Questions — WJEC (504 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
WJEC Unit 3 Specimen Q3
8 marks Moderate -0.3
  1. Sketch the graph of \(y = x^2 + 6x + 13\), identifying the stationary point. [2]
  2. The function \(f\) is defined by \(f(x) = x^2 + 6x + 13\) with domain \((a,b)\).
    1. Explain why \(f^{-1}\) does not exist when \(a = -10\) and \(b = 10\). [1]
    2. Write down a value of \(a\) and a value of \(b\) for which the inverse of \(f\) does exist and derive an expression for \(f^{-1}(x)\). [5]
WJEC Unit 3 Specimen Q4
4 marks Moderate -0.8
  1. Expand \((1-x)^{-\frac{1}{2}}\) in ascending power of \(x\) as far as the term in \(x^2\). State the range of \(x\) for which the expansion is valid. [2]
  2. By taking \(x = \frac{1}{10}\), find an approximation for \(\sqrt{10}\) in the form \(\frac{a}{b}\), where \(a\) and \(b\) are to be determined. [2]
WJEC Unit 3 Specimen Q5
5 marks Standard +0.3
Aled decides to invest £1000 in a savings scheme on the first day of each year. The scheme pays 8% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by $$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08^2 + 1000 \times 1 \cdot 08^3$$ Calculate, to the nearest pound, the amount of savings at the end of thirty years. [5]
WJEC Unit 3 Specimen Q6
4 marks Standard +0.3
The lengths of the sides of a fifteen-sided plane figure form an arithmetic sequence. The perimeter of the figure is 270 cm and the length of the largest side is eight times that of the smallest side. Find the length of the smallest side. [4]
WJEC Unit 3 Specimen Q7
16 marks Standard +0.8
The curve \(y = ax^4 + bx^3 + 18x^2\) has a point of inflection at \((1, 11)\).
  1. Show that \(2a + b + 6 = 0\). [2]
  2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \((3, 27)\). [8]
  3. Sketch the curve, identifying all the stationary points including their nature. [6]
WJEC Unit 3 Specimen Q8
14 marks Standard +0.3
  1. Integrate
    1. \(e^{-3x+5}\) [2]
    2. \(x^2 \ln x\) [4]
  2. Use an appropriate substitution to show that $$\int_0^{\frac{1}{2}} \frac{x^2}{\sqrt{1-x^2}} dx = \frac{\pi}{12} - \frac{\sqrt{3}}{8}.$$ [8]
WJEC Unit 3 Specimen Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram above shows a sketch of the curves \(y = x^2 + 4\) and \(y = 12 - x^2\). Find the area of the region bounded by the two curves. [6]
WJEC Unit 3 Specimen Q10
15 marks Standard +0.3
The equation $$1 + 5x - x^4 = 0$$ has a positive root \(\alpha\).
  1. Show that \(\alpha\) lies between 1 and 2. [2]
  2. Use the iterative sequence based on the arrangement $$x = \sqrt[4]{1+5x}$$ with starting value 1.5 to find \(\alpha\) correct to two decimal places. [3]
  3. Use the Newton-Raphson method to find \(\alpha\) correct to six decimal places. [6]
WJEC Unit 3 Specimen Q11
11 marks Standard +0.3
  1. The curve \(C\) is given by the equation $$x^4 + x^2 y + y^2 = 13.$$ Find the value of \(\frac{dy}{dx}\) at the point \((-1, 3)\). [4]
  2. Show that the equation of the normal to the curve \(y^2 = 4x\) at the point \(P(p^2, 2p)\) is $$y + px = 2p + p^3.$$ Given that \(p \neq 0\) and that the normal at \(P\) cuts the \(x\)-axis at \(B(b, 0)\), show that \(b > 2\). [7]
WJEC Unit 3 Specimen Q12
9 marks Standard +0.3
  1. Differentiate \(\cos x\) from first principles. [5]
  2. Differentiate the following with respect to \(x\), simplifying your answer as far as possible.
    1. \(\frac{3x^2}{x^3+1}\) [2]
    2. \(x^3 \tan 3x\) [2]
WJEC Unit 3 Specimen Q13
12 marks Standard +0.3
  1. Solve the equation $$\operatorname{cosec}^2 x + \cot^2 x = 5$$ for \(0^{\circ} \leq x \leq 360^{\circ}\). [5]
    1. Express \(4\sin \theta + 3\cos \theta\) in the form \(R\sin(\theta + \alpha)\), where \(R > 0\) and \(0^{\circ} \leq \alpha \leq 90^{\circ}\). [4]
    2. 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. [3]
WJEC Unit 3 Specimen Q14
10 marks Standard +0.3
  1. A cylindrical water tank has base area 4 m\(^2\). The depth of the water at time \(t\) seconds is \(h\) metres. Water is poured in at the rate 0.004 m\(^3\) per second. Water leaks from a hole in the bottom at a rate of 0.0008\(h\) m\(^3\) per second. Show that $$5000 \frac{dh}{dt} = 5 - h.$$ [2] [Hint: the volume, \(V\), of the cylindrical water tank is given by \(V = 4h\).]
  2. Given that the tank is empty initially, find \(h\) in terms of \(t\). [7]
  3. Find the depth of the water in the tank when \(t = 3600\) s, giving your answer correct to 2 decimal places. [1]
WJEC Unit 3 Specimen Q15
3 marks Moderate -0.5
Prove by contradiction the following proposition. When \(x\) is real and positive, $$4x + \frac{9}{x} \geq 12.$$ The first line of the proof is given below. Assume that there is a positive and a real value of \(x\) such that $$4x + \frac{9}{x} < 12.$$ [3]
WJEC Unit 4 2018 June Q1
7 marks Easy -1.2
An architect bids for two construction projects. He estimates the probability of winning bid \(A\) is \(0 \cdot 6\), the probability of winning bid \(B\) is \(0 \cdot 5\) and the probability of winning both is \(0 \cdot 2\).
  1. Show that the probability that he does not win either bid is \(0 \cdot 1\). [2]
  2. Find the probability that he wins exactly one bid. [2]
  3. Given that he does not win bid \(A\), find the probability that he wins bid \(B\). [3]
WJEC Unit 4 2018 June Q2
7 marks Moderate -0.8
  1. Marie is an athlete who competes in the high jump. In a certain competition she is allowed two attempts to clear each height, but if she is successful with the first attempt she does not jump again at this height. The probability that she is successful with her first jump at a height of \(1 \cdot 7\) m is \(p\). The probability that she is successful with her second jump is also \(p\). The probability that she clears \(1 \cdot 7\) m is \(0 \cdot 64\). Find the value of \(p\). [4]
  2. The following table shows the numbers of male and female athletes competing for Wales in track and field events at a competition.
    TrackField
    Male139
    Female74
    Two athletes are chosen at random to participate in a drugs test. Given that the first athlete is male, find the probability that both are field athletes. [3]
WJEC Unit 4 2018 June Q3
10 marks Standard +0.3
Antonio arrives at a train station at a random point in time. The trains to his desired destination are scheduled to depart at 12-minute intervals.
  1. Assume that Antonio gets on the next train.
    1. Suggest an appropriate distribution to model his waiting time and give the parameters.
    2. State the mean and the variance of this distribution.
    3. State an assumption you have made in suggesting this distribution. [4]
  2. Now assume that the probability that Antonio misses the next available train because he is distracted by his smartphone is \(0 \cdot 12\). If he misses the next available train, he is sure to get on the one after that.
    1. Find the probability that he waits between 9 and 19 minutes.
    2. Given that he waits between 9 and 19 minutes, find the probability that he gets on the first train. [6]
WJEC Unit 4 2018 June Q4
8 marks Moderate -0.8
Arwyn collects data about household expenditure on food. He records the weekly expenditure on food for 80 randomly selected households from across Wales.
Cost, \(x\) (£)\(x < 40\)\(40 \leqslant x<50\)\(50 \leqslant x<60\)\(60 \leqslant x<70\)\(70 \leqslant x<80\)\(80 \leqslant x<90\)\(x \geqslant 90\)
Number of households51116181596
  1. Explain why a normal distribution may be an appropriate model for the weekly expenditure on food for this sample. [1]
Arwyn uses the distribution N(64, 15²) to model expenditure on food.
  1. Find the number of households in the sample that this model would predict to have weekly food expenditure in the range
    1. \(60 \leqslant x < 70\),
    2. \(x \geqslant 90\). [4]
  2. Use your answers to part (b)
    1. to comment on the suitability of this model,
    2. to explain how Arwyn could improve the model by changing one of its parameters. [2]
  3. Arwyn's friend Colleen wishes to use the improved model to predict household expenditure on food in Northern Ireland. Comment on this plan. [1]
WJEC Unit 4 2018 June Q5
8 marks Moderate -0.8
Rebecca is a farmer who is monitoring prices for products to use on her farm. She records the prices of two products made from different grains, wheat and oats, at random points in time, to investigate whether there is any correlation. \includegraphics{figure_1} The product moment correlation coefficient for the data is \(0 \cdot 244\). There are 12 data points, and the \(p\)-value is \(0 \cdot 4447\).
  1. Comment on the correlation between the prices of Feed Wheat and Feed Oats. [2]
Rebecca also records the prices of two wheat products at random points in time, to investigate whether there is any correlation. \includegraphics{figure_2} The product moment correlation coefficient for the data is \(0 \cdot 653\). There are 12 data points.
  1. Stating your hypotheses clearly, test at the 5% level of significance whether there is any evidence of correlation between the prices of these two products. [5]
  2. Without referring to the positioning of the points on the graphs, suggest why the product moment correlation coefficient is higher for the second set of data. [1]
WJEC Unit 4 2018 June Q6
4 marks Moderate -0.3
The diagram shows a uniform plank \(AB\) of length 4 m supported in horizontal equilibrium by means of a central pivot. On the plank there are three objects of masses 8 kg, 2 kg and 15 kg placed in positions \(C\), \(D\) and \(E\) respectively. The distance \(AC\) is \(0 \cdot 6\) m and the distance \(AE\) is \(2 \cdot 8\) m. \includegraphics{figure_3} Find the distance \(AD\). [4]
WJEC Unit 4 2018 June Q7
11 marks Standard +0.3
An object of mass \(0 \cdot 5\) kg is thrown vertically upwards with initial speed \(24\) ms\(^{-1}\). The velocity of the object at time \(t\) seconds is \(v\) ms\(^{-1}\). During the upward motion, the object experiences a resistance to motion \(RN\), where \(R\) is proportional to \(v\). When the velocity of the object is \(0 \cdot 2\) ms\(^{-1}\) the resistance to motion is \(0 \cdot 08\) N.
  1. Show that the upward motion of the object satisfies the differential equation $$\frac{\mathrm{d}v}{\mathrm{d}t} = -9 \cdot 8 - 0 \cdot 8\,v.$$ [3]
  2. Find an expression for \(v\) at time \(t\). [6]
  3. Determine the value of \(t\) when the object is at the highest point of the motion. [2]
WJEC Unit 4 2018 June Q8
9 marks Moderate -0.3
An object of mass 60 kg is on a rough plane inclined at an angle of 20° to the horizontal. The coefficient of friction between the object and the plane is \(0 \cdot 3\). Initially, the object is held at rest. A force which is parallel to the plane and of magnitude \(T\) N is applied to the object in an upward direction along the line of greatest slope. The object is then released.
  1. Given that \(T = 15\), calculate the acceleration of the object down the plane. [6]
  2. Given that \(T = 350\), determine whether or not the object moves up the plane. Give a reason for your answer. [3]
WJEC Unit 4 2018 June Q9
10 marks Standard +0.8
Points \(A\) and \(B\) lie on horizontal ground. At time \(t = 0\) seconds, an object \(P\) is projected from \(A\) towards \(B\) such that \(AB\) is the range of \(P\). The speed of projection is \(24 \cdot 5\) ms\(^{-1}\) in a direction which is 30° above the horizontal.
  1. Calculate the range \(AB\) of the object \(P\). [5]
At time \(t = 1\) second, another object \(Q\) is projected from \(B\) towards \(A\) with the same speed of projection \(24 \cdot 5\) ms\(^{-1}\) and in a direction which is also 30° above the horizontal.
  1. Determine the height above the ground at which \(P\) and \(Q\) collide. [5]
WJEC Unit 4 2018 June Q10
6 marks Moderate -0.3
A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$
  1. Find the magnitude of the acceleration of the particle. [3]
  2. 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. [3]
WJEC Unit 4 2019 June Q1
5 marks Moderate -0.8
Val buys electrical components from one of 3 suppliers \(A\), \(B\), \(C\), in the ratio \(2:1:7\). The probability that the component is faulty is \(0.33\) for \(A\), \(0.45\) for \(B\) and \(0.05\) for \(C\). Val selects a component at random.
  1. Find the probability that the component works. [3]
  2. Given that the component works, find the probability that Val bought the component from supplier \(B\). [2]
WJEC Unit 4 2019 June Q2
10 marks Standard +0.3
Four children are playing a game in order to win a calculator. They take turns, starting with Alex, followed by Ben, then Caroline, then Danielle, rolling a fair six-sided dice until someone obtains a 6. This player then wins a calculator.
  1. Find the probability that
    1. Danielle wins the calculator on her first turn, [1]
    2. Ben wins the calculator on his first or second turn, [3]
    3. Caroline rolls the dice exactly twice. [3]
  2. Show that the probability that Alex wins the calculator is \(\frac{216}{671}\). [3]