The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$, such that the sum of their first and last digits is not more than $8$, is

1

The sum of all local minimum values of the function $\mathrm{f}(x)=\left\{\begin{array}{lr} 1-2 x, & x2 \end{array}\rig…

Topic: JEE Main 2025 (28 January Morning Shift)

2

Let, for some function $y=f(x)$, $\displaystyle \int_{0}^{x} t,f(t),dt = x^{2}f(x)$ for $x>0$ and $f(2)=3$. Then $f(6)$…

Topic: JEE Main 2025 (28 January Morning Shift)

3

If the image of the point $(4,4,3)$ in the line $\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-1}{3}$ is $(\alpha,\beta,\gamma…

Topic: JEE Main 2025 (28 January Morning Shift)

4

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)

5

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)

6

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)

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

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)

9

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)

10

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

Topic: JEE Main 2025 (28 January Morning Shift)

1

Let the equation of the circle, which touches $x$-axis at the point $(a,0)$, $a>0$, and cuts off an intercept of length…

Topic: JEE Main 2025 (28 January Morning Shift)

2

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=(2+3a)x^{2}+\dfrac{a+2}{a-1}x+b$, $a\ne1$. If $f(x+y)=f(…

Topic: JEE Main 2025 (28 January Morning Shift)

3

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2\sqrt{2},i$, the point $B$ $(z_2)$ be such that $\sqrt{3},|z_2|=…

Topic: JEE Main 2025 (28 January Morning Shift)

4

The area (in sq. units) of the region ${(x,y): 0\le y\le 2|x|+1,; 0\le y\le x^{2}+1,; |x|\le 3}$ is

Topic: JEE Main 2025 (28 January Morning Shift)