NTA JEE Main 9th January 2020 Shift 2

The number of terms common to the two A.P.'s 3, 7, 11, ..., 407 and 2, 9, 16, ..., 709 is ___________.

If $$C_r \equiv {}^{25}C_r$$ and $$C_0 + 5 \cdot C_1 + 9 \cdot C_2 + \ldots + (101) \cdot C_{25} = 2^{25} \cdot k$$, then $$k$$ is equal to ___________.

If the curves, $$x^2 - 6x + y^2 + 8 = 0$$ and $$x^2 - 8y + y^2 + 16 - k = 0$$, $$(k \gt 0)$$ touch each other at a point, then the largest value of $$k$$ is ___________.

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three vectors such that $$|\vec{a}| = \sqrt{3}$$, $$|\vec{b}| = 5$$, $$\vec{b} \cdot \vec{c} = 10$$ and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\frac{\pi}{3}$$. If $$\vec{a}$$ is perpendicular to the vector $$\vec{b} \times \vec{c}$$, then $$\left|\vec{a} \times (\vec{b} \times \vec{c})\right|$$ is equal to ___________.

If the distance between the plane, $$23x - 10y - 2z + 48 = 0$$ and the plane containing the lines $$\frac{x+1}{2} = \frac{y-3}{4} = \frac{z+1}{3}$$ and $$\frac{x+3}{2} = \frac{y+2}{6} = \frac{z-1}{\lambda}$$ $$(\lambda \in R)$$ is equal to $$\frac{k}{\sqrt{633}}$$, then $$k$$ is equal to ___________.