NTA JEE Mains 24th Jan 2025 Shift 1

Let circle $$C$$ be the image of $$x^2 + y^2 - 2x + 4y - 4 = 0$$ in the line $$2x - 3y + 5 = 0$$ and $$A$$ be the point on $$C$$ such that $$OA$$ is parallel to the $$x$$-axis and $$A$$ lies on the right hand side of the centre $$O$$ of $$C$$. If $$B(\alpha, \beta)$$, with $$\beta < 4$$, lies on $$C$$ such that the length of the arc $$AB$$ is $$\left(1/6\right)^{\text{th}}$$ of the perimeter of $$C$$, then $$\beta + \sqrt{3}\alpha$$ is equal to

Let in a $$\triangle ABC$$, the length of the side $$AC$$ be $$6$$, the vertex $$B$$ be $$(1, 2, 3)$$ and the vertices $$A, C$$ lie on the line $$ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. $$ Then the area (in sq. units) of $$\triangle ABC$$ is:

Let the product of the focal distances of the point $$\left( \sqrt{3}, \frac{1}{2} \right)$$ on the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad (a > b),$$ be $$\frac{7}{4}.$$ Then the absolute difference of the eccentricities of two such ellipses is

$$ \text{If the system of equations } \begin{aligned} 2x - y + z &= 4, \\ 5x + \lambda y + 3z &= 12, \\ 100x - 47y + \mu z &= 212 \end{aligned} \text{ has infinitely many solutions, then } \mu - 2\lambda \text{ is equal to: } $$

For some $$n \ne 10,$$ let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $$(1+x)^{n+4}$$ be in A.P. Then the largest coefficient in the expansion of  $$(1+x)^{n+4}$$ is:

The product of all the rational roots of the equation $$(x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3$$ is equal to:

$$ \text{Let the line passing through the points } (-1,2,1) \text{ and parallel to the line } \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4} \text{ intersect the line } \frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1} \text{ at the point } P. \text{ Then the distance of } P \text{ from the point } Q(4,-5,1) \text{ is:} $$

$$ \text{Let the lines } 3x-4y-\alpha=0,\; 8x-11y-33=0,\; \text{ and } 2x-3y+\lambda=0$$ $$\text{ be concurrent. If the image of the point } (1,2) \text{ in the line } 2x-3y+\lambda=0 $$ $$\text{ is } \left(\frac{57}{13},-\frac{40}{13}\right), \text{ then } |\alpha\lambda| \text{ is equal to:} $$

$$ \text{If } \alpha \text{ and } \beta \text{ are the roots of the equation } 2z^2-3z-2i=0,\; \text{ where } i=\sqrt{-1}, \text{ then } 16 \cdot \operatorname{Re}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{Im}\!\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \text{ is equal to:} $$

For a statistical data $$x_1,x_2,\ldots,x_{10}$$ of 10 values, a student obtained the mean as  5.5 and  $$\sum_{i=1}^{10} x_i^2 = 371. $$  He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values  6 and 8  respectively. The variance of the corrected data is: