Suppose that the number of terms in an A.P. is $2k$, $k\in\mathbb{N}$. If the sum of all odd terms of the A.P. is $40$, the sum of all even terms is $55$ and the last term exceeds the first term by $27$, then $k$ is equal to:

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Let the curve $z(1+i)+\overline{z}(1-i)=4,\ z\in\mathbb{C}$, divide the region $|z-3|\le 1$ into two parts of areas $\a…

Topic: JEE Main 2025 (22 January Evening Shift)

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Let $A=\{1,2,3,4\}$ and $B=\{1,4,9,16\}$. Then the number of many-one functions $f:A\to B$ such that $1\in f(A)$ is equ…

Topic: JEE Main 2025 (22 January Evening Shift)

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Let $f(x)=\displaystyle \int_{0}^{e^{x^{2}}}\frac{t^{2}-8t+15}{e^{t}}\,dt,\ x\in\mathbb{R}$. Then the numbers of loca…

Topic: JEE Main 2025 (22 January Evening Shift)

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If $\displaystyle \lim_{x\to\infty}\left(\frac{e}{1-e}\left(\frac{1}{e}-\frac{x}{1+x}\right)\right)^{x}=\alpha$, then t…

Topic: JEE Main 2025 (22 January Evening Shift)

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In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys …

Topic: JEE Main 2025 (22 January Evening Shift)

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The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is :

Topic: JEE Main 2025 (22 January Evening Shift)

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Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a…

Topic: JEE Main 2025 (22 January Evening Shift)

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If $\displaystyle \int e^{x}\!\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{\sin^{-1}x}{(1-x^{2})^{3/2}}+\frac{x}{1-x^…

Topic: JEE Main 2025 (22 January Evening Shift)

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For a $3\times3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ …

Topic: JEE Main 2025 (22 January Evening Shift)

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If $x=f(y)$ is the solution of the differential equation $(1+y^{2})+\big(x-2e^{\tan^{-1}y}\big)\dfrac{dy}{dx}=0,\ y\in\…

Topic: JEE Main 2025 (22 January Evening Shift)

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Let $P(4,4\sqrt{3})$ be a point on the parabola $y^{2}=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ a…

Topic: JEE Main 2025 (22 January Evening Shift)

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Let $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$ and $H:\ \dfrac{x^{2}}{A^{2}}-\dfrac{y^{2}}{B^{2}}=1$. Let …

Topic: JEE Main 2025 (22 January Evening Shift)