For the following questions answer them individually
The number of ways of getting a sum 16 on throwing a dice four times is ______
Let $$a_1, a_2, a_3, \ldots$$ be in an arithmetic progression of positive terms. Let $$A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{2k-1}^2 - a_{2k}^2$$. If $$A_3 = -153, A_5 = -435$$ and $$a_1^2 + a_2^2 + a_3^2 = 66$$, then $$a_{17} - A_7$$ is equal to ______
If the constant term in the expansion of $$\left(1 + 2x - 3x^3\right)\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9$$ is $$p$$, then $$108p$$ is equal to ______
Suppose $$AB$$ is a focal chord of the parabola $$y^2 = 12x$$ of length $$l$$ and slope $$m < \sqrt{3}$$. If the distance of the chord $$AB$$ from the origin is $$d$$, then $$ld^2$$ is equal to ______
Let $$f$$ be a differentiable function in the interval $$(0, \infty)$$ such that $$f(1) = 1$$ and $$\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1$$ for each $$x > 0$$. Then $$2f(2) + 3f(3)$$ is equal to ______
From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. If the variance of $$X$$ is $$\sigma^2$$, then $$96\sigma^2$$ is equal to ______
The number of distinct real roots of the equation $$|x||x + 2| - 5|x + 1| - 1 = 0$$ is ______
If $$S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$ and $$\{t\}$$ represents the fractional part of $$t$$, then $$72\sum_{a \in S} a$$ is equal to ______
The area of the region enclosed by the parabolas $$y = x^2 - 5x$$ and $$y = 7x - x^2$$ is ______
Let $$\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}, \vec{b} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})$$. If $$\vec{a} \cdot \vec{c} = 130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to ______
