For the following questions answer them individually
Let $$\lambda \in \mathbb{R}$$ and let the equation $$E$$ be $$|x|^2 - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set $$S = \{x + \lambda : x \text{ is an integer solution of } E\}$$ is ______
A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is
The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______
The 4th term of GP is 500 and its common ratio is $$\frac{1}{m}$$, $$m \in \mathbb{N}$$. Let $$S_n$$ denote the sum of the first $$n$$ terms of this GP. If $$S_6 > S_5 + 1$$ and $$S_7 < S_6 + \frac{1}{2}$$, then the number of possible values of $$m$$ is ______
Suppose $$\sum_{r=0}^{2023} r^2 \cdot {^{2023}C_r} = 2023 \times \alpha \times 2^{2022}$$, then the value of $$\alpha$$ is
Let a tangent to the curve $$9x^2 + 16y^2 = 144$$ intersect the coordinate axes at the points $$A$$ and $$B$$. Then, the minimum length of the line segment $$AB$$ is ______
Let $$C$$ be the largest circle centred at $$(2, 0)$$ and inscribed in the ellipse $$\frac{x^2}{36} + \frac{y^2}{16} = 1$$. If $$(1, \alpha)$$ lies on $$C$$, then $$10\alpha^2$$ is equal to ______
The value of $$\frac{8}{\pi}\int_0^{\pi/2} \frac{\cos x^{2023}}{\sin x^{2023} + \cos x^{2023}} dx$$ is ______.
The value of $$12\int_0^3 x^2 - 3x + 2 dx$$ is ______
The shortest distance between the lines $$\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2}$$ and $$\frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0}$$ is equal to ______
