Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x\in\mathbb{R}$, $\displaystyle\int_{0}^{2} x F''(x),dx=6$ and $\displaystyle\int_{0}^{2} x^{2} F''(x),dx=40$, then $F'(2)+\displaystyle\int_{0}^{2} F(x),dx$ is equal to:

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Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 whit…

Topic: JEE Main 2025 (28 January Evening Shift)

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The area of the region bounded by the curves $x(1+y^{2})=1$ and $y^{2}=2x$ is:

Topic: JEE Main 2025 (28 January Evening Shift)

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If the midpoint of a chord of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is $\left(\sqrt{2},,\dfrac{4}{3}\right)…

Topic: JEE Main 2025 (28 January Evening Shift)

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The square of the distance of the point $\left(\dfrac{15}{7},,\dfrac{32}{7},,7\right)$ from the line $\dfrac{x+1}{3}=\d…

Topic: JEE Main 2025 (28 January Evening Shift)

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Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then t…

Topic: JEE Main 2025 (28 January Evening Shift)

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Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ …

Topic: JEE Main 2025 (28 January Evening Shift)

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If the components of $\vec a=\alpha,\hat i+\beta,\hat j+\gamma,\hat k$ along and perpendicular to $\vec b=3\hat i+\hat …

Topic: JEE Main 2025 (28 January Evening Shift)

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Let $f:\mathbb{R}-{0}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\l…

Topic: JEE Main 2025 (28 January Evening Shift)

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Let [x] denote the greatest integer less than or equal to x. Then the domain of $ f(x) = \sec^{-1}(2[x] + 1) $ is:

Topic: JEE Main 2025 (28 January Evening Shift)

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Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, o…

Topic: JEE Main 2025 (28 January Evening Shift)

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If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y…

Topic: JEE Main 2025 (28 January Evening Shift)

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If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+…

Topic: JEE Main 2025 (28 January Evening Shift)