Questions — WJEC (325 questions)

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WJEC Unit 3 Specimen Q13
13. (a) Solve the equation $$\operatorname { cosec } ^ { 2 } x + \cot ^ { 2 } x = 5$$ for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).
(b) (i) Express \(4 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } \leq \alpha \leq 90 ^ { \circ }\).
(ii) Solve the equation $$4 \sin \theta + 3 \cos \theta = 2$$ for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\), giving your answer correct to the nearest degree
WJEC Unit 3 Specimen Q14
14. (a) A cylindrical water tank has base area \(4 \mathrm {~m} ^ { 2 }\). The depth of the water at time \(t\) seconds is \(h\) metres. Water is poured in at the rate \(0.004 \mathrm {~m} ^ { 3 }\) per second. Water leaks from a hole in the bottom at a rate of \(0.0008 \mathrm { hm } ^ { 3 }\) per second. Show that $$5000 \frac { \mathrm {~d} h } { \mathrm {~d} t } \equiv 5 - h$$ [Hint: the volume, \(V\), of the cylindrical water tank is given by \(V = 4 h\).]
(b) Given that the tank is empty initially, find \(h\) in terms of \(t\).
(c) Find the depth of the water in the tank when \(t = 3600 \mathrm {~s}\), giving your answer correct to 2 decimal places.
WJEC Unit 3 Specimen Q15
15. Prove by contradiction the following proposition. When \(x\) is real and positive, $$4 x + \frac { 9 } { x } \geq 12$$ The first line of the proof is given below.
Assume that there is a positive and a real value of \(x\) such that $$4 x + \frac { 9 } { x } < 12$$
WJEC Unit 4 2018 June Q1
1 \(\mathbf { 0 }\) A particle of mass 2 kg moves under the action of a constant force \(\mathbf { F N }\), where \(\mathbf { F }\) is given by $$\mathbf { F } = - 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k }$$ a) Find the magnitude of the acceleration of the particle.
b) Given that at time \(t = 0\) seconds, the position vector of the particle is \(2 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k }\) and it is moving with velocity \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\), find the position vector of the particle when \(t = 2\) seconds. \section*{END OF PAPER}
WJEC Unit 4 2019 June Q1
\(\mathbf { 1 }\) | \(\mathbf { 0 }\) A tennis ball is projected with velocity vector \(( 30 \mathbf { i } - 1 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from a point \(P\) which is at a height of 2.4 m vertically above a horizontal tennis court. The ball then passes over a net of height 0.9 m , before hitting the ground after \(\frac { 4 } { 7 } \mathrm {~s}\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. The origin \(O\) lies on the ground directly below the point \(P\). The base of the net is \(x \mathrm {~m}\) from \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{7d864f3d-0bb8-41ff-8e98-fee9776c037e-8_456_1655_689_203}
a) Find the speed of the ball when it first hits the ground, giving your answer correct to one decimal place.
b) After \(\frac { 2 } { 5 } \mathrm {~s}\), the ball is directly above the net.
i) Find the position vector of the ball after \(\frac { 2 } { 5 } \mathrm {~s}\).
ii) Hence determine the value of \(x\) and show that the ball clears the net by approximately 16 cm .
c) In fact, the ball clears the net by only 4 cm .
i) Explain why the observed value is different from the value calculated in (b)(ii).
ii) Suggest a possible improvement to this model.
WJEC Unit 4 2024 June Q1
  1. The table below shows the destination from school of 180 year 11 pupils. Most pupils either continued education, in school or college, or went into some form of employment.
\cline { 2 - 6 } \multicolumn{1}{c|}{}SchoolCollegeEmploymentOtherTotal
Boys334982\(\mathbf { 9 2 }\)
Girls404071\(\mathbf { 8 8 }\)
Total\(\mathbf { 7 3 }\)\(\mathbf { 8 9 }\)\(\mathbf { 1 5 }\)\(\mathbf { 3 }\)\(\mathbf { 1 8 0 }\)
A reporter selects two pupils at random to interview. Given that the first pupil is in school or college, find the probability that both pupils are girls.
WJEC Unit 4 2024 June Q2
2. The smallest angle \(\theta\), in degrees, of a right-angled triangle with hypotenuse 8 cm , is uniformly distributed across all possible values.
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-04_419_696_479_687}
  1. Find the mean and standard deviation of \(\theta\).
  2. The shortest side of the triangle is of length \(X \mathrm {~cm}\). Find the probability that \(X\) is greater than 5 .
WJEC Unit 4 2024 June Q3
3. Awena has a large data set of body measurements, and she wants to investigate relationships between body dimensions. In this particular investigation, she is testing for a correlation between forearm girth and bicep girth. The diagrams below show how to measure these. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-06_499_590_534_276} \captionsetup{labelformat=empty} \caption{Forearm girth}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-06_499_591_534_1194} \captionsetup{labelformat=empty} \caption{Bicep girth}
\end{figure}
  1. Why is it appropriate for Awena to use a one-tailed test?
    Awena takes a random sample of size 11 from her data set and plots the following scatter diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-07_937_1431_420_312}
  2. Using the computer output above, carry out a one-tailed significance test on the sample product moment correlation coefficient at the \(0 \cdot 5 \%\) level.
  3. Blodwen also has access to the same large data set. She decides to do the same test using all of the 507 available data points. Her results are shown below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Forearm girth versus Bicep girth} \includegraphics[alt={},max width=\textwidth]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-08_933_1504_477_276}
    \end{figure}
    1. State the problem Blodwen will encounter when attempting to use statistical tables for her test.
    2. How should Blodwen deal with this problem?
      \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 4 2024 June Q4
  1. Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean \(2 \cdot 2\) and standard deviation \(0 \cdot 3\).
    1. Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .
    Jake delivers the lightest \(80 \%\) of parcels on his bike. The rest he puts in his car and delivers by car.
  2. Find the mass of the heaviest parcel he would deliver by bike.
  3. He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .
  4. In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the \(5 \%\) level of significance. Jake delivers each parcel to one of three areas, \(A , B\) or \(C\). The probabilities that a parcel has destination area \(A , B\) and \(C\) are \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. All parcels are considered to be independent.
  5. On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.
  6. On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.
WJEC Unit 4 2024 June Q5
7 marks
  1. The diagram below shows four coplanar horizontal forces of magnitude \(F \mathrm {~N} , 12 \mathrm {~N} , 16 \mathrm {~N}\) and 20 N acting at a point \(P\) in the directions shown.
    \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-14_792_862_593_607}
Given that the forces are in equilibrium, calculate the value of \(F\) and the size of the angle \(\alpha\). [7]
WJEC Unit 4 2024 June Q6
6. A ball is projected with velocity \(( 4 w \mathbf { i } + 7 w \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the top of a vertical tower. After 5 seconds, the ball hits the ground at a point that is 60 m horizontally from the foot of the tower. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.
  1. Find the value of \(w\) and hence determine the height of the tower.
  2. Determine the proportion of the 5 seconds for which the ball is on its way down.
WJEC Unit 4 2024 June Q7
3 marks
7. As part of a design for a new building, an architect wants to support a wooden beam in a horizontal position. The beam is suspended using a vertical steel cable and a smooth fixed support on its underside. The diagram below shows the architect's diagram and the adjacent table shows the categories of steel cable available.
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-18_504_1699_559_191} You may use the following modelling assumptions.
  • The wooden beam is a rigid uniform rod of mass 100 kg .
  • The force exerted on the beam by the support is vertical.
  • The steel cable is inextensible.
\section*{SAFETY REQUIREMENT} Both the steel cable and the support must be capable of withstanding forces of at least four times those present in the architect's diagram above. The wooden beam is held in horizontal equilibrium.
[0pt]
    1. Given that the support is capable of withstanding loads of up to 2000 N , show that the force exerted on the beam by the support satisfies the safety requirement. [3]
    2. Determine which categories of steel cable in the table opposite could meet the safety requirements.
  2. State how you have used the modelling assumption that the beam is a uniform rod. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Unit 4 2024 June Q8
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are acting on an object of mass 3 kg such that
$$\begin{aligned} & \mathbf { F } _ { 1 } = ( \mathbf { i } + 8 c \mathbf { j } + 11 c \mathbf { k } ) \mathrm { N } ,
& \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } - c \mathbf { j } - 12 \mathbf { k } ) \mathrm { N } ,
& \mathbf { F } _ { 3 } = ( ( 15 c + 1 ) \mathbf { i } + 2 c \mathbf { j } - 5 c \mathbf { k } ) \mathrm { N } , \end{aligned}$$ where \(c\) is a constant. The acceleration of the object is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).
  1. Find the value of the constant \(c\) and hence show that the acceleration of the object is \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\).
  2. When \(t = 0\) seconds, the object has position vector \(\mathbf { r } _ { 0 } \mathrm {~m}\) and is moving with velocity \(( - 17 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When \(t = 4\) seconds, the object has position vector \(( - 13 \mathbf { i } + 84 \mathbf { j } ) \mathrm { m }\). Find the vector \(\mathbf { r } _ { 0 }\).
WJEC Unit 4 2024 June Q9
1 marks
9. The diagram below shows a parcel, of mass \(m \mathrm {~kg}\), sliding down a rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 7 } { 25 }\).
\includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-24_394_906_497_584} The coefficient of friction between the parcel and the slope is \(\frac { 1 } { 12 }\). In addition to friction, the parcel experiences a variable resistive force of \(m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the parcel at time \(t\) seconds.
  1. Show that the motion of the parcel satisfies the differential equation $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$
    numberAdditional page, if required.Examiner only
    \multirow{6}{*}{}
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 2019 June Q1
  1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 7
    - 2 & 0 \end{array} \right)\), \(\mathbf { B } = \left( \begin{array} { l l } 5 & 1
    0 & 4 \end{array} \right)\).
The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).
WJEC Further Unit 1 2019 June Q2
2. The position vectors of the points \(A , B , C , D\) are given by
\(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\),
\(\mathbf { b } = 4 \mathbf { j } + 5 \mathbf { k }\),
\(\mathbf { c } = 7 \mathbf { i } - 3 \mathbf { k }\),
\(\mathbf { d } = - 3 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }\),
respectively.
  1. Find the vector equations of the lines \(A B\) and \(C D\).
  2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.
WJEC Further Unit 1 2019 June Q3
3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.
  1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
  2. Find the complex number \(\frac { z } { w }\).
WJEC Further Unit 1 2019 June Q4
4. Prove, by mathematical induction, that \(9 ^ { n } + 15\) is a multiple of 8 for all positive integers \(n\).
WJEC Further Unit 1 2019 June Q5
5. Given that \(x = - \frac { 1 } { 2 }\) and \(x = - 3\) are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.
WJEC Further Unit 1 2019 June Q6
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.
WJEC Further Unit 1 2019 June Q7
7. (a) Find an expression for \(\sum _ { r = 1 } ^ { 2 m } ( r + 2 ) ^ { 2 }\) in the form \(\frac { 1 } { 3 } m \left( a m ^ { 2 } + b m + c \right)\), where \(a , b , c\) are integers whose values are to be determined.
(b) Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).
WJEC Further Unit 1 2019 June Q8
8. The plane \(\Pi\) contains the three points \(A ( 3,5,6 ) , B ( 5 , - 1,7 )\) and \(C ( - 1,7,0 )\). Find the vector equation of the plane \(\Pi\) in the form r.n \(= d\).
Express this equation in Cartesian form.
WJEC Further Unit 1 2019 June Q9
9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$
  1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).
WJEC Further Unit 1 2019 June Q10
10. The quadratic equation \(p x ^ { 2 } + q x + r = 0\) has roots \(\alpha\) and \(\beta\), where \(p , q , r\) are non-zero constants.
  1. A cubic equation is formed with roots \(\alpha , \beta , \alpha + \beta\). Find the cubic equation with coefficients expressed in terms of \(p , q , r\).
  2. Another quadratic equation \(p x ^ { 2 } - q x - r = 0\) has roots \(2 \alpha\) and \(\gamma\). Show that \(\beta = - 2 \gamma\).
WJEC Further Unit 1 2022 June Q1
  1. The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).
    1. (i) Find the modulus and argument of \(z w\).
      (ii) Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
    2. The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
    3. The complex conjugate of \(v\) is denoted by \(\bar { v }\).
    Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.