Questions — WJEC Further Unit 1 (52 questions)

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WJEC Further Unit 1 2022 June Q9
9. (a) Given that \(A _ { r } = \frac { 1 } { r + 1 } - \frac { 2 } { r + 2 } + \frac { 1 } { r + 3 }\), show that \(A _ { r }\) can be expressed as \(\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) }\).
(b) Hence, show that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { ( n + 2 ) ( n + 3 ) }\).
(c) Find the ratio of \(\sum _ { r = 1 } ^ { 5 } A _ { r } : \sum _ { r = 1 } ^ { 10 } A _ { r }\), giving your answer in its simplest form.
WJEC Further Unit 1 2023 June Q1
  1. The complex number \(z\) is given by \(z = 3 + \lambda \mathrm { i }\), where \(\lambda\) is a positive constant. The complex conjugate of \(z\) is denoted by \(\bar { z }\).
Given that \(z ^ { 2 } + \bar { z } ^ { 2 } = 2\), find the value of \(\lambda\).
WJEC Further Unit 1 2023 June Q2
2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 2 & - 1
4 & - 7 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { c c c } 2 & 0 & 9
4 & - 20 & 13 \end{array} \right]\).
  1. Find the inverse of \(\mathbf { A }\).
  2. Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \mathbf { B }\).
WJEC Further Unit 1 2023 June Q3
3. Given that \(5 - \mathrm { i }\) is a root of the equation \(x ^ { 4 } - 10 x ^ { 3 } + 10 x ^ { 2 } + 160 x - 416 = 0\),
  1. write down another root of the equation,
  2. find the remaining roots.
WJEC Further Unit 1 2023 June Q4
4. The transformation \(T\) in the plane consists of a translation in which the point \(( x , y )\) is transformed to the point ( \(x + 2 , y - 2\) ), followed by a reflection in the line \(y = x\).
  1. Determine the \(3 \times 3\) matrix which represents \(T\).
  2. Determine how many invariant points exist under the transformation \(T\).
WJEC Further Unit 1 2023 June Q5
5. The points \(A\) and \(B\) have coordinates \(( 3,4 , - 2 )\) and \(( - 2,0,7 )\) respectively. The equation of the plane \(\Pi\) is given by \(2 x + 3 y + 3 z = 27\).
  1. Show that the vector equation of the line \(A B\) may be expressed in the form $$\mathbf { r } = ( 3 - 5 \lambda ) \mathbf { i } + ( 4 - 4 \lambda ) \mathbf { j } + ( - 2 + 9 \lambda ) \mathbf { k }$$
  2. Find the coordinates of the point of intersection of the line \(A B\) and the plane \(\Pi\).
WJEC Further Unit 1 2023 June Q6
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + \mathrm { i } | = 2 | z - 5 - 2 \mathrm { i } |$$ show that the locus of \(P\) is a circle and write down the coordinates of its centre.
WJEC Further Unit 1 2023 June Q7
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
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
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
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
  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
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
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
  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
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
  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. 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
  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
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 Specimen Q1
  1. Use mathematical induction to prove that \(4 ^ { n } + 2\) is divisible by 6 for all positive integers \(n\).
  2. Solve the equation \(2 z + i \bar { z } = \frac { - 1 + 7 i } { 2 + i }\).
    1. Give your answer in Cartesian form
    2. Give your answer in modulus-argument form.
    3. 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.
WJEC Further Unit 1 Specimen Q4
4. The roots of the equation $$x ^ { 3 } - 4 x ^ { 2 } + 14 x - 20 = 0$$ are denoted by \(\alpha , \beta , \gamma\).
  1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12$$ Explain why this result shows that exactly one of the roots of the above cubic equation is real.
  2. Given that one of the roots is \(1 + 3 \mathrm { i }\), find the other two roots. Explain your method for each root.
WJEC Further Unit 1 Specimen Q5
5. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram and $$| z - 3 | = 2 | z + \mathrm { i } |$$ Show that the locus of \(P\) is a circle and determine its radius and the coordinates of its centre.
WJEC Further Unit 1 Specimen Q6
6. The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \(( x , y )\) is transformed to the point \(( x + 1 , y - 2 )\),followed by an anticlockwise rotation through \(90 ^ { \circ }\) about the origin.
  1. Find the \(3 \times 3\) matrix representing \(T\).
  2. Show that \(T\) has no fixed points.
WJEC Further Unit 1 Specimen Q7
7. The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and $$w = z ( 1 + z )$$
  1. Show that $$v = y ( 1 + 2 x )$$ and find an expression for \(u\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = a u ^ { 2 } + b u\), where \(a\) and \(b\) are constants whose values are to be determined.