No questions found
12 students are each given a spelling test. Here is a list of the scores.
9 5 10 9 11 7 7 6 6 7 8 11
Find
(a) the range, .................................................. [1]
(b) the mode, .................................................. [1]
(c) the median, .................................................. [1]
(d) the upper quartile, .................................................. [1]
(e) the inter-quartile range, .................................................. [1]
(f) the mean. .................................................. [1]
505
544
583
(a) Increase 4.5 kg by 16%.
................................................... kg [2]
(b) Find the percentage profit when the cost price of a book is $8.50 and the selling price is $11.05.
................................................... % [3]
(c) The price of a loaf of bread increases by $0.06. This is a 5% increase.
Find the original price of this loaf of bread.
$................................................... [2]
502
577
514
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-3$ and $3$. [3]
(b) Write down the range of $f(x)$ for $-3 \leq x \leq 0$. ................................................ [2]
(c) On the same diagram, sketch the graph of $y = x^2$ for $-2 \leq x \leq 2$. [1]
(d) (i) Solve the equation $\frac{1}{1-x^3} = x^2$. $x =$ ................................................. [1]
(ii) The equation $\frac{1}{1-x^3} = x^2$ can be written in the form $x^u - x^w + 1 = 0$.
Find the value of $u$ and the value of $w$.
$u = ................................................$
$w = ................................................$ [2]
545
598
579
(a) Describe fully the \textit{single} transformation that maps triangle $T$ onto
(i) triangle $A$,
..................................................................................................................
.................................................................................................................. [2]
(ii) triangle $B$,
..................................................................................................................
.................................................................................................................. [3]
(iii) triangle $C$.
..................................................................................................................
.................................................................................................................. [3]
(b) Stretch triangle $T$ by a factor of 2 with the $y$-axis invariant. [2]
548
507
539
Each year the value of a motor bike decreases by 10\% of its value at the start of the year. At the start of 2019, the value of the motor bike was $2025.
(a) Find the value at the end of 4 years. Give your answer correct to the nearest dollar.
$\text{.............................}\ [4]
(b) Find the value at the start of 2017.
$\text{.............................}\ [2]
(c) Find the number of complete years it takes for the value of $2025 to decrease to a value less than $500.
\text{.............................}\ [4]
519
569
564
The diagram shows a six-sided die and a coin.
The numbers on the faces of the die are 1, 1, 1, 2, 2, 3.
When the die is rolled it is equally likely for any of the six faces to be on the top. When the coin is spun it is equally likely to land showing heads or tails.
(a) Abi rolls the die.
Write down the probability that it shows the number 3 on the top.
................................................ [1]
(b) Beatrice rolls the die and spins the coin.
(i) Find the probability that the die shows the number 2 on the top and the coin shows heads.
................................................ [2]
(ii) Find the probability that the die shows the number 2 on the top or the coin shows heads or both.
................................................ [2]
(c) Carl spins the coin 3 times.
Find the probability that the coin shows heads at least once.
................................................ [2]
(d) Drew rolls the die 3 times and records the numbers on the top.
Find the probability that the die shows each of the numbers, 1, 2 and 3, once.
................................................ [3]
(e) Eva spins the coin $n$ times.
The probability that the coin shows tails each time is $\frac{1}{64}$.
Find the value of $n$.
$n = .............................................$ [1]
(f) Frank rolls the die twice and records the two numbers.
The probability of these two numbers occurring is $\frac{1}{3}$.
Find these two numbers.
............... and ............... [2]
600
542
593
The diagram shows four villages $A, B, C$ and $D$ and five straight roads connecting them.
$B$ is $10$ km due east of $A$.
$C$ is $12$ km from $B$ on a bearing of $150^\circ$.
$D$ is $21$ km from $C$ and $18$ km from $A$.
(a) Calculate the distance $AC$ and show that your answer rounds to $19.08$ km, correct to $2$ decimal places. [4]
(b) Using the sine rule, calculate angle $ACB$ and show that your answer rounds to $27.0^\circ$, correct to $1$ decimal place. [3]
(c) Calculate the bearing of $D$ from $C$. [4]
(d) A straight path, $BP$, connects $B$ to the closest point, $P$, on $AC$.
Calculate the length of this path. [2]
(e) The area within triangle $ABC$ is grassland.
Calculate the area of this grassland. [2]
535
540
506
(a) 200 people took part in a charity walk. They each recorded how far, \( d \) metres, they walked in one hour. The table shows the results.
[Table_1]
| Distance (\( d \) metres) | 1000 < \( d \leq 2000 \) | 2000 < \( d \leq 2500 \) | 2500 < \( d \leq 3000 \) | 3000 < \( d \leq 4000 \) |
|---|---|---|---|---|
| Number of people | 40 | 60 | 80 | 20 |
[Graph_1]
(i) Complete the cumulative frequency curve. [3]
(ii) Use your curve to find the inter-quartile range.
................................................. \( \text{m} \) [2]
(iii) Use your curve to estimate the number of people who walked further than 3500 m.
................................................. [2]
(b) 2000 people took part in a "NO FOOD FOR 6 HOURS" day. They each recorded the reduction in their mass, \( m \) grams, at the end of the day. The histogram shows their results.
[Histogram_1]
(i) Complete the frequency table. [2]
| Reduction in mass (\( m \) grams) | 0 < \( m \leq 50 \) | 50 < \( m \leq 100 \) | 100 < \( m \leq 200 \) | 200 < \( m \leq 400 \) |
|---|---|---|---|---|
| Number of people | 500 |
(ii) Calculate an estimate of the mean.
................................................. \( \text{g} \) [2]
542
567
566
(a) Lionel runs 10.6 km in 94 minutes.
Calculate his average speed in km/h.
............................................. km/h [2]
(b) Monika walks 2 km at a speed of 4 km/h and then 3 km at a speed of 3 km/h.
Calculate Monika’s overall average speed.
............................................. km/h [3]
(c) A train is travelling at $v$ metres per second.
Find an expression, in terms of $v$, for the speed of the train in kilometres per hour.
Give your answer in its simplest form.
............................................. km/h [2]
(d) (i) A car travels 50 km at $x$ km/h and then 80 km at $(x + 10)$ km/h.
Find an expression, in terms of $x$, for the total time taken, $T$ hours.
Give your answer as a single fraction, in its simplest form.
$$T = ext{.............................................}$$ h [3]
(ii) When $T = 2$, show that $x^2 - 55x - 250 = 0$.
[2]
(iii) When $T = 2$, find the value of $x$.
$$x = ext{.............................................}$$ [3]
554
563
527
Given the functions:
$f(x) = 2x + 3$
$g(x) = \frac{1}{x}, x \neq 0$
$h(x) = 2^x$
$j(x) = \log_3 x$
(a) Find
(i) $f(−2)$,
......................................................... [1]
(ii) $g\left(\frac{1}{2}\right)$.
......................................................... [1]
(b) Find $g(f(1))$.
......................................................... [2]
(c) Find $x$ when $h(x) = \frac{1}{8}$.
$x = $ ......................................................... [1]
(d) Find $j(81)$.
......................................................... [1]
(e) Find $f(f(x))$ in its simplest form.
......................................................... [2]
(f) Find $f(x) \times f(x) + f(x) + 1$ in its simplest form.
......................................................... [3]
(g) Find $j^{-1}(x)$.
$j^{-1}(x) = $ ......................................................... [2]
561
589
506
Given $f(x) = 3\sin(3x^\circ)$,
(a) On the diagram, sketch the graph of $y = f(x)$ for $0 \leq x \leq 180$. [2]
(b) Write down the amplitude and the period of $f(x)$.
Amplitude = ..................................................
Period = .................................................. [2]
(c) Solve the inequality $f(x) < -1.5$ for $0 \leq x \leq 180$.
.................................................. [2]
Given $g(x) = 3\sin(x^\circ)$,
(d) (i) On the same diagram, sketch the graph of $y = g(x)$ for $0 \leq x \leq 180$. [1]
(ii) On the diagram, shade the regions where $f(x) \geq g(x)$. [1]
(iii) Describe fully the single transformation that maps the graph of $y = g(x)$ onto the graph of $y = f(x)$.
.................................................................................................................................
.................................................................................................................................
................................................................................................................................. [3]
557
588
553
