to

\( \displaystyle \lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{x}}}{x} \) is equal to

If the variance of the frequency distribution

is \( 160 \), then the value of \( c \in \mathbb{N} \) is

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \( 2\sin^{-1}x + 3\cos^{-1}x = \frac{2\pi}{5} \), is _________.

If \(\displaystyle \left( \frac{1}{\alpha+1} + \frac{1}{\alpha+2} + \cdots + \frac{1}{\alpha+1012} \right) - \left( \frac{1}{2\cdot 1} + \frac{1}{4\cdot 3} + \frac{1}{6\cdot 5} + \cdots + \frac{1}{2024\cdot 2023} \right) = \frac{1}{2024} \), then \( \alpha \) is equal to _________.

Let the set of all values of \( p \), for which \( f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \) does not have any critical point, be the interval \( (a, b) \). Then \( 16ab \) is equal to ________.

Let \( A \), \( B \) and \( C \) be three points on the parabola \( y^2 = 6x \) and let the line segment \( AB \) meet the line \( L \) through \( C \) parallel to the \( x \)-axis at the point \( D \). Let \( M \) and \( N \) respectively be the feet of perpendiculars from \( A \) and \( B \) on \( L \). Then \( \left(\frac{AM \cdot BN}{CD}\right)^2 \) is equal to ________.

For a differentiable function \( f:\mathbb{R} \to \mathbb{R} \), suppose \( f'(x) = 3f(x) + \alpha \), where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \) and \(\displaystyle \lim_{x \to -\infty} f(x) = 7 \). Then \( 9 f(\ln 3) \) is equal to ________.

The number is integers, between \( 100 \) and \( 1000 \) having the sum of their digits equals to \( 14 \), is ________.

Consider the circle \( C: x^2 + y^2 = 4 \) and the parabola \( P: y^2 = 8x \). If the set of all values of \( \alpha \), for which three chords of the circle \( C \) on three distinct lines passing through the point \( (\alpha, 0) \) are bisected by the parabola \( P \) is the interval \( (p, q) \), then \( (2q - p)^2 \) is equal to ________.

Let \( A = \{(x,y): 2x + 3y = 23, \, x,y \in \mathbb{N}\} \) and \( B = \{ x : (x,y) \in A \} \). Then the number of one-one functions from \( A \) to \( B \) is equal to ________.

Consider the matrices:
. Let the set of all \( m \), for which the system of equations \( AX=B \) has a negative solution (i.e., \( x<0 \) and \( y<0 \)), be the interval \( (a,b) \). Then \(\displaystyle 8\int_{a}^{b}\lvert A\rvert\,dm \) is equal to ________.

The square of the distance of the image of the point \( (6,1,5) \) in the line \( \dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4} \), from the origin is_________.

JEE Main 2024 9th April Evening Shift

Mathematics Physics Chemistry

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If the variance of the frequency distribution

is \( 160 \), then the value of \( c \in \mathbb{N} \) is