No questions found
REMAINING SHAPES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the area of the remaining shape when part of a larger shape is removed.
The side difference is the difference between the lengths of the sides of the two shapes.
In this investigation all lengths are in centimetres.
A square is removed from a larger square.
The side difference is 1.
A square of side 2 is removed from a larger square of side 3.
The area of the remaining shape is 5.
A square of side 3 is removed from a larger square of side 4.
This is the remaining shape.
(a) On this grid, draw the remaining shape for a larger square of side 5.
[1]
(b) Complete the table.
[Table_1]
Side of larger square Area of remaining shape
2
3 5
4
5
6
[3]
(c) Find an expression, in terms of $n$, for the area of the remaining shape when the larger square has side $n$.
........................................... [2]
(d) The area of a remaining shape is 381.
Work out the length of the side of the larger square.
........................................... [2]
581
559
511
REMAINING SHAPES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the area of the remaining shape when part of a larger shape is removed. The side difference is the difference between the lengths of the sides of the two shapes. In this investigation all lengths are in centimetres.
The side length of the square removed is now 2 less than the side length of the larger square. The side difference is 2.
A square of side 2 is removed from a larger square of side 4. The area of the remaining shape is 12.
(a) Complete the table. You may use the grid below to help you.
| Side of larger square | Area of remaining shape |
|----------------------|-------------------------|
| 3 | |
| 4 | 12 |
| 5 | |
| 6 | |
[3 marks]
(b) Complete the expression for the area of the remaining shape when the larger square has side n and the side difference is 2.
4n ........................................... [1]
580
527
508
REMAINING SHAPES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the area of the remaining shape when part of a larger shape is removed. The \textit{side difference} is the difference between the lengths of the sides of the two shapes. In this investigation all lengths are in centimetres.
(a) Complete the table using expressions in the form $an + b$, where $a$ and $b$ are integers.
Use your expressions from Question 1(c) and Question 2(b). You may use the grid below to help you.
| Side difference (d) | Area of remaining shape for larger square of side n |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 |
[4]
(b) Find an expression, in terms of $n$ and $d$, for the area of the remaining shape when the side difference is $d$.
..................................................
[2]
(c) The area of a remaining shape is 343. The side difference is 7.
Find the length of the side of the larger square.
..................................................
[2]
505
528
558
REMAINING SHAPES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at the area of the remaining shape when part of a larger shape is removed. The side difference is the difference between the lengths of the sides of the two shapes. In this investigation all lengths are in centimetres.
This question looks at the area of a remaining shape for an equilateral triangle.
(a) Show that the area of an equilateral triangle with side of length $n$ is \( \frac{n^2 \sqrt{3}}{4} \).
[1]
(b) An equilateral triangle is removed from a larger equilateral triangle. The side difference is 1.
An equilateral triangle with a side of length 3 is removed from a larger equilateral triangle with a side of length 4.
Show that the area of the remaining shape is \( \frac{7 \sqrt{3}}{4} \).
[1]
(c) Find the area of the remaining shape for a larger equilateral triangle with a side of length 5 and a side difference of 1.
Give your answer in the form \( \frac{k \sqrt{3}}{4} \).
[2]
(d) Complete the table.
[Table_1]
[3]
(e) Find an expression, in terms of $n$ and $d$, for the area of the remaining shape for a larger equilateral triangle with side of length $n$ when the side difference is $d$.
Give an exact answer in its simplest form.
[3]
507
588
500
ESCALATORS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at going up and down on escalators.
(a) Every step of an escalator travels at a speed of 1.8 km/h. Show that the distance a step moves in 1 second is 0.5 m. [2]
(b) The length of the escalator is 40 m.
Find the time, in seconds, for a step to travel the length of the escalator. [2]
577
584
535
ESCALATORS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at going up and down on escalators.
Matt travels up a different escalator. The length of this escalator is 24 metres.
(a) (i) The escalator is not moving. It takes Matt 50 seconds to walk up the length of the escalator.
Find his speed.
[2]
(ii) The escalator is now moving. When Matt stands on the escalator it takes him 30 seconds to move up the length of the escalator.
Matt now walks up the moving escalator at the same speed as in part (a)(i).
Show that his speed walking up the moving escalator is 1.28m/s.
[2]
(iii) Find how long it will take him to walk up the length of the moving escalator.
[2]
(b) (i) Matt walks down the escalator when it is not moving. It takes him 24 seconds to walk down the length of the escalator.
The escalator starts moving upwards at the same speed as in part (a). Matt now walks down the moving escalator.
Show that his speed walking down the moving escalator is 0.2m/s.
[2]
(ii) Find how long it will take Matt to walk down the length of the escalator when it is moving upwards at the same speed as before.
[2]
548
553
537
ESCALATORS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at going up and down on escalators.
Another escalator has a length of 6.2 m. The escalator travels upwards at a speed of $v$ metres per second.
(a) With the escalator travelling upwards, Matt walks up the escalator and then immediately turns around and walks down the escalator.
His walking speed is 0.8 m/s. The total time for Matt to walk up and down the moving escalator is $T$ seconds.
(i) Give reasons why the model for $T$ is $T = \frac{6.2}{0.8 + v} + \frac{6.2}{0.8 - v}$.
.......................................................................................................................................................... ..........................................................................................................................................................
.......................................................................................................................................................... [3]
(ii) Show that the model simplifies to $T = \frac{9.92}{0.64 - v^2}$.
...................................................................................................................... [3]
(iii) When the speed of the escalator is 0.4 m/s, find how long it takes Matt to walk up and down the escalator.
................................................................ [1]
(b) Matt continues to walk at 0.8 m/s. He walks up the moving escalator, waits for 10 seconds, then walks down the escalator.
(i) Change the model in part (a)(ii) to find a new model for $T$. You do not need to simplify your model.
................................................................ [1]
(ii) Sketch the graph of $T$ for $0 \leq v \leq 5$.
[4]
(iii) The total time for Matt to walk up, wait, and walk down the moving escalator is 138 seconds.
Find the speed of the escalator.
................................................................ [2]
(iv) Give a reason why the model is not valid when $v = 5$ .
.......................................................................................................................................................... [1]
(v) For what range of values of $v$ is the model valid?
................................................................ [1]
517
526
518
