No questions found
(a) These are the first four terms of a geometric sequence.
4\ \ \ 16\ \ \ 64\ \ \ 256
(i) Work out the next term.
...................................................... \ [1]
(ii) Work out the 8th term.
...................................................... \ [2]
(iii) Find the $n^{\text{th}}$ term.
...................................................... \ [1]
(b) The $n^{\text{th}}$ term of another geometric sequence is $3 \times 2^{n-1}$.
(i) Show that the first term of this sequence is 3.
\[1]
(ii) Complete the first four terms of this sequence.
3 \ , \ ............ \ , \ ............ \ , \ ............ \ [1]
(c) The $n^{\text{th}}$ term of a different geometric sequence is $36 \times 0.5^{n-1}$.
Work out the first four terms of this sequence.
............ \ , \ ............ \ , \ ............ \ , \ ............ \ [2]
(d) These are the first four terms of another sequence.
5 \ \ \ 15 \ \ \ 45 \ \ \ 135
Find the $n^{\text{th}}$ term.
...................................................... \ [2]
596
511
526
The first two terms of a geometric sequence are 16 and 12.
Show that the 3rd term is 9.
594
505
503
(a) A geometric sequence has:
2nd term = 12.5
3rd term = 62.5.
(i) Work out the 1st term.
................................................... [3]
(ii) Find the $n$th term.
................................................... [1]
(b) Another geometric sequence has:
2nd term = 24
5th term = 81.
Work out the 6th term.
................................................... [4]
587
533
536
These are the first three terms of a geometric sequence.
x \quad 2x + 3 \quad 11x - 6
(a) (i) Give a reason why \(\frac{2x+3}{x} = \frac{11x-6}{2x+3}\).
...............................................................................................................................
............................................................................................................................... [1]
(ii) Show that \(7x^2 - 18x - 9 = 0\). [3]
(b) Solve \(7x^2 - 18x - 9 = 0\) by factorisation.
Give your answer as an integer and a fraction.
\(x = \text{..............................................}\quad x = \text{..............................................}\) [2]
(c) There are two different geometric sequences where the first three terms are \(x, 2x+3, 11x-6\).
Use your answer to part (b) to find the fourth term of each of these sequences.
.........................................................................
......................................................................... [4]
564
536
525
FILLING CONTAINERS (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at models for filling containers with water.
In this task all containers are empty when the water starts to flow into them.
A container is a cuboid 20 cm long, 20 cm wide and 25 cm high.
(a) Calculate the volume of the container.
\( \text{................................................} \) [2]
(b) Water flows into the container at a constant rate of \( 50 \, \text{cm}^3 \) per second.
(i) Show that it takes 8 seconds to fill the container to a height of 1 cm.
\( \text{..............................................................} \) [2]
(ii) How many seconds does it take to completely fill the container?
\( \text{................................................} \) [1]
(c) (i) Explain why the model for the time to fill the container is
\[ t = 8h \]
where \( t \) seconds is the time taken and \( h \) cm is the height of water.
............................................................................................................................
......................................................................................................................... [1]
(ii) Work out the height of the water in the container after 1 minute.
\( \text{................................................} \) [2]
(iii) Sketch the model for the time to fill the container.
[2]
599
502
591
A container is in the shape of a cone.
The circular top has a radius of 16 cm and the container has a height of 24 cm.
(a) Calculate the volume of the container.
Give your answer correct to 3 significant figures.
[2]
(b) Water is poured into the container to a height of 3 cm.
The surface of the water is a circle with radius $x$ cm.
(i) Show that $x = 2$.
[2]
(ii) Work out the volume of water in the container.
[2]
(iii) Water flows into the empty container at a constant rate of 20 cm$^3$ per second.
Work out the time it takes to fill the container to a height of 3 cm.
[2]
(c) The container is emptied.
Water is now poured into the container to a height of $h$ cm.
The surface of the water is a circle with radius $r$ cm.
(i) Water flows into the container at a constant rate of 20 cm$^3$ per second.
The time taken to fill the container to a height of $h$ cm is $t$ seconds.
Show that the model for the time to fill the container is $t = \frac{\pi h^3}{135}$.
[3]
(ii) How many seconds does it take to completely fill the container?
[2]
(iii) Sketch the model for the time to fill the container.
[2]
(d) The container is emptied.
Water is now poured into the container at a different rate.
The model for the time to fill the container is now $t = \frac{\pi h^3}{189}$.
Find the rate of flow for this model.
[2]
600
576
553
The containers in \textbf{Question 5} and \textbf{Question 6} are both empty. Water flows into the container in \textbf{Question 5} at a rate of 50 cm$^3$ per second. At the same time water flows into the container in \textbf{Question 6} at a rate of 20 cm$^3$ per second.
(a) Find the height of water when it is the same in both containers.
....................................................... [2]
(b) Find the time it takes for the water to reach this height.
....................................................... [1]
568
575
553
