The value of the integral $\int _ { 1 / 2 } ^ { 2 } \frac { \tan ^ { - 1 } x } { x } d x$ is equal to (1) $\pi \log _ { e } 2$ (2) $\frac { 1 } { 2 } \log _ { e } 2$ (3) $\frac { \pi } { 4 } \log _ { \mathrm { e } } 2$ (4) $\frac { \pi } { 2 } \log _ { \mathrm { e } } 2$

If $\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7}\, dx = \frac{1}{l} \cdot 11^{m/n}$ where $l, m, n \in \mathbb{N}$, $m$ and $n$ are co-prime, then $l + m + n$ is equal to $\_\_\_\_$.

The integral $\int \frac { x ^ { 8 } - x ^ { 2 } d x } { x ^ { 12 } + 3 x ^ { 6 } + 1 \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } }$ is equal to :
(1) $\log \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 3 } } + C$
(2) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { \frac { 1 } { 2 } } + C$
(3) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + \frac { 1 } { x ^ { 3 } } + C$
(4) $\log _ { e } \tan ^ { - 1 } x ^ { 3 } + { \frac { 1 } { x ^ { 3 } } } ^ { 3 } + C$

If $\int \frac { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } { \sqrt { \sin ^ { 3 } x \cos ^ { 3 } x \sin ( x - \theta ) } } d x = A \sqrt { \cos \theta \tan x - \sin \theta } + B \sqrt { \cos \theta - \sin \theta \cot x } + C$, where $C$ is the integration constant, then $AB$ is equal to
(1) $4 \operatorname { cosec } ( 2 \theta )$
(2) $4 \sec \theta$
(3) $2 \sec \theta$
(4) $8 \operatorname { cosec } ( 2 \theta )$

$\int _ { 0 } ^ { \pi / 4 } \frac { \cos ^ { 2 } x \sin ^ { 2 } x } { \left( \cos ^ { 3 } x + \sin ^ { 3 } x \right) ^ { 2 } } d x$ is equal to
(1) $1 / 6$
(2) $1 / 3$
(3) $1 / 12$
(4) $1 / 9$

If $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + x } + \sqrt { 1 + x } } d x = a + b \sqrt { 2 } + c \sqrt { 3 }$, where $a , b , c$ are rational numbers, then $2 a + 3 b - 4 c$ is equal to:
(1) 4
(2) 10
(3) 7
(4) 8

For $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$, if $y ( x ) = \int \frac { \operatorname { cosec } x + \sin x } { \operatorname { cosec } x \sec x + \tan x \sin ^ { 2 } x } d x$ and $\lim _ { x \rightarrow \left( \frac { \pi } { 2 } \right) ^ { - } } y ( x ) = 0$ then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(2) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(3) $- \frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right)$
(4) $\frac { 1 } { \sqrt { 2 } } \tan ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$

Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2 . If the roots of the equation $| A - x I | = 0$ be - 1 and 3 , then the sum of the diagonal elements of the matrix $A ^ { 2 }$ is $\_\_\_\_$ .

If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$

If $f ( x ) = \int \frac { 1 } { x ^ { 1 / 4 } \left( 1 + x ^ { 1 / 4 } \right) } \mathrm { d } x , f ( 0 ) = - 6$, then $f ( 1 )$ is equal to :
(1) $4 \left( \log _ { e } 2 - 2 \right)$
(2) $2 - \log _ { e ^ { 2 } } 2$
(3) $\log _ { e } 2 + 2$
(4) $4 \left( \log _ { e } 2 + 2 \right)$

The value of $\int _ { e ^ { 2 } } ^ { e ^ { 4 } } \frac { 1 } { x } \left( \frac { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } + e ^ { \left( \left( 6 - \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } \right) d x$ is
(1) 2
(2) $\log _ { e } 2$
(3) 1
(4) $e ^ { 2 }$

Let $\mathrm { I } ( x ) = \int \frac { d x } { ( x - 11 ) ^ { \frac { 11 } { 13 } } ( x + 15 ) ^ { \frac { 15 } { 13 } } }$. If $\mathrm { I } ( 37 ) - \mathrm { I } ( 24 ) = \frac { 1 } { 4 } \left( \frac { 1 } { \mathrm {~b} ^ { \frac { 1 } { 13 } } } - \frac { 1 } { \mathrm { c } ^ { \frac { 1 } { 13 } } } \right) , \mathrm { b } , \mathrm { c } \in \mathrm { N }$, then $3 ( \mathrm {~b} + \mathrm { c } )$ is equal to
(1) 22
(2) 39
(3) 40
(4) 26

If $\int \frac{2x^{2} + 5x + 9}{\sqrt{x^{2} + x + 1}} \, \mathrm{d}x = x\sqrt{x^{2} + x + 1} + \alpha\sqrt{x^{2} + x + 1} + \beta \log_{e}\left| x + \frac{1}{2} + \sqrt{x^{2} + x + 1} \right| + \mathrm{C}$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to $\_\_\_\_$.

Q67. $\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { \int _ { x ^ { 3 } } ^ { ( \pi / 2 ) ^ { 3 } } \left( \sin \left( 2 t ^ { 1 / 3 } \right) + \cos \left( t ^ { 1 / 3 } \right) \right) d t } { \left( x - \frac { \pi } { 2 } \right) ^ { 2 } } \right)$ is equal to
(1) $\frac { 5 \pi ^ { 2 } } { 9 }$
(2) $\frac { 9 \pi ^ { 2 } } { 8 }$
(3) $\frac { 11 \pi ^ { 2 } } { 10 }$
(4) $\frac { 3 \pi ^ { 2 } } { 2 }$

Q73. Let $\int \frac { 2 - \tan x } { 3 + \tan x } \mathrm {~d} x = \frac { 1 } { 2 } \left( \alpha x + \log _ { \mathrm { e } } | \beta \sin x + \gamma \cos x | \right) + C$, where $C$ is the constant of integration. Then $\alpha + \frac { \gamma } { \beta }$ is equal to :
(1) 7
(2) 4
(3) 1
(4) 3

Q74. If $\int \frac { 1 } { \mathrm { a } ^ { 2 } \sin ^ { 2 } x + \mathrm { b } ^ { 2 } \cos ^ { 2 } x } \mathrm {~d} x = \frac { 1 } { 12 } \tan ^ { - 1 } ( 3 \tan x ) +$ constant, then the maximum value of $\mathrm { a } \sin x + \mathrm { b } \cos x$, is :
(1) $\sqrt { 40 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 39 }$
(4) $\sqrt { 42 }$

Q75. $\int _ { 0 } ^ { \pi / 4 } \frac { \cos ^ { 2 } x \sin ^ { 2 } x } { \left( \cos ^ { 3 } x + \sin ^ { 3 } x \right) ^ { 2 } } d x$ is equal to
(1) $1 / 6$
(2) $1 / 3$
(3) $1 / 12$
(4) $1 / 9$

Q76. The integral $\int _ { 0 } ^ { \pi / 4 } \frac { 136 \sin x } { 3 \sin x + 5 \cos x } d x$ is equal to :
(1) $3 \pi - 50 \log _ { e } 2 + 20 \log _ { e } 5$
(2) $3 \pi - 25 \log _ { e } 2 + 10 \log _ { e } 5$
(3) $3 \pi - 10 \log _ { e } ( 2 \sqrt { 2 } ) + 10 \log _ { e } 5$
(4) $3 \pi - 30 \log _ { e } 2 + 20 \log _ { e } 5$

Q88. If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$

If $F ( t ) = \int \frac { 1 - \sin ( \ln t ) } { 1 - \cos ( \ln t ) } d t$ and $F \left( e ^ { \pi / 2 } \right) = - e ^ { \pi / 2 }$ then $F \left( e ^ { \pi / 4 } \right)$ is:
(A) $( - 1 - \sqrt { 2 } ) \mathrm { e } ^ { \frac { \pi } { 4 } }$
(B) $( 1 - \sqrt { 2 } ) e ^ { \frac { \pi } { 4 } }$
(D) $( - 2 - \sqrt { 2 } ) e ^ { \frac { \pi } { 4 } }$

Consider the definite integral $S = \int _ { 0 } ^ { a } x \sqrt { \frac { 1 } { 3 } x + 2 } \, d x$.
(1) Set $t = \sqrt { \frac { 1 } { 3 } x + 2 }$. Then we have
$$\begin{aligned} \int x \sqrt { \frac { 1 } { 3 } x + 2 } \, d x & = \mathbf { N O } \int \left( t ^ { \mathbf { P } } - \mathbf { Q } t ^ { \mathbf { R } } \right) d t \\ & = \mathbf { S } + C , \end{aligned}$$
where $C$ is the integral constant.
(2) Using the result in (1), we have
$$S = \mathbf { T } .$$
Thus we obtain
$$\lim _ { a \rightarrow \infty } \frac { S } { a ^ { \frac { \mathbf { U } } { \mathbf{V} } } } = \frac { \mathbf { W } \sqrt { \mathbf { X } } } { \mathbf { Y Z } }$$
For $\mathbf{S}$ and $\mathbf{T}$, choose the appropriate expression from among the choices (0) $\sim$ (9) below.
(0) $\frac { 6 } { 5 } t ^ { 5 } \left( 3 t ^ { 2 } - 10 \right)$
(1) $\frac { 6 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 10 \right)$
(2) $\frac { 12 } { 5 } t ^ { 5 } \left( 3 t ^ { 2 } - 5 \right)$
(3) $\frac { 12 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 5 \right)$
(4) $\frac { 6 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 5 \right)$
(5) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 5 } ( a - 4 ) + 8 \sqrt { 2 } \right\}$
(6) $\frac { 12 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 2 ) + 4 \sqrt { 2 } \right\}$
(7) $\frac { 12 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 5 } ( a - 2 ) + 4 \sqrt { 2 } \right\}$
(8) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 4 ) + 8 \sqrt { 2 } \right\}$
(9) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 2 ) + 8 \sqrt { 2 } \right\}$

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science and Computer Science \& Philosophy applicants should turn to page 16.
In this question we shall investigate when functions are close approximations to each other. We define $| x |$ to be equal to $x$ if $x \geqslant 0$ and to $- x$ if $x < 0$. With this notation we say that a function $f$ is an excellent approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 320 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
we say that $f$ is a good approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 100 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
For example, any function $f$ is an excellent approximation to itself. If $f$ is an excellent approximation to $g$ then $f$ is certainly a good approximation to $g$, but the converse need not hold.
(i) Give an example of two functions $f$ and $g$ such that $f$ is a good approximation to $g$ but $f$ is not an excellent approximation to $g$.
(ii) Show that if
$$f ( x ) = x \quad \text { and } \quad g ( x ) = x + \frac { \sin \left( 4 x ^ { 2 } \right) } { 400 }$$
then $f$ is an excellent approximation to $g$. For the remainder of the question we are going to a try to find a good approximation to the exponential function. This function, which we shall call $h$, satisfies the following equation
$$h ( x ) = 1 + \int _ { 0 } ^ { x } h ( t ) \mathrm { d } t \quad \text { whenever } \quad x \geqslant 0$$
You may not use any other properties of the exponential function during this question, and any attempt to do so will receive no marks.
Let
$$f ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 6 }$$
(iii) Show that if
$$g ( x ) = 1 + \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$$
then $f$ is an excellent approximation to $g$.
(iv) Show that for $x \geqslant 0$
$$h ( x ) - f ( x ) = g ( x ) - f ( x ) + \int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t$$
(v) You are given that $h ( x ) - f ( x )$ has a maximum value on the interval $0 \leqslant x \leqslant 1 / 2$ at $x = x _ { 0 }$. Explain why
$$\int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t \leqslant \frac { 1 } { 2 } \left( h \left( x _ { 0 } \right) - f \left( x _ { 0 } \right) \right) \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
(vi) You are also given that $f ( x ) \leqslant h ( x )$ for all $0 \leqslant x \leqslant \frac { 1 } { 2 }$. Show that $f$ is a good approximation to $h$ when $0 \leqslant x \leqslant \frac { 1 } { 2 }$.
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Given the function $f ( x ) = \frac { x ^ { 2 } + x + 6 } { x - 2 }$, it is requested:
a) ( 0.5 points) Determine its domain and vertical asymptotes.
b) ( 0.5 points) Calculate $\lim _ { x \rightarrow \infty } \frac { f ( x ) } { x }$.
c) (1 point) Calculate $\int _ { 3 } ^ { 5 } f ( x ) d x$.

a) (1.25 points) Calculate, if they exist, the value of the following limits:
a.1) $(0.5$ points) $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } ( 1 - 2 x ) } { x - 2 x ^ { 2 } - \operatorname { sen } x }$
a.2) (0.75 points) $\lim _ { x \rightarrow \infty } \frac { 1 } { x } \left( \frac { 3 } { x } - \frac { 2 } { \operatorname { sen } \frac { 1 } { x } } \right)$
(Hint: use the change of variable $t = 1 / x$ where necessary).
b) (1.25 points) Calculate the following integrals:
b.1) (0.5 points) $\int \frac { x } { x ^ { 2 } - 1 } d x$
b.2) (0.75 points) $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x } d x$

It is given that
$$\frac{dV}{dt} = \frac{24\pi(t-1)}{(1+\sqrt{t})} \text{ for } t \geq 1$$
and $V = 7$ when $t = 1$. Find the value of $V$ when $t = 9$.