NTA JEE Mains 22nd Jan 2025 Shift 1

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $$16\left(\left(\sec^{-1}x\right)^{2}\left(\cosec^{-1}x\right)^{2}\right) \text{is :} $$

$$ \text{Let }f: \mathbb{R} \rightarrow \mathbb{R} \text{ be a twice differentiable function such that } f(x+y)=f(x)f(y)\text{ for all } x,y \in R. \text{ If } f^{'}(0)=4a \text{ and } f \text{ satisfies } f^{''}(x)-3af^{'}(x)=0,a>0, \text{then the area of the region } R= \left\{(x,y) \mid 0\leq y\leq f(ax), 0\leq x \leq2 \right\}$$

$$ \text{The area of the region, inside the circle }(x-2\sqrt{3})^{2}+y^{2}=12 \text{ and outside the parabola } y^{2}=2\sqrt{3}x \text{ is :} $$

$$ \text{Let the foci of a hyperbola be }(1, 4) \text{ and } (1, -12). \text{ If it passes through the point}(1, 6)$$, then the length of its latus-rectum is :

$$ \text{If } \sum_{r=1}^n T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64} \text{ then } \lim_{n \rightarrow \infty} \sum_{r=1}^n\left( \frac {1}{T_r}\right) \text{is equal to:} $$

A coin is tossed three times. Let  $$X$$ denote the number of times a tail follows a head. If $$ \mu \text{ and } \sigma^{2} \text{denote the mean and variance of } X, \text{ then the value of }64(\mu+\sigma^{2}) \text{ is :} $$

$$ \text{The number of non-empty equivalence relations on the set }\left\{1, 2, 3\right\} \text{ is :} $$

A circle $$C$$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $$ r$$ be the radius of a circle that has centre at the point  (2, 5)  and intersects the circle $$ C $$ at exactly two points. If the set of all possible values of r is the interval $$(\alpha, \beta), \text{ then } 3\beta - 2\alpha \text{ is equal to :} $$

$$ \text{Let } A=\left\{1, 2, 3,....,10\right\} \text{ and }B=\left\{ \frac {m}{n},n \in A,m < n \text{ and }gcd(m,n)=1\right\}.$$ Then n(B) is equal to:

Let $$ z_1,z_2 \text{ and } z_3$$ be three complex numbers on the circle $$ \mid z \mid = 1 $$ with $$ arg(z_1)=\frac{-\pi}{4},arg(z_2)=0 \text{ and } arg(z_3)=\frac{\pi}{4}$$.  If $$\mid z_1\overline{z}_2+z_2\overline{z}_3+z_3\overline{z}_1 \mid^{2}= \alpha+ \beta \sqrt{2}, \alpha, \beta \in Z$$, then the value of $$ \alpha^{2}+\beta^{2} \text{ is :} $$