Let $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .

Let $[ \mathrm { x } ]$ denote the greatest integer $\leq \mathrm { x }$. Consider the function $\mathrm { f } ( \mathrm { x } ) = \max \left\{ \mathrm { x } ^ { 2 } , 1 + [ x ] \right\}$. Then the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ is :
(1) $\frac { 5 + 4 \sqrt { 2 } } { 3 }$
(2) $\frac { 8 + 4 \sqrt { 2 } } { 3 }$
(3) $\frac { 1 + 5 \sqrt { 2 } } { 3 }$
(4) $\frac { 4 + 5 \sqrt { 2 } } { 3 }$

Let $[ t ]$ denote the greatest integer $\leq t$. Then $\frac { 2 } { \pi } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } ( 8 [ \operatorname { cosec } x ] - 5 [ \cot x ] ) d x$ is equal to $\_\_\_\_$

If $\int _ { - 0.15 } ^ { 0.15 } \left| 100 x ^ { 2 } - 1 \right| d x = \frac { k } { 3000 }$, then $k$ is equal to $\_\_\_\_$ .

Let $[ t ]$ denote the largest integer less than or equal to $t$. If $$\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$$ where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$

If $24 \int _ { 0 } ^ { \frac { \pi } { 4 } } \left( \sin \left| 4 x - \frac { \pi } { 12 } \right| + [ 2 \sin x ] \right) \mathrm { d } x = 2 \pi + \alpha$, where $[ \cdot ]$ denotes the greatest integer function, then $\alpha$ is equal to $\_\_\_\_$ .

Q75. Let $f ( x ) = \left\{ \begin{array} { l l } - 2 , & - 2 \leq x \leq 0 \\ x - 2 , & 0 < x \leq 2 \end{array} \right.$ and $h ( x ) = f ( | x | ) + | f ( x ) |$. Then $\int _ { - 2 } ^ { 2 } h ( x ) \mathrm { d } x$ is equal to :
(1) 1
(2) 6
(3) 4
(4) 2

Q87. Let $[ t ]$ denote the largest integer less than or equal to $t$. If $\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$, where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$

The value of $\int_{0}^{\pi/2} |\sin x + \sin 2x + \sin 3x|dx$ is
(A) 17 (B) 16 (C) 15 (D) 14

$\int _ { \frac { - \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { [ x ] + 5 }$ is equal to: ([.] denotes greatest integer function)
(A) $\frac { \pi } { 4 } - \frac { 1 } { 10 }$
(B) $\frac { \pi } { 4 } - \frac { 1 } { 20 }$
(C) $\frac { \pi } { 12 } - \frac { 1 } { 10 }$
(D) $\frac { \pi } { 4 } - \frac { 1 } { 5 }$

Let the area bounded by the curve $\mathrm { f } ( \mathrm { x } ) = \max \{ \sin x , \cos x \}$ and x -axis is\ $A$ where $x \in \left[ 0 , \frac { 3 \pi } { 2 } \right]$. Find $A + A ^ { 2 }$

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 12 ( 3 + [ x ] ) } { 3 + [ \sin x ] + [ \cos x ] } d x$ is equal to (A) $3 + 10 \pi$ (B) $11 \pi + 2$ (C) $10 \pi + 2$

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 applicants should turn to page 14. Let
$$f ( x ) = \left\{ \begin{array} { c c } x + 1 & \text { for } 0 \leqslant x \leqslant 1 ; \\ 2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leqslant x \leqslant 2 . \end{array} \right.$$
(i) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leqslant x \leqslant 2$, labelling any turning points and the values attained at $x = 0,1,2$.
(ii) For $1 \leqslant t \leqslant 2$, define
$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$
Express $g ( t )$ as a cubic in $t$.
(iii) Calculate and factorize $g ^ { \prime } ( t )$.
(iv) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leqslant t \leqslant 2$ ? [Figure]

Evaluate
$$\int_{-1}^{3} |x|(1-x) \, dx$$

The function f is such that, for every integer $n$
$$\int _ { n } ^ { n + 1 } \mathrm { f } ( x ) \mathrm { d } x = n + 1$$
Evaluate
$$\sum _ { r = 1 } ^ { 8 } \left( \int _ { 0 } ^ { r } \mathrm { f } ( x ) \mathrm { d } x \right)$$
A 36 B 84 C 120 D 165 E 204 F 288

A sequence of functions $f _ { 1 } , f _ { 2 } , f _ { 3 } , \ldots$ is defined by
$$\begin{aligned} \mathrm { f } _ { 1 } ( x ) & = | x | \\ \mathrm { f } _ { n + 1 } ( x ) & = \left| \mathrm { f } _ { n } ( x ) + x \right| \quad \text { for } n \geq 1 \end{aligned}$$
Find the value of
$$\int _ { - 1 } ^ { 1 } \mathrm { f } _ { 99 } ( x ) \mathrm { d } x$$
A 0
B 0.5
C 1
D 49.5
E 50 F 99 G 99.5 H 100

For any integer $n \geq 0$,
$$\int _ { n } ^ { n + 1 } f ( x ) \mathrm { d } x = n + 1$$
Evaluate
$$\int _ { 0 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 1 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 2 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 4 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 5 } ^ { 3 } f ( x ) \mathrm { d } x$$

The ceiling of $x$, written $[ x ]$, is defined to be the value of $x$ rounded up to the nearest integer. For example: $\quad \lceil \pi \rceil = 4 , \quad \lceil 2.1 \rceil = 3 , \quad \lceil 8 \rceil = 8$ What is the value of the following integral?
$$\int _ { 0 } ^ { 99 } 2 ^ { \lceil x \rceil } d x$$
A $2 ^ { 99 }$ B $\quad 2 ^ { 99 } - 1$ C $2 ^ { 99 } - 2$ D $2 ^ { 100 }$ E $\quad 2 ^ { 100 } - 1$ F $\quad 2 ^ { 100 } - 2$

$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$
What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$?
A) 2
B) 4
C) 6
D) 8
E) 10

$f ( x ) = \begin{cases} 2 x - 4 , & \text{if } 0 \leq x < 1 \\ - 2 , & \text{if } 1 \leq x < 4 \\ x - 6 , & \text{if } 4 \leq x \leq 6 \end{cases}$
Given this, what is the value of the integral $\int _ { 0 } ^ { 6 } f ( x ) d x$?
A) - 11
B) - 10
C) - 9
D) - 8
E) - 7