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WJEC Unit 3 2024 June Q10
14 marks Standard +0.3
The function \(f\) has domain \([4, \infty)\) and is defined by $$f(x) = \frac{2(3x + 1)}{x^2 - 2x - 3} + \frac{x}{x + 1}.$$
  1. Show that \(f(x) = \frac{x + 2}{x - 3}\). [4]
  2. Determine the range of \(f(x)\). [2]
  3. Find an expression for \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\). [4]
  4. Find the value of \(x\) when \(f(x) = f^{-1}(x)\). [4]
WJEC Unit 3 2024 June Q11
10 marks Standard +0.3
A curve is defined parametrically by $$x = 2\theta + \sin 2\theta, \quad y = 1 + \cos 2\theta.$$
  1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(-\tan\theta\). [6]
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]
WJEC Unit 3 2024 June Q12
6 marks Standard +0.3
  1. Given that \(\theta\) is small, show that \(2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2\). [2]
  2. Hence, when \(\theta\) is small, show that $$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$ where \(a\), \(b\) are constants to be found. [4]
WJEC Unit 3 2024 June Q13
3 marks Standard +0.8
The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}
  1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
  2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]
WJEC Unit 3 2024 June Q14
7 marks Standard +0.3
  1. Given that \(y = \frac{1 + \ln x}{x}\), show that \(\frac{dy}{dx} = \frac{-\ln x}{x^2}\). [2]
  2. Hence, solve the differential equation $$\frac{dx}{dt} = \frac{x^2 t}{\ln x},$$ given that \(t = 3\) when \(x = 1\). Give your answer in the form \(t^2 = g(x)\), where \(g\) is a function of \(x\). [5]
WJEC Unit 3 2024 June Q15
5 marks Standard +0.3
Robert wants to deposit \(£P\) into a savings account. He has a choice of two accounts. • Account \(A\) offers an annual compound interest rate of \(1\%\). • Account \(B\) offers an interest rate of \(5\%\) for the first year and an annual compound interest rate of \(0.6\%\) for each subsequent year. After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\). [5]
WJEC Unit 3 Specimen Q1
4 marks Standard +0.3
Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]
WJEC Unit 3 Specimen Q2
3 marks Standard +0.3
Air is pumped into a spherical balloon at the rate of 250 cm\(^3\) per second. When the radius of the balloon is 15 cm, calculate the rate at which the radius is increasing, giving your answer to three decimal places [3]
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]