NTA JEE Mains 29th Jan 2025 Shift 2

Let the area enclosed between the curves $$|y|= 1-x^{2}$$ and $$x^{2}+y^{2}=1$$ be $$\alpha$$. If $$9\alpha$$ = $$\beta \pi + \gamma; \beta,\gamma$$ are integers, then the value of $$|\beta - \gamma |$$ equals.

The remainder, when $$7^{103}$$ is divided by 23 , is equal to :

If $$\alpha x+ \beta y = 109$$ is the equation of the chord of the ellipse $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$, whose mid point is $$(\frac{5}{2},\frac{1}{2})$$ , then $$\alpha + \beta$$
is equal to :

If the domain of the function $$\log_{5}(18x - x^{2} - 77)$$ is $$(\alpha,\beta)$$ and the domain of the function $$\log_{(x-1)}\left(\frac{2x^{2}+3x-2}{x^{2}-3x-4}\right)$$ is $$(\gamma,\delta)$$, then $$\alpha^{2}+\beta^{2}+\gamma^{2}$$ is equal to :

Let a circle C pass through the points (4,2) and (0,2) , and its centre lie on 3x+2y+2=0. Then the length of the chord, of the circle C, whose mid-point is (1,2), is:

Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines $$\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$$ and $$\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}.$$ If the line L intersects the yz-plane at the point Q , then the distance between the points P and Q is :

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/45 , then n is equal to :

Let $$\alpha, \beta (\alpha \neq \beta)$$ be the values of m , for which the equations x + y + z = 1, x + 2y + 4z = m and x + 4y + 10z = $$m^{2}$$ have infinitely many solutions. Then the value of $$\sum_{n=1}^{10} \left(n^{\alpha} + n^{\beta}\right)$$ is equal to :

Let $$S = \mathbb{N} \cup \{0\}$$. Define a relation R from S to $$\mathbb{R}$$ by : $$R = \{(x,y) : \log_{e} y = x \log_e\left(\frac{2}{5}\right),\ x \in S,\ y \in \mathbb{R}\}$$ Then, the sum of all the elements in the range of $$\mathbb{R}$$ is equal to :

If $$\sin x + \sin^2 x = 1$$, $$x \in (0, \tfrac{\pi}{2})$$ then $$(\cos^{12}x+\tan^{12}x)+3(\cos^{10}x+\tan^{10}x+\cos^{8}x+\tan^{8} x)+(\cos^{6}x+\tan^{6}x)$$
is equal to :