Instructions

For the following questions answer them individually

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

## Let E denote the parabola $$y^{2} = 8x$$. Let P = (−2, 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 ?

Instructions

Consider the region $$R = \left\{(𝑥, 𝑦) \epsilon R \times R ∶ x \geq 0 and y^{2} \leq 4 − x \right\}$$. Let F be the family of all circles that are contained in 𝑅 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 where the circle C meets the curve $$y^{2} = 4 − x$$.

Question 7

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Question 8

## The value of $$\alpha$$ is ___ .

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Instructions

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 $$a_{1}, a_{2}, … , a_{n}, \prod_{}{}^{n}i=1 a_{i}$$ denotes the product of $$a_{1}, a_{2}, … , a_{n}$$. 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 ($$0, \infty$$).

Question 9

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Question 10

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