### CAT Content

Instructions

Let $$g_{i} : \left[\frac{\pi}{8},\frac{3\pi}{8}\right] \rightarrow R, i = 1,2$$, and $$f:\left[\frac{\pi}{8},\frac{3\pi}{8}\right] \rightarrow R$$ be function such that
$$g_{1}(x) = 1, g_{2}(x) = |4x-\pi|$$ and $$f(x) = \sin^{2} x$$, for all $$x \epsilon \left[\frac{\pi}{8},\frac{3\pi}{8}\right]$$
Define
$$S_{i} = \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} f(x)\cdot g_{i}(x) dx, i- 1, 2$$

Question 11

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Question 12

## he value of $$\frac{48S_{2}}{\pi^{2}}$$ is _______.

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Instructions

Paragraph
let $$M = \left\{(x, y) \epsilon R \times R ∶ x^{2} + y^{2} \leq r^{2} \right\}$$

where r > 0. Consder the geometric progression $$a_{n} = \frac{1}{2^{n-1}}, n = 1, 2, 3, ...$$ Let $$S_{0}=0$$ and, for $$n \geq 1$$, let $$S_{n}$$ denote the sume of the first n terms of this progression . For $$n \geq 1$$, Let$$C_{n}$$ denote the circle with center $$\left(S_{n-1},S_{n-1}\right)$$ and radius $$a_{n}$$.

Question 13

Question 14

## Consider $$M$$ with $$r = \frac{(2^{199}-1)\sqrt{2}}{2^{198}}$$. The number of all those circles $$D_{n}$$ that are inside M is

Instructions

Let $$\psi: [0, \infty) \rightarrow R, \psi: [0, \infty) \rightarrow R, f:[0, \infty) \rightarrow R$$ and $$g:[0, \infty) \rightarrow R$$ be functions such that $$f(0)=g(0)=0$$,
$$\psi:(x)=e^{-x} + x, x \geq 0$$,
$$\psi:(x)=e^{2} - 2x - 2e^{-x} + 2 x \geq 0$$,
$$f(x)= \int_{-x}^{x} (|t| - t^{2})e^{-t^{2}} dt, x > 0$$,
and
$$g(x) = \int_{0}^{x^{2}} \sqrt{t} e^{-t} dt, x > 0.$$

Question 15

Question 16

## Which of the following statements is TRUE ?

Instructions

For the following questions answer them individually

Question 17

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Question 18

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Question 19

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Question 20