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WJEC Unit 3 2018 June Q16
11 marks Moderate -0.3
  1. Differentiate the following functions with respect to \(x\), simplifying your answer wherever possible.
    1. \(e^{3\tan x}\),
    2. \(\frac{\sin 2x}{x^2}\). [5]
  2. A function is defined implicitly by $$3x^2y + y^2 - 5x = 5.$$ Find the equation of the normal at the point \((1, 2)\). [6]
WJEC Unit 3 2018 June Q17
6 marks Moderate -0.3
By drawing suitable graphs, show that \(x - 1 = \cos x\) has only one root. Starting with \(x_0 = 1\), use the Newton-Raphson method to find the value of this root correct to two decimal places. [6]
WJEC Unit 3 2023 June Q1
5 marks Moderate -0.8
The 12th term of an arithmetic series is 41 and the sum of the first 16 terms is 488. Find the first term and the common difference of the series. [5]
WJEC Unit 3 2023 June Q2
13 marks Moderate -0.3
  1. Differentiate each of the following with respect to \(x\).
    1. \(\left(\sin x + x^2\right)^5\) [2]
    2. \(x^3 \cos x\) [2]
    3. \(\frac{e^{3x}}{\sin 2x}\) [3]
  2. Find the equation of the tangent to the curve $$4y^2 - 7xy + x^2 = 12$$ at the point \((2, 4)\). [6]
WJEC Unit 3 2023 June Q3
11 marks Standard +0.3
  1. Express \(\frac{9}{(1-x)(1+2x)^2}\) in terms of partial fractions. [4]
  2. Using your answer from part (a), find the expansion of \(\frac{9}{(1-x)(1+2x)^2}\) in ascending powers of \(x\) as far as the term in \(x^2\). State the values of \(x\) for which the expansion is valid. [7]
WJEC Unit 3 2023 June Q4
8 marks Standard +0.3
A function \(f\) with domain \((-\infty,\infty)\) is defined by \(f(x) = 6x^3 + 35x^2 - 7x - 6\).
  1. Determine the number of roots of the equation \(f(x) = 0\) in the interval \([-1, 1]\). [2]
  2. Use the Newton-Raphson method to find a root of the equation \(f(x) = 0\). Starting with \(x_0 = 1\),
    1. write down the value of \(x_1\),
    2. determine the value of the root correct to one decimal place. [4]
  3. It is suggested that another iterative sequence $$x_{n+1} = \sqrt{\frac{7x_n + 6 - 6x_n^3}{35}},$$ starting with \(x_0 = -3\), could be used to find a root of the equation \(f(x) = 0\). Explain why this method fails. [2]
WJEC Unit 3 2023 June Q5
6 marks Moderate -0.8
A tree is 80 cm in height when it is planted. In the first year, the tree grows in height by 32 cm. In each subsequent year, the tree grows in height by 90% of the growth of the previous year.
  1. Find the height of the tree 10 years after it was planted. [4]
  2. Determine the maximum height of the tree. [2]
WJEC Unit 3 2023 June Q6
15 marks Standard +0.3
  1. Using the trigonometric identity \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), show that the exact value of \(\cos 75°\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\). [3]
  2. Solve the equation \(2\cot^2 x + \cosec x = 4\) for values of \(x\) between \(0°\) and \(360°\). [6]
    1. Express \(7\cos\theta - 24\sin\theta\) in the form \(R\cos(\theta + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\).
    2. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$7\cos\theta - 24\sin\theta = 5.$$ [6]
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]