Questions - WJEC - A-Level Maths

The point $( c , 1 )$ lies on $y = f ( x )$. Find the value of $c$.
A point $P$ lies on $y = f ( x )$ and has $x$-coordinate $\alpha$. Another point $Q$ lies on $y = g ( x )$ and also has $x$-coordinate $\alpha$. The distance between $P$ and $Q$ is 1.2 units. Find the value of $\alpha$, giving your answer correct to three decimal places.

WJEC Unit 1 2024 June Q18

18. (a) A circle $C$ has centre $( - 3 , - 1 )$ and radius $\sqrt { 5 }$. Show that the equation of $C$ can be written as $x ^ { 2 } + y ^ { 2 } + 6 x + 2 y + 5 = 0$.
(b) (i) Find the equations of the tangents to $C$ that pass through the origin $O$.
(ii) Determine the coordinates of the points where the tangents touch the circle.
Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Unit 1 Specimen Q1

The circle $C$ has centre $A$ and equation

$$x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 15 = 0 .$$

Find the coordinates of $A$ and the radius of $C$.
The point $P$ has coordinates ( $4 , - 7$ ) and lies on $C$. Find the equation of the tangent to $C$ at $P$.

WJEC Unit 1 Specimen Q2

2. Find all values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$ satisfying $$7 \sin ^ { 2 } \theta + 1 = 3 \cos ^ { 2 } \theta - \sin \theta$$

WJEC Unit 1 Specimen Q3

\begin{enumerate} \setcounter{enumi}{2} \item Given that $y = x ^ { 3 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ from first principles. \item The cubic polynomial $f ( x )$ is given by $f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are constants. The graph of $f ( x )$ intersects the $x$-axis at the points with coordinates $( - 3,0 ) , ( 2 \cdot 5,0 )$ and $( 4,0 )$. Find the coordinates of the point where the graph of $f ( x )$ intersects the $y$-axis. \item The points $A ( 0,2 ) , B ( - 2,8 ) , C ( 20,12 )$ are the vertices of the triangle $A B C$. The point $D$ is the mid-point of $A B$.

Show that $C D$ is perpendicular to $A B$.
Find the exact value of $\tan C \hat { A } B$.
Write down the geometrical name for the triangle $A B C$. \item In each of the two statements below, $c$ and $d$ are real numbers. One of the statements is true while the other is false.
A Given that $( 2 c + 1 ) ^ { 2 } = ( 2 d + 1 ) ^ { 2 }$, then $c = d$.
B Given that $( 2 c + 1 ) ^ { 3 } = ( 2 d + 1 ) ^ { 3 }$, then $c = d$.

WJEC Unit 1 Specimen Q8

8. The circle $C$ has radius 5 and its centre is the origin. The point $T$ has coordinates $( 11,0 )$.
The tangents from $T$ to the circle $C$ touch $C$ at the points $R$ and $S$.

Write down the geometrical name for the quadrilateral ORTS.
Find the exact value of the area of the quadrilateral ORTS. Give your answer in its simplest form.

WJEC Unit 1 Specimen Q9

9. The quadratic equation $4 x ^ { 2 } - 12 x + m = 0$, where $m$ is a positive constant, has two distinct real roots.
Show that the quadratic equation $3 x ^ { 2 } + m x + 7 = 0$ has no real roots.

WJEC Unit 1 Specimen Q10

10. (a) Use the binomial theorem to express $( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 }$ in the form $a \sqrt { 3 } + b \sqrt { 2 }$, where $a , b$ are integers whose values are to be found.
(b) Given that $( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 } \approx 0$, use your answer to part (a) to find an approximate value for $\sqrt { 6 }$ in the form $\frac { c } { d }$, where $c$ and $d$ are positive integers whose values are to be found.

WJEC Unit 1 Specimen Q11

11.
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-4_609_951_1541_605} The diagram shows a sketch of the curve $y = 6 + 4 x - x ^ { 2 }$ and the line $y = x + 2$. The point $P$ has coordinates ( $a , b$ ). Write down the three inequalities involving $a$ and $b$ which are such that the point $P$ will be strictly contained within the shaded area above, if and only if, all three inequalities are satisfied.

WJEC Unit 1 Specimen Q12

12. Prove that $$\log _ { 7 } a \times \log _ { a } 19 = \log _ { 7 } 19$$ whatever the value of the positive constant $a$.

WJEC Unit 1 Specimen Q13

13. In triangle $A B C , B C = 12 \mathrm {~cm}$ and $\cos A \hat { B } C = \frac { 2 } { 3 }$. The length of $A C$ is 2 cm greater than the length of $A B$.

Find the lengths of $A B$ and $A C$.
Find the exact value of $\sin B \hat { A } C$. Give your answer in its simplest form.

WJEC Unit 1 Specimen Q14

14. The diagram below shows a closed box in the form of a cuboid, which is such that the length of its base is twice the width of its base. The volume of the box is $9000 \mathrm {~cm} ^ { 3 }$. The total surface area of the box is denoted by $S \mathrm {~cm} ^ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-5_357_915_1190_543}

Show that $S = 4 x ^ { 2 } + \frac { 27000 } { x }$, where $x \mathrm {~cm}$ denotes the width of the base.
Find the minimum value of $S$, showing that the value you have found is a minimum value.

WJEC Unit 1 Specimen Q15

15. The size $N$ of the population of a small island at time $t$ years may be modelled by $N = A \mathrm { e } ^ { k t }$, where $A$ and $k$ are constants. It is known that $N = 100$ when $t = 2$ and that $N = 160$ when $t = 12$.

Interpret the constant $A$ in the context of the question.
Show that $k = 0 \cdot 047$, correct to three decimal places.
Find the size of the population when $t = 20$.

WJEC Unit 1 Specimen Q16

16. Find the range of values of $x$ for which the function $$f ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 13$$ is an increasing function.

WJEC Unit 1 Specimen Q17

17.
\includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-6_705_1130_623_438} The diagram above shows a sketch of the curve $y = 3 x - x ^ { 2 }$. The curve intersects the $x$-axis at the origin and at the point $A$. The tangent to the curve at the point $B ( 2,2 )$ intersects the $x$-axis at the point C .

Find the equation of the tangent to the curve at $B$.
Find the area of the shaded region.

WJEC Unit 1 Specimen Q18

18. (a) The vectors $\mathbf { u }$ and $\mathbf { v }$ are defined by $\mathbf { u } = 2 \mathbf { i } - 3 \mathbf { j } , \mathbf { v } = - 4 \mathbf { i } + 5 \mathbf { j }$.

Find the vector $4 \mathbf { u } - 3 \mathbf { v }$.
The vectors $\mathbf { u }$ and $\mathbf { v }$ are the position vectors of the points $U$ and $V$, respectively. Find the length of the line UV.
(b) Two villages $A$ and $B$ are 40 km apart on a long straight road passing through a desert. The position vectors of $A$ and $B$ are denoted by $\mathbf { a }$ and $\mathbf { b }$, respectively.
Village $C$ lies on the road between $A$ and $B$ at a distance 4 km from $B$. Find the position vector of $C$ in terms of $\mathbf { a }$ and $\mathbf { b }$.
Village $D$ has position vector $\frac { 2 } { 9 } \mathbf { a } + \frac { 5 } { 9 } \mathbf { b }$. Explain why village $D$ cannot possibly be on the straight road passing through $A$ and $B$.

WJEC Unit 2 2018 June Q11

11 A vehicle moves along a straight horizontal road. Points $A$ and $B$ lie on the road. As the vehicle passes point $A$, it is moving with constant speed $15 \mathrm {~ms} ^ { - 1 }$. It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point $B$.
a) Find the distance $A B$. The vehicle then reverses with a constant acceleration of $2 \mathrm {~ms} ^ { - 2 }$ for 8 seconds, followed by a constant deceleration of $1.6 \mathrm {~ms} ^ { - 2 }$, coming to rest at the point $C$, which is between $A$ and $B$.
b) Calculate the time it takes for the vehicle to reverse from $B$ to $C$.
c) Sketch a velocity-time graph for the motion of the vehicle.
d) Determine the distance $A C$.

WJEC Unit 2 2022 June Q1

1 In this question, $\mathbf { i }$ and $\mathbf { j }$ represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

the shop has position vector $\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }$,
the park is 2 km away on a bearing of $060 ^ { \circ }$.
a) Show that the position vector of the park relative to the house is $( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }$.
b) Determine the total distance walked by Sarah from her house to the park.
c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

$\mathbf { 1 }$

A particle $P$ moves along the $x$-axis so that its velocity $v \mathrm {~ms} ^ { - 1 }$ at time $t$ seconds $( t \geqslant 0 )$ is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of $t$ when $P$ is instantaneously at rest.
b) Calculate the total distance travelled by the particle $P$ whilst its velocity is decreasing.

WJEC Unit 2 2024 June Q1

An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.
1. What is the intended population in this context?
2. What type of sampling is this?
3. How could the sampling process be improved?
4. A baker sells $3 \cdot 5$ birthday cakes per hour on average.
5. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution.
6. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period.
  （c）Calculate the probability that，during a randomly selected 3 －hour period，the baker sells more than 10 birthday cakes．
  （d）The baker sells a birthday cake at 9：30 a．m．Calculate the probability that the baker will sell the next birthday cake before 10：00 a．m．
7. Select one of the assumptions in part (a) and comment on its reasonableness.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Unit 2 2024 June Q3

The following Venn diagram shows the participation of 100 students in three activities, $A , B$, and $C$, which represent athletics, baseball and climbing respectively.
\includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-08_641_1050_477_511}

For these 100 students, participation in athletics and participation in climbing are independent events.

Show that $x = 10$ and find the value of $y$.
Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing.

WJEC Unit 2 2024 June Q4

4. A company produces sweets of varying colours. The company claims that the proportion of blue sweets is $13.6 \%$. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

State suitable hypotheses for the test.
1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, $5 \%$.
2. Given that there are 6 blue sweets in the sample of 80, complete the significance test.
Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion?
Now suppose that the proportion of blue sweets is $7 \%$. Find the probability of a Type II error. Interpret your answer in context.

WJEC Unit 2 2024 June Q5

4 marks

5. In March 2020, the coronavirus pandemic caused major disruption to the lives of individuals across the world. A newspaper published the following graph from the \href{http://gov.uk}{gov.uk} website, along with an article which included the following excerpt.
"The daily number of vaccines administered continues to fall. In order to get control of the virus, we need the number of people receiving a second dose of the vaccine to keep rocketing. The fear is it will start to drop off soon, which will leave many people still unprotected." \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Number of people (1000s) who received 2nd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-12_531_1525_906_260}

\end{figure}

By referring to the graph, explain how the quote could be misleading.
The daily numbers of second dose vaccines, in thousands, over the period April 1st 2021 to May 31st 2021 are shown in the table below.

Daily number of 2nd dose vaccines (1000s)	Midpoint $x$	Frequency $f$	Percentage
$0 \leqslant v < 100$	50	2	3.3
$100 \leqslant v < 200$	150	8	13.1
$200 \leqslant v < 300$	250	10	16.4
$300 \leqslant v < 400$	350	13	21.3
$400 \leqslant v < 500$	450	26	$42 \cdot 6$
$500 \leqslant v < 600$	550	2	3.3
Total		61	100

1. Calculate estimates of the mean and standard deviation for the daily number of second dose vaccines given over this period. You may use $\sum x ^ { 2 } f = 8272500$. [4]
2. Comment on the skewness of these data.
  A second graph in the article shows the number of people receiving a third dose of the vaccine. This graph has a repeated pattern of rising then falling. An extract is shown below. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Number of people (millions) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-14_556_1115_571_466}
  \end{figure}
Give a possible reason for the pattern observed in this graph.
Another extract shows the number of people who received the third dose of the vaccine between 27th March 2022 and 25th April 2022. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Number of people (1000s) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-15_537_1246_571_406}
\end{figure}
State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect.

WJEC Unit 2 2024 June Q6

2 marks

A ship $S$ is moving with constant velocity $( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively.
Find the speed and direction of $S$, giving the direction as a three-figure bearing, correct to the nearest degree.
The diagram below shows a forklift truck being used to raise two boxes, $P$ and $Q$, vertically. Box $Q$ rests on horizontal forks and box $P$ rests on top of box $Q$. Box $P$ has mass 25 kg and box $Q$ has mass 55 kg .
\includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-17_504_814_504_589}
1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box $Q$ is 820 N . Calculate the magnitude of the acceleration.
2. Calculate the reaction of box $Q$ on box $P$ when they are moving vertically upwards with constant speed.
3. A particle, of mass 4 kg , moves in a straight line under the action of a single force $F \mathrm {~N}$, whose magnitude at time $t$ seconds is given by
$$F = 12 \sqrt { t } - 32 \text { for } t \geqslant 0 .$$
Find the acceleration of the particle when $t = 9$.
Given that the particle has velocity $- 1 \mathrm {~ms} ^ { - 1 }$ when $t = 4$, find an expression for the velocity of the particle at $t \mathrm {~s}$.
Determine whether the speed of the particle is increasing or decreasing when $t = 9$. [2]

WJEC Unit 2 2024 June Q9

9. The diagram below shows an object $A$, of mass $2 m \mathrm {~kg}$, lying on a horizontal table. It is connected to another object $B$, of mass $m \mathrm {~kg}$, by a light inextensible string, which passes over a smooth pulley $P$, fixed at the edge of the table. Initially, object $A$ is held at rest so that object $B$ hangs freely with the string taut.
\includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-20_589_871_593_605} Object $A$ is then released.

When object $B$ has moved downwards a vertical distance of 0.4 m , its speed is $1.2 \mathrm {~ms} ^ { - 1 }$. Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of $B$ is $1.8 \mathrm {~ms} ^ { - 2 }$.
During the motion, object $A$ experiences a constant resistive force of 22 N . Find the value of $m$ and hence determine the tension in the string.
What assumption did the word 'inextensible' in the description of the string enable you to make in your solution?
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

Daily number of 2nd dose vaccines (1000s)	Midpoint \(x\)	Frequency \(f\)	Percentage
\(0 \leqslant v < 100\)	50	2	3.3
\(100 \leqslant v < 200\)	150	8	13.1
\(200 \leqslant v < 300\)	250	10	16.4
\(300 \leqslant v < 400\)	350	13	21.3
\(400 \leqslant v < 500\)	450	26	\(42 \cdot 6\)
\(500 \leqslant v < 600\)	550	2	3.3
Total		61	100

Questions — WJEC (325 questions)

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