Questions — WJEC (504 questions)

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WJEC Unit 3 2023 June Q7
10 marks Moderate -0.3
  1. The graphs of \(y = 5x - 3\) and \(y = 2x + 3\) intersect at the point A. Show that the coordinates of A are \((2, 7)\). [2]
  2. On the same set of axes, sketch the graphs of \(y = |5x - 3|\) and \(y = |2x + 3|\), clearly indicating the coordinates of the points of intersection of the two graphs and the points where the graphs touch the \(x\)-axis. [4]
  3. Calculate the area of the region satisfying the inequalities $$y \geqslant |5x - 3| \quad \text{and} \quad y \leqslant |2x + 3|.$$ [4]
WJEC Unit 3 2023 June Q8
7 marks Standard +0.3
The function \(f\) is defined by \(f(x) = \frac{4x^2 + 12x + 9}{2x^2 + x - 3}\), where \(x > 1\).
  1. Show that \(f(x)\) can be written as \(2 + \frac{5}{x-1}\). [3]
  2. Hence find the exact value of \(\int_3^7 f(x)\,dx\). [4]
WJEC Unit 3 2023 June Q9
8 marks Moderate -0.3
The aerial view of a patio under construction is shown below. \includegraphics{figure_9} The curved edge of the patio is described by the equation \(9x^2 + 16y^2 = 144\), where \(x\) and \(y\) are measured in metres. To construct the patio, the area enclosed by the curve and the coordinate axes is to be covered with a layer of concrete of depth 0.06 m.
  1. Show that the volume of concrete required for the construction of the patio is given by \(0.015 \int_0^4 \sqrt{144 - 9x^2}\,dx\). [3]
  2. Use the trapezium rule with six ordinates to estimate the volume of concrete required. [4]
  3. State whether your answer in part (b) is an overestimate or an underestimate of the volume required. Give a reason for your answer. [1]
WJEC Unit 3 2023 June Q10
8 marks Moderate -0.8
Two real functions are defined as $$f(x) = \frac{8}{x-4} \quad \text{for} \quad (-\infty < x < 4) \cup (4 < x < \infty),$$ $$g(x) = (x-2)^2 \quad \text{for} \quad -\infty < x < \infty.$$
    1. Find an expression for \(fg(x)\). [2]
    2. Determine the values of \(x\) for which \(fg(x)\) does not exist. [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
WJEC Unit 3 2023 June Q11
7 marks Standard +0.3
A curve C has equation \(f(x) = 5x^3 + 2x^2 - 3x\).
  1. Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary. [5]
  2. Determine the range of values of \(x\) for which C is concave. [2]
WJEC Unit 3 2023 June Q12
7 marks Moderate -0.3
The rate of change of a variable \(y\) with respect to \(x\) is directly proportional to \(y\).
  1. Write down a differential equation satisfied by \(y\). [1]
  2. When \(x = 1\) and \(y = 0.5\), the rate of change of \(y\) with respect to \(x\) is 2. Find \(y\) when \(x = 3\). [6]
WJEC Unit 3 2023 June Q13
7 marks Standard +0.3
The curve \(C_1\) has parametric equations \(x = 3p + 1\), \(y = 9p^2\). The curve \(C_2\) has parametric equations \(x = 4q\), \(y = 2q\). Find the Cartesian coordinates of the points of intersection of \(C_1\) and \(C_2\). [7]
WJEC Unit 3 2023 June Q14
8 marks Moderate -0.3
  1. Use integration by parts to evaluate \(\int_0^1 (3x-1)e^{2x}\,dx\). [4]
  2. Use the substitution \(u = 1 - 2\cos x\) to find \(\int \frac{\sin x}{1 - 2\cos x}\,dx\). [4]
WJEC Unit 3 2024 June Q1
11 marks Standard +0.3
The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$
  1. Express \(f(x)\) in terms of partial fractions. [4]
  2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
  3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]
WJEC Unit 3 2024 June Q2
11 marks Standard +0.3
  1. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$3\cot\theta + 4\cosec^2\theta = 5.$$ [5]
  2. By writing \(24\cos x - 7\sin x\) in the form \(R\cos(x + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\), solve the equation $$24\cos x - 7\sin x = 16$$ for values of \(x\) between \(0°\) and \(360°\). [6]
WJEC Unit 3 2024 June Q3
7 marks Standard +0.3
The diagram below shows a badge \(ODC\). The shape \(OAB\) is a sector of a circle centre \(O\) and radius \(r\) cm. The shape \(ODC\) is a sector of a circle with the same centre \(O\). The length \(AD\) is \(5\) cm and angle \(AOB\) is \(\frac{\pi}{5}\) radians. The area of the shaded region, \(ABCD\), is \(\frac{13\pi}{2}\) cm\(^2\). \includegraphics{figure_3}
  1. Determine the value of \(r\). [4]
  2. Calculate the perimeter of the shaded region. [3]
WJEC Unit 3 2024 June Q4
6 marks Moderate -0.8
A function \(f\) is given by \(f(x) = |3x + 4|\).
  1. Sketch the graph of \(y = f(x)\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis. [3]
  2. On a separate set of axes, sketch the graph of \(y = \frac{1}{2}f(x) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A'\) and \(B'\). Clearly label the coordinates of the points \(A'\) and \(B'\). [3]
WJEC Unit 3 2024 June Q5
4 marks Standard +0.3
Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]
WJEC Unit 3 2024 June Q6
13 marks Standard +0.8
  1. Differentiate \(\cos x\) from first principles. [5]
  2. Differentiate \(e^{3x}\sin 4x\) with respect to \(x\). [3]
  3. Find \(\int x^2\sin 2x dx\). [5]
WJEC Unit 3 2024 June Q7
7 marks Moderate -0.8
Showing all your working, evaluate
  1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
  2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]
WJEC Unit 3 2024 June Q8
7 marks Standard +0.3
The function \(f\) is defined by $$f(x) = x^3 + 4x^2 - 3x - 1.$$
  1. Show that the equation \(f(x) = 0\) has a root in the interval \([0, 1]\). [1]
  2. Using the Newton-Raphson method with \(x_0 = 0 \cdot 8\),
    1. write down in full the decimal value of \(x_1\) as given in your calculator,
    2. determine the value of this root correct to six decimal places. [4]
  3. Explain why the Newton-Raphson method does not work if \(x_0 = \frac{1}{3}\). [2]
WJEC Unit 3 2024 June Q9
9 marks Standard +0.3
The diagram below shows a sketch of the curve \(C_1\) with equation \(y = -x^2 + \pi x + 1\) and a sketch of the curve \(C_2\) with equation \(y = \cos 2x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\). \includegraphics{figure_9} Calculate the area of the shaded region enclosed by \(C_1\), \(C_2\) and the \(x\)-axis. Give your answer in terms of \(\pi\). [9]
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]