NTA JEE Mains 22nd Jan 2025 Shift 1

Let $$a_1,a_2,a_3,...$$ be a G.P. of increasing positive terms. If $$a_1a_5 = 28$$ and $$a_2+a_4 = 29$$, then $$a_6$$ is equal to:

Let $$x=x(y)$$ be the solution of the differential equation$$ y^{2}dx+\left( x-\frac{1}{y}\right)dy=0 $$ . If $$x(1)=1$$, then $$\frac{1}{2}$$ is :

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $$\frac{m}{n}$$, where $$gcd(m,n)=1$$, then $$m+n$$ is equal to :

The product of all solutions of the equation $$ e^{5(\log_e x)^{2}+3}=x^{8},x>0 $$, is :

Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line $$x+2y=2$$. If the centroid of $$ \triangle PQR $$ is the point $$ (\alpha, \beta) $$, then $$ 15(\alpha - \beta) $$ is equal to :

$$ \text{Let for }f(x)=7 \tan^{8}x + 7\tan^{6}x-3\tan^{4}x-3\tan^{2}x \text{ } I_1=\int_{0}^{\pi/4}f(x)dx \text{ and }I_2=\int_{0}^{\pi/4}xf(x)dx. \text{ Then } 7I_1+12T_2 \text{ is equal to :} $$

Let the parabola $$ y=x^{2}+px-3 $$, meet the coordinate axes at the points P, Q and R . If the circle C with centre at (-1, -1) passes through the points P, Q and R, then the area of $$ \triangle PQR $$ is :

Let $$ L_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \text{ and } L_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5} $$ be two lines. Then which of the following points lies on the line of the shortest distance between $$L_1 \text{ and } L_2 $$ ?

$$ \text{Let } f(x) \text{be a real differentiable function such that } f(0)=1 \text{ and } f(x+y)=f(x)f^{'}(y)+f^{'}(x)f(y) \text{ for all } x,y \in \mathbb{R}. \text{ Then } \sum_{n=1}^{100} \log_e f(n) \text{ is equal to :} $$

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :