NTA JEE Mains 28th Jan 2025 Shift 1

Let $$A(x,y,Z)$$ be a point in $$xy-plain$$,which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, −1) and C = (2, 0, −2). Then among the statements $$(SI): \triangle ABC$$ is an isosceles right angled triangle, and (SI):the area of $$\triangle ABC$$ is $$\frac{9\sqrt{2}}{2}$$,

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

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

Let $$T_{r}$$ be the $$r^{th}$$ term of an A.P. If for some m,$$T_{m}=\frac{1}{25},T_{25}=\frac{1}{25}$$, and $$20\sum_{r=1}^{25}T_{r}=13$$,then $$5m\sum_{r=m}^{2m}T_{r}$$ is equal to

Three defective oranges are accidently 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 $$x$$ denote the number of defective oranges, then the variance of $$x$$ is

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

If $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{96x^{2}\cos^{2}x}{(1+e^{x})}dx=\pi (\alpha \pi^{2}+\beta),\alpha,\beta \in \mathbb{Z}$$,then $$(\alpha +\beta)^{2}$$ equals

Let $$\langle a_{n}\rangle$$ be a sequence such that $$a_{0}=0,a_{1}=\frac{1}{2}$$ and $$2a_{n+2}=5a_{n+1}-3a_{n},n=0,1,2,3,....$$ Then $$\sum_{k=1}^{100}a_{k}$$ is equal to

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 should not be more than 8 , is

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