Questions — WJEC Further Unit 1 (55 questions)

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WJEC Further Unit 1 2023 June Q7
7 marks Standard +0.8
7. Using mathematical induction, prove that $$\left[ \begin{array} { l l } 2 & 5 \\ 0 & 2 \end{array} \right] ^ { n } = \left[ \begin{array} { c c } 2 ^ { n } & 2 ^ { n - 1 } \times 5 n \\ 0 & 2 ^ { n } \end{array} \right]$$ for all positive integers \(n\).
WJEC Further Unit 1 2023 June Q8
9 marks Challenging +1.2
8. The roots of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 2 x + 8 = 0\) are denoted by \(\alpha , \beta , \gamma\). Determine the cubic equation whose roots are \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.
WJEC Further Unit 1 2023 June Q9
12 marks Standard +0.8
9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).
  1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
  3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).
WJEC Further Unit 1 2023 June Q10
8 marks Challenging +1.2
10. Gareth is investigating a series involving cube numbers. His series is $$1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 } - 6 ^ { 3 } + 7 ^ { 3 } - \ldots$$ Gareth continues his series and ends with an odd number.
Find and simplify an expression for the sum of Gareth's series in terms of \(k\), where \(k\) is the number of odd numbers in his series.
WJEC Further Unit 1 2024 June Q1
5 marks Moderate -0.5
  1. The complex numbers \(z , v\) and \(w\) are related by the equation
$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
WJEC Further Unit 1 2024 June Q2
3 marks Moderate -0.5
2. Given that \(x ^ { 2 } + 4 x + 5\) is a factor of \(x ^ { 3 } + x ^ { 2 } - 7 x - 15\), solve the equation \(x ^ { 3 } + x ^ { 2 } - 7 x - 15 = 0\).
WJEC Further Unit 1 2024 June Q3
6 marks Standard +0.8
3. The quadratic equation \(x ^ { 2 } + p x + q = 0\) has a repeated root \(\alpha\). A new quadratic equation has a repeated root \(\frac { 1 } { \alpha }\) and is of the form \(x ^ { 2 } + m x + m = 0\).
Find the values of \(p\) and \(q\) in the original equation.
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WJEC Further Unit 1 2024 June Q4
10 marks Standard +0.8
  1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and
$$w = \frac { z } { 1 - z }$$
  1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).
WJEC Further Unit 1 2024 June Q5
7 marks Standard +0.8
5. Given that $$\sum _ { r = k } ^ { 76 } ( r - 31 ) = 980$$ show that there are two possible values of \(k\).
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WJEC Further Unit 1 2024 June Q6
12 marks Standard +0.3
  1. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram.
Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), are given by: $$\begin{aligned} & L _ { 1 } : | z - 2 + \mathrm { i } | = | z + 2 - 3 \mathrm { i } | \\ & L _ { 2 } : | z - 2 + \mathrm { i } | = \sqrt { 10 } \end{aligned}$$
  1. Find the coordinates of the points of intersection of these loci.
  2. On the same Argand diagram, sketch the loci \(L _ { 1 }\) and \(L _ { 2 }\). Clearly label the coordinates of the points of intersection.
WJEC Further Unit 1 2024 June Q7
7 marks Standard +0.3
7. Prove, by mathematical induction, that \(13 ^ { ( 2 n - 1 ) } + 8\) is a multiple of 7 for all positive integers \(n\).
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WJEC Further Unit 1 2024 June Q8
12 marks Challenging +1.8
  1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where
$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13 \\ - 13 & 84 \end{array} \right]$$
  1. Determine the value of \(k\).
    Find the invariant points of \(T\).
WJEC Further Unit 1 2024 June Q9
8 marks Standard +0.3
9. Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), are defined by $$\begin{aligned} & \Pi _ { 1 } : 4 x - 3 y + 2 z = 5 \\ & \Pi _ { 2 } : 6 x + y + z = 9 \end{aligned}$$
  1. Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Give your answer correct to three significant figures.
  2. Find the perpendicular distance from the point \(A ( 5 , - 2 , - 6 )\) to the plane \(\Pi _ { 1 }\).
    1. Show that the point \(B ( 5,5,0 )\) lies on \(\Pi _ { 1 }\) and that the point \(C ( 1,3,0 )\) lies on \(\Pi _ { 2 }\).
    2. State an equation of a plane that contains the points \(B\) and \(C\).
      Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 1 2018 June Q1
6 marks Moderate -0.8
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that \(\mathbf{A} = \begin{bmatrix} 4 & 2 \\ -1 & -3 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 4 & 2 \\ 2 & 1 \end{bmatrix}\).
  1. Explain why \(\mathbf{B}\) has no inverse. [1]
    1. Find the inverse of \(\mathbf{A}\). [3]
    2. Hence, find the matrix \(\mathbf{X}\), where \(\mathbf{AX} = \begin{bmatrix} -4 \\ 1 \end{bmatrix}\) [2]
WJEC Further Unit 1 2018 June Q2
6 marks Standard +0.3
Prove, by mathematical induction, that \(\sum_{r=1}^{n} r(r+3) = \frac{1}{3}n(n+1)(n+5)\) for all positive integers \(n\). [6]
WJEC Further Unit 1 2018 June Q3
8 marks Standard +0.3
A cubic equation has roots \(\alpha\), \(\beta\), \(\gamma\) such that $$\alpha + \beta + \gamma = -9, \quad \alpha\beta + \beta\gamma + \gamma\alpha = 20, \quad \alpha\beta\gamma = 0.$$
  1. Find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
  2. Find the cubic equation with roots \(3\alpha\), \(3\beta\), \(3\gamma\). Give your answer in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), \(d\) are constants to be determined. [4]
WJEC Further Unit 1 2018 June Q4
7 marks Moderate -0.3
A complex number is defined by \(z = -3 + 4\mathrm{i}\).
    1. Express \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi \leqslant \theta \leqslant \pi\).
    2. Express \(\bar{z}\), the complex conjugate of \(z\), in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [4]
Another complex number is defined as \(w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)\).
  1. Express \(zw\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [3]
WJEC Further Unit 1 2018 June Q5
8 marks Standard +0.3
  1. Show that \(\frac{2}{n-1} - \frac{2}{n+1}\) can be expressed as \(\frac{4}{(n^2-1)}\). [1]
  2. Hence, find an expression for \(\sum_{r=2}^{n} \frac{4}{(r^2-1)}\) in the form \(\frac{(an+b)(n+c)}{n(n+1)}\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [6]
  3. Explain why \(\sum_{r=1}^{100} \frac{4}{(r^2-1)}\) cannot be calculated. [1]
WJEC Further Unit 1 2018 June Q6
7 marks Moderate -0.3
  1. Show that \(1 - 2\mathrm{i}\) is a root of the cubic equation \(x^3 + 5x^2 - 9x + 35 = 0\). [3]
  2. Find the other two roots of the equation. [4]
WJEC Further Unit 1 2018 June Q7
5 marks Standard +0.3
The complex number \(z\) is represented by the point \(P(x, y)\) in the Argand diagram and $$|z - 4 - \mathrm{i}| = |z + 2|.$$
  1. Find the equation of the locus of \(P\). [4]
  2. Give a geometric interpretation of the locus of \(P\). [1]
WJEC Further Unit 1 2018 June Q8
9 marks Standard +0.8
The transformation \(T\) in the plane consists of a translation in which the point \((x, y)\) is transformed to the point \((x - 1, y + 1)\), followed by a reflection in the line \(y = x\).
  1. Determine the \(3 \times 3\) matrix which represents \(T\). [4]
  2. Find the equation of the line of fixed points of \(T\). [2]
  3. Find \(T^2\) and hence write down \(T^{-1}\). [3]
WJEC Further Unit 1 2018 June Q9
14 marks Standard +0.3
The line \(L_1\) passes through the points \(A(1, 2, -3)\) and \(B(-2, 1, 0)\).
    1. Show that the vector equation of \(L_1\) can be written as $$\mathbf{r} = (1 - 3\lambda)\mathbf{i} + (2 - \lambda)\mathbf{j} + (-3 + 3\lambda)\mathbf{k}.$$
    2. Write down the equation of \(L_1\) in Cartesian form. [4]
The vector equation of the line \(L_2\) is given by \(\mathbf{r} = 2\mathbf{i} - 4\mathbf{j} + \mu(4\mathbf{j} + 7\mathbf{k})\).
  1. Show that \(L_1\) and \(L_2\) do not intersect. [5]
  2. Find a vector in the direction of the common perpendicular to \(L_1\) and \(L_2\). [5]
WJEC Further Unit 1 Specimen Q1
7 marks Standard +0.8
Use mathematical induction to prove that \(4^n + 2\) is divisible by 6 for all positive integers \(n\). [7]
WJEC Further Unit 1 Specimen Q2
11 marks Standard +0.3
Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).
  1. Give your answer in Cartesian form [7]
  2. Give your answer in modulus-argument form. [4]
WJEC Further Unit 1 Specimen Q3
6 marks Challenging +1.2
Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series $$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$ Express your answer as a product of linear factors. [6]