Questions — WJEC Unit 3 (82 questions)

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WJEC Unit 3 2018 June Q5
8 marks Moderate -0.3
  1. Show that $$\frac{3x}{(x-1)(x-4)^2} = \frac{A}{(x-1)} + \frac{B}{(x-4)} + \frac{C}{(x-4)^2},$$ where \(A\), \(B\) and \(C\) are constants to be found. [3]
  2. Evaluate \(\int_5^7 \frac{3x}{(x-1)(x-4)^2} \, dx\), giving your answer correct to 3 decimal places. [5]
WJEC Unit 3 2018 June Q6
5 marks Moderate -0.3
Write down the first three terms in the binomial expansion of \((1-4x)^{-\frac{1}{2}}\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac{1}{13}\) in your expansion, find an approximate value for \(\sqrt{13}\) in the form \(\frac{a}{b}\), where \(a\), \(b\) are integers. [5]
WJEC Unit 3 2018 June Q7
3 marks Standard +0.3
Use small angle approximations to find the small negative root of the equation $$\sin x + \cos x = 0.5.$$ [3]
WJEC Unit 3 2018 June Q8
5 marks Moderate -0.3
Find seven numbers which are in arithmetic progression such that the last term is 71 and the sum of all of the numbers is 329. [5]
WJEC Unit 3 2018 June Q9
10 marks Moderate -0.3
  1. Explain why the sum to infinity of a geometric series with common ratio \(r\) only converges when \(|r| < 1\). [1]
  2. A geometric progression \(V\) has first term 2 and common ratio \(r\). Another progression \(W\) is formed by squaring each term in \(V\). Show that \(W\) is also a geometric progression. Given that the sum to infinity of \(W\) is 3 times the sum to infinity of \(V\), find the value of \(r\). [6]
  3. At the beginning of each year, a man invests £5000 in a savings account earning compound interest at the rate of 3% per annum. The interest is added at the end of each year. Find the total amount of his savings at the end of the 20th year correct to the nearest pound. [3]
WJEC Unit 3 2018 June Q10
14 marks Standard +0.3
The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2\theta, \quad y = \cos\theta.$$
  1. Find the Cartesian equation of \(C\). [2]
  2. Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac{\pi}{3}\), and at the point \(Q\), where \(\theta = \frac{\pi}{2}\). Write down the coordinates of \(P\) and \(Q\). [5]
  3. Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. [7]
WJEC Unit 3 2018 June Q11
4 marks Standard +0.8
Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac{\pi}{2}\), $$\sin x + \cos x \geqslant 1.$$ [4]
WJEC Unit 3 2018 June Q12
10 marks Moderate -0.8
  1. Given that \(f\) is a function,
    1. state the condition for \(f^{-1}\) to exist,
    2. find \(ff^{-1}(x)\). [2]
  2. The functions \(g\) and \(h\), are given by $$g(x) = x^2 - 1,$$ $$h(x) = e^x + 1.$$
    1. Suggest a domain for \(g\) such that \(g^{-1}\) exists.
    2. Given the domain of \(h\) is \((-\infty, \infty)\), find an expression for \(h^{-1}(x)\) and sketch, using the same axes, the graphs of \(h(x)\) and \(h^{-1}(x)\). Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
    3. Determine an expression for \(gh(x)\) in its simplest form. [8]
WJEC Unit 3 2018 June Q13
8 marks Standard +0.3
  1. Express \(8\sin\theta - 15\cos\theta\) in the form \(R\sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$8\sin\theta - 15\cos\theta - 7 = 0.$$ [3]
  3. Determine the greatest value and the least value of the expression $$\frac{1}{8\sin\theta - 15\cos\theta + 23}.$$ [2]
WJEC Unit 3 2018 June Q14
12 marks Standard +0.3
Evaluate
  1. \(\int_0^2 x^3 \ln x \, dx\). [6]
  2. \(\int_0^1 \frac{2+x}{\sqrt{4-x^2}} \, dx\). [6]
WJEC Unit 3 2018 June Q15
5 marks Moderate -0.3
The variable \(y\) satisfies the differential equation $$2\frac{dy}{dx} = 5 - 2y, \quad \text{where } x \geqslant 0.$$ Given that \(y = 1\) when \(x = 0\), find an expression for \(y\) in terms of \(x\). [5]
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]