NTA JEE Mains 22nd Jan 2025 Shift 2

If $$\lim_{x \rightarrow \infty}((\frac{e}{1-e})(\frac{1}{e}-\frac{x}{1+x}))^{x}=\alpha$$ then the value of $$\frac{\log_{e}^{\alpha}}{1+\log_{e}^{\alpha}}$$ equals :

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

Suppose that the number of terms in an A.P is $$2k, k \in N$$. 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 of the A.P. exceeds the first term by 27, then k is equal to :

The perpendicular distance, of the line $$\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$$ from the point P(2,−10, 1), is :

If the system of linear equations : $$x+y+2z=6\\2x+3y+az=a+1\\-x-3y+bz=2b$$ where $$a,b \in R$$, has infinitely many solutions, then 7a + 3b is equal to :

If x = f(y) is the solution of the differential equation $$\left(1+y^{2}\right)+\left(x-2e^{\tan^{-1}y}\right)\frac{dy}{dx}=0,y \in (-\frac{\pi}{2},\frac{\pi}{2})$$ with f(0) = 1, then $$f(\frac{1}{\sqrt{3}})$$ is equal to :

Let $$\alpha_{\theta}$$ anf $$\beta_{\theta}$$ be the distinct roots of $$2x^{2}+(\cos \theta)x-1=0,\theta \in (0,2\pi)$$. If m and M are the minimum and the maximum values of $$\alpha_{\theta}^{4}+\beta_{\theta}^{4}$$, then 16(M+m) equals :

The sum of all values of $$\theta \in [0,2\pi]$$ satisfying $$2\sin^{2}\theta =\cos2\theta \text{ and }2\cos^{2}\theta =3\sin\theta$$ is

Let the curve $$z(1+i)+\overline{z}(1-i)=4,z \in C$$,divide the region $$|z-3|\leq 1$$ into two parts of areas $$\alpha$$ and $$\beta$$. Then $$|\alpha - \beta |$$ equals:

Let $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,a > b$$ and $$H: \frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1$$.Let the distance between the foci of E and the foci of H be $$2sqrt{3}$$. If a-A=2, and the ratio of the eccentricities of E and H is $$\frac{1}{3}$$, then the sum of the lengths of their latus rectums is equal to: