Three defective oranges are accidentally mixed with seven good ones and, on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denotes the number of defective oranges, then the variance of $x$ is

1

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)

2

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)

3

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)

4

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)

5

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

Topic: JEE Main 2025 (28 January Morning Shift)

6

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)

7

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)

8

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)

9

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…

Topic: JEE Main 2025 (28 January Morning Shift)

10

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

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)

2

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)

3

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)

4

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)

5

The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$, s…

Topic: JEE Main 2025 (28 January Morning Shift)

6

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)

7

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)

8

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)

9

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)