If $f(x)=\dfrac{2^x}{,2^x+\sqrt{2},},; x\in\mathbb{R}$, then $\displaystyle \sum_{k=1}^{81} f!\left(\dfrac{k}{82}\right)$ is equal to

1

et $A(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $(0,3,2)$, $(2,0,3)$ and $(0,0,1)$. Let …

Topic: JEE Main 2025 (28 January Morning Shift)

1

The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is

Topic: JEE Main 2025 (28 January Morning Shift)

2

The relation $R={(x,y): x,y\in\mathbb{Z}\ \text{and}\ x+y\ \text{is even}}$ is:

Topic: JEE Main 2025 (28 January Morning Shift)

3

Let $(a_n)$ be a sequence such that $a_0=0$, $a_1=\dfrac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_n,; n=0,1,2,\ldots$. Then $\d…

Topic: JEE Main 2025 (28 January Morning Shift)

4

$\displaystyle \cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$ is equal to:

Topic: JEE Main 2025 (28 January Morning Shift)

5

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos ^2 x}{\left(1+e^x\right)} \mathrm{d} x=\pi\left(\alpha \pi^…

Topic: JEE Main 2025 (28 January Morning Shift)

6

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\m…

Topic: JEE Main 2025 (28 January Morning Shift)

7

Let $\binom{n}{r-1}=28$, $\binom{n}{r}=56$ and $\binom{n}{r+1}=70$. Let $A(4\cos t,,4\sin t)$, $B(2\sin t,,-2\cos t)$ a…

Topic: JEE Main 2025 (28 January Morning Shift)

8

Two numbers $k_{1}$ and $k_{2}$ are randomly chosen from the set of natural numbers. Then, the probability that the val…

Topic: JEE Main 2025 (28 January Morning Shift)

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Three defective oranges are accidentally mixed with seven good ones and, on looking at them, it is not possible to diff…

Topic: JEE Main 2025 (28 January Morning Shift)

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Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^{2}=4x$. Let the sides $AD$ and $BC$ of the trapezium b…

Topic: JEE Main 2025 (28 January Morning Shift)