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For the following questions answer them individually
Let $$S_{1} = \left\{\left(i, j, k\right) : i, j, k \epsilon \left\{1, 2, ... , 10\right\}\right\}$$,
$$S_{2} = \left\{\left(i, j\right) : 1 \leq i < j + 2 \leq 10, i, j \epsilon \left\{1, 2, ..., 10\right\}\right\}$$,
$$S_{3} = \left\{\left(i,j, k, l\right) : 1 \leq i < j < k < l, i,j, k, l \epsilon \left\{1, 2, ..., 10\right\}\right\}$$
and
$$S_{4} = $${ $$\left(i, j, k, l \right) : i, j, k$$ and l are distinct elements in $$\left\{1, 2, ..., 10 \right\}$$}.
If the total number of elements in the set $$S_{r}$$ is $$n_{r},= 1, 2, 3, 4$$, then which of the following statements is (are) TRUE?
Consider a triangle ๐๐๐ having sides of lengths ๐, ๐ and ๐ opposite to the angles ๐,๐ and ๐ , respectively. Then which of the following statements is (are) TRUE ?
Let $$f : \left[-\frac{\pi}{2}, \frac{\pi}{2} \right] \rightarrow R$$ be a continuous function such that
$$f(0)= 1$$ and $$\int_{0}^{\frac{\pi}{3}} f\left(t\right)dt = 0$$
Then which of the following statements is (are) TRUE ?
For any real numbers $$\alpha$$ and $$\beta$$, let $$y_{\alpha, \beta} \left(x\right), x \epsilon R$$, be the solution of the differential equation
$$\frac{dy}{dx} + \alpha y = x e^{\beta x}, y \left(1\right) = 1$$.
Let $$S = \left\{y_{\alpha, \beta} \left(x \right) : \alpha, \beta \epsilon R \right\}$$. Then which of the following function belong(s) to the set S ?
Let $$O$$ be the origin and $$\overrightarrow{OA} = 2\hat{i} + 2\hat{j} + \hat{k}, \overrightarrow{OB} = \hat{i} โ 2\hat{j} + 2\hat{k}$$ and $$\overrightarrow{OC} = \frac{1}{2}(\overrightarrow{OB} โ\lambda\overrightarrow{OA})$$ for some $$\lambda > 0$$. If $$|\overrightarrow{OB} \times \overrightarrow{OC}| = \frac{9}{2}$$, then which of the following statements is (are) TRUE ?
Let E denote the parabola $$y^{2} = 8x$$. Let P = (โ2, 4), and let Q and $$Q^{'}$$ be two distinct points on E such that the lines PQ and $$PQ^{'}$$ are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE ?
Consider the region $$R = \left\{(๐ฅ, ๐ฆ) \epsilon R \times R โถ x \geq 0ย andย ย y^{2} \leq 4 โ x \right\}$$. Let F be the family of all circles that are contained in ๐ and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let ($$\alpha, \beta$$) be a point whereย the circle C meets the curve $$y^{2} = 4 โ x$$.
Let $$f_{1}:(0, \infty) \rightarrow$$ R and $$f_{2}:(0, \infty) \rightarrow$$ be defined by
$$f_{1}(x): \int_{0}^{x} \prod_{j=1}^{21}(t-1)^{j}$$ dt, $$x> 0$$ and
$$f_{3}(x) = 98(x-1)^{50}-600(x-1)^{49} + 2450, x > 0$$,
where, for any positive integer n and real numbers $$a_{1}, a_{2}, โฆ , a_{n}, \prod_{}{}^{n}i=1 a_{i}$$ denotes the product of $$a_{1}, a_{2}, โฆ , a_{n}$$. Let $$m_{i}$$ and $$n_{i}$$, respectively, denote the number of points of local minima and the number of points of local maxima of function $$f{i}$$, i = 1, 2, in the interval ($$0, \infty$$).
The value of $$2m_{1} + 3n_{1} + m_{1}n_{1}$$ is ___ .
The value of $$6m_{2} + 4n_{2} + 8m_{2}n_{2}$$ is ___ .
