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(a) On the diagram below, draw the square in Position 4, Position 5 and Position 6.
(b) Complete this table to show the x-coordinate of the centre of the square in each position. You may use the diagram to help you.
[Table_1]
Position (n): 1, 2, 3, 4, 5, 6, n
x-coordinate: 0.5, 1.5, 2.5
[2]
(c) Find the x-coordinate of the centre of the square in Position 92.
............................................................. [2]
(d)(i) The square rolls from Position 1 to Position n. The centre has moved a distance equal to the circumference of 1 circle. The radius, $r$, of the circle is half the diagonal of the square.
(a) Write down the number of rolls needed.
............................................................. [1]
(b) Write down the value of $n$.
............................................................. [1]
(ii) $\begin{align} \text{Circumference, C, of circle, radius } r. & C = 2\pi r\\ \text{Hypotenuse, r, of a right-angled triangle with sides } r, x \text{ and } y. & r^2 = x^2 + y^2 \end{align}$
(a) Show that the radius of the circle is 0.707 cm, correct to 3 decimal places.
............................................................. [2]
(b) Find the length of the arc that the centre of the square moves along from Position 1 to Position 2.
............................................................. [2]
556
573
593
The side of the square is now 2 cm.
The square rolls along the x-axis in the same way as in Question 1.
(a) Complete the table of x-coordinates of the centre of the square in different positions.
\( \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Position (n)} & 1 & 2 & 3 & 4 & 5 & 6 & n \\ \hline \text{x-coordinate} & 1 & 3 & & & & & \\ \hline \end{array} \) ......
[3]
(b) Find the coordinates of the centre of the square in Position 35.
( ..................... , ..................... ) [3]
503
543
555
(a) The side of the square is now 3 cm.
Complete the table of $x$-coordinates of the centre of the square in different positions. You may use the diagram below to help you.
$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Position } (n) & 1 & 2 & 3 & 4 & 5 & 6 & n \\ \hline x\text{-coordinate} & 1.5 & & & & & & \\ \hline \end{array} $$
(b) The side of the square is now 4 cm.
Complete the table of $x$-coordinates of the centre of the square in different positions.
$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Position } (n) & 1 & 2 & 3 & 4 & 5 & 6 & n \\ \hline x\text{-coordinate} & 2 & & & & & & \\ \hline \end{array} $$
553
528
586
Write your expressions from Questions 1(b), 2(a) and 3 in the table below. Complete the table using any patterns you notice.
[Table_1]
596
558
593
A square of side $w$ cm rolls from Position 1 to Position 120.
At Position 120, the x-coordinate of the centre of the square is 2151.
Find the value of $w$.
583
514
563
A square of side $a$ cm is in Position 1.
The coordinates of the centre of the square are $(11, k)$.
(a) Find the value of $k$ and the value of $a$.
$k = \text{..................................................}$
$a = \text{.............................................}$ [2]
(b) Find the coordinates of the top right corner of the square.
$(\text{...................} , \text{...................})$ [1]
(c) Write down the $y$-coordinate of the centre of the square in Position 400.
$\text{....................................................}$ [1]
598
515
511
A square rolls along the x-axis.
For the top left corner give a reason why
total distance moved in 2 rolls = total distance moved in 3 rolls.
You may use this grid.
599
574
508
