Let $f:\mathbb{R}-{0}\to\mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}.$ If $\displaystyle \lim_{x\to 0}\left(\frac{1}{x} + f(x)\right) = \beta,\ \alpha, \beta \in \mathbb{R},$ then $\alpha + 2\beta$ is equal to :

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If $\displaystyle \lim_{x \to 0} \csc x \left( \sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4} \right)$ is :

Topic: JEE Main 2025 (24 January Morning Shift)

1

The product of all the rational roots of the equation $ (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 $ is equal to

Topic: JEE Main 2025 (24 January Morning Shift)

2

For some $ n \ne 10 $, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $ (1 + x)^{n+4} …

Topic: JEE Main 2025 (24 January Morning Shift)

3

Let $f(x) = \dfrac{9x^2 + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \big( f\left(\dfrac{1}{15}\right) + f\lef…

Topic: JEE Main 2025 (24 January Morning Shift)

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If $I(m,n) = \displaystyle \int_0^1 x^{m-1}(1-x)^{n-1} dx, ; m,n > 0$, then $I(9,14) + I(10,13)$ is

Topic: JEE Main 2025 (24 January Morning Shift)

5

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text…

Topic: JEE Main 2025 (24 January Morning Shift)

6

Let the product of the focal distances of the point $\left( \sqrt{3}, \dfrac{1}{2} \right)$ on the ellipse $\dfrac{x^2}…

Topic: JEE Main 2025 (24 January Morning Shift)

7

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1 + x^2}\right)dx + (1 + x^2)dy = 0$,…

Topic: JEE Main 2025 (24 January Morning Shift)

8

Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie…

Topic: JEE Main 2025 (24 January Morning Shift)

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Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \d…

Topic: JEE Main 2025 (24 January Morning Shift)

10

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $5$ before $B$ throws a sum of $8$, and $B…

Topic: JEE Main 2025 (24 January Morning Shift)