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For \( \alpha, \beta \in \) \(\mathbb{R}\) and a natural number \( n \), let . Then \( 2A_{10} - A_8 \) is:

A circle is inscribed in an equilateral triangle of side of length \( 12 \). If the area and perimeter of any square inscribed in this circle are \( m \) and \( n \), respectively, then \( m + n^2 \) equal to:

Let the relations \( R_1 \) and \( R_2 \) on the set \( X = \{1,2,3,\ldots,20\} \) be given by \( R_1 = \{(x,y): 2x - 3y = 2\} \) and \( R_2 = \{(x,y): -5x + 4y = 0\} \). If \( M \) and \( N \) be the minimum number of elements required to be added in \( R_1 \) and \( R_2 \), respectively in order to make the relations symmetric, then \( M + N \) equals.

If \( A(3,1,-1), B\!\left(\frac{5}{3},\frac{7}{3},\frac{1}{3}\right) \), \( C(2,2,1) \) and \( D\!\left(\frac{10}{3},\frac{2}{3},-\frac{1}{3}\right) \) are the vertices of a quadrilateral \( ABCD \), then its area is:

Let \( f : (-\infty, \infty) - \{0\} \to \) \(\mathbb{R}\) be a differentiable function such that \( f'(1) = \lim_{a \to \infty} a^2 f\!\left(\frac{1}{a}\right) \). Then \( \lim_{a \to \infty} \left[ \dfrac{a(a+1)}{2} \tan^{-1}\!\left(\dfrac{1}{a}\right) \right] + a^2 - 2 \log_e a \) is equal to:

The interval in which the function \( f(x) = x^x, \ x > 0 \), is strictly increasing is:

Let \( \alpha, \beta \) be the distinct roots of the equation \( x^2 - (t^2 - 5t + 6)x + 1 = 0, \; t \in \mathbb{R} \) and \( a_n = \alpha^n + \beta^n \). Then the minimum value of \( \dfrac{a_{2023} + a_{2025}}{a_{2024}} \) is:

\( \displaystyle \int_{0}^{\pi/4} \frac{\cos^2 x \,\sin^2 x}{\left(\cos^3 x + \sin^3 x\right)^2}\, dx \) is equal to:

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is:

Let the area of the region enclosed by the curves \( y = 3x \), \( 2y = 27 - 3x \) and \( y = 3x - x\sqrt{x} \) be \( A \). Then \( 10A \) is equal to:

Let a variable line of slope \( m > 0 \) passing through the point \( (4,-9) \) intersect the coordinate axes at the points \( A \) and \( B \). The minimum value of the sum of the distance of \( A \) and \( B \) from the origin is:

Let \( y = y(x) \) be the solution of the differential equation \( (2x \log_e x)\,\frac{dy}{dx} + 2y = \frac{3}{x}\log_e x ,\ x>0 \) and \( y(e^{-1}) = 0 \). Then, \( y(e) \) is equal to:

A company has two plants \( A \) and \( B \) to manufacture motorcycles. 60% motorcycles are manufactured at plant \( A \) and the remaining are manufactured at plant \( B \). 80% of the motorcycle manufactured at plant \( A \) are rated of the standard quality, while 90% of the motorcycles manufactured at plant \( B \) are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If \( p \) is the probability that it was manufactured at plant \( B \), then \( 126p \) is:

The shortest distance between the lines \( \frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \) and \( \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3} \) is:

Let $A = \{n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of } 3 \text{ nor a multiple of } 4\}$. Then the number of elements in $A$ is:

The function \( f(x)=\frac{x^{2}+2x-15}{x^{2}-4x+9},\ x\in \mathbb{R} \) is:

The mean and standard deviation of \( 20 \) observation are found to be \( 10 \) and \( 2 \), respectively. On rechecking, it was found that an observation by mistake was taken \( 8 \) instead of \( 12 \). The correct standard deviation is:

Let \( y = y(x) \) be the solution of the differential equation \( (1 + x^2)\frac{dy}{dx} + y = e^{\tan^{-1} x}, \; y(1) = 0 \). Then \( y(0) \) is:

Let \( C \) be the circle of minimum area touching the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3}\,|x| \). Then, which one of the following points lies on the circle \( C \)?

Let \( L_1, L_2 \) be the lines passing through the point \( P(0,1) \) and touching the parabola \( 9x^2 + 12x + 18y - 14 = 0 \). Let \( Q \) and \( R \) be the points on the lines \( L_1 \) and \( L_2 \) such that the \( \triangle PQR \) is an isosceles triangle with base \( QR \). If the slopes of the lines \( QR \) are \( m_1 \) and \( m_2 \), then \( 16\left(m_1^2 + m_2^2\right) \) is equal to ____.

Let a conic \( C \) pass through the point \( (4,-2) \) and \( P(x,y) \), \( x \ge 3 \), by any point on \( C \). Let the slope of the line touching the conic \( C \) only at a single point \( P \) be half the slope of the line joining the points \( P \) and \( (3,-5) \). If the focal distance of the point \( (7,1) \) on \( C \) is \( d \), then \( 12d \) equals:

Let \( \alpha \beta \gamma = 45; \alpha, \beta, \gamma \in \; \)\(\mathbb{R}\). If \( x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0) \) for some \( x, y, z \in \; \)\(\mathbb{R}\), \( xyz \ne 0 \), then \( 6\alpha + 4\beta + \gamma \) is equal to __________.

Let \( P \) be the point \( (10, -2, -1) \) and \( Q \) be the foot of the perpendicular drawn from the point \( R(1, 7, 6) \) on the line passing through the points \( (2, -5, 11) \) and \( (-6, 7, -5) \). Then the length of the line segment \( PQ \) is equal to:

For \( n \in \)\(\mathbb{N}\) , if \( \cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \frac{\pi}{4} \), then \( n \) is equal to ________.

Let the first term of a series be \( T_1 = 6 \) and its \( r^{\text{th}} \) term \( T_r = 3T_{r-1} + 6^r, \; r = 2, 3, \ldots, n \). If the sum of the first \( n \) terms of this series is \( \frac{1}{5}\left(n^2 - 12n + 39\right)\!\left(4\cdot 6^n - 5\cdot 3^n + 1\right) \), then \( n \) is equal to ________.

Let \( \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \), \( \vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k} \) and a vector \( \vec{c} \) be such that \( \vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k} \). If \( \vec{a} \cdot \vec{c} = 13 \), then \( \left(24 - \vec{b} \cdot \vec{c}\right) \) is equal to ________.

If the second, third and fourth terms in the expansion of \( (x + y)^n \) are \( 135 \), \( 30 \) and \( \frac{10}{3} \), respectively, then \( 6(n^3 + x^2 + y) \) is equal to ______.

Let $r_k = \frac{\int_{0}^{1}\left(1-x^7\right)^k dx}{\int_{0}^{1}\left(1-x^7\right)^{k+1} dx}, k \in \mathbb{N}$. Then the value of $\displaystyle \sum_{k=1}^{10}\frac{1}{7\left(r_k-1\right)}$ is equal to:

Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \( 4x^{4} + 8x^{3} - 17x^{2} - 12x + 9 = 0 \) and \( \left(4 + x_{1}^{2}\right) \left(4 + x_{2}^{2}\right) \left(4 + x_{3}^{2}\right) \left(4 + x_{4}^{2}\right) = \dfrac{125}{16}\, m \). Then the value of \( m \) is:

JEE Main 2024 6th April Morning Shift

Mathematics Physics Chemistry

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If the second, third and fourth terms in the expansion of \( (x + y)^n \) are \( 135 \), \( 30 \) and \( \frac{10}{3} \), respectively, then \( 6(n^3 + x^2 + y) \) is equal to ______.