Questions — WJEC Further Unit 1 (52 questions)

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WJEC Further Unit 1 2018 June Q1
  1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 4 & 2
    - 1 & - 3 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { l l } 4 & 2
    2 & 1 \end{array} \right]\).
    1. Explain why \(\mathbf { B }\) has no inverse.
      1. Find the inverse of \(\mathbf { A }\).
      2. Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \left[ \begin{array} { c } - 4
        1 \end{array} \right]\).
    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 \(n\).
    4. A cubic equation has roots \(\alpha , \beta , \gamma\) such that
    $$\alpha + \beta + \gamma = - 9 , \quad \alpha \beta + \beta \gamma + \gamma \alpha = 20 , \quad \alpha \beta \gamma = 0$$
WJEC Further Unit 1 2018 June Q4
4. 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 \(r ( \cos \theta + \mathrm { i } \sin \theta )\). Another complex number is defined as \(w = \sqrt { 5 } ( \cos 2 \cdot 68 + \mathrm { i } \sin 2 \cdot 68 )\).
  2. Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
WJEC Further Unit 1 2018 June Q5
5. (a) Show that \(\frac { 2 } { n - 1 } - \frac { 2 } { n + 1 }\) can be expressed as \(\frac { 4 } { \left( n ^ { 2 } - 1 \right) }\).
(b) Hence, find an expression for \(\sum _ { r = 2 } ^ { n } \frac { 4 } { \left( r ^ { 2 } - 1 \right) }\) in the form \(\frac { ( a n + b ) ( n + c ) } { n ( n + 1 ) }\), where \(a , b , c\) are integers whose values are to be determined.
(c) Explain why \(\sum _ { r = 1 } ^ { 100 } \frac { 4 } { \left( r ^ { 2 } - 1 \right) }\) cannot be calculated.
WJEC Further Unit 1 2018 June Q6
6. (a) Show that \(1 - 2 \mathrm { i }\) is a root of the cubic equation \(x ^ { 3 } + 5 x ^ { 2 } - 9 x + 35 = 0\).
(b) Find the other two roots of the equation.
WJEC Further Unit 1 2018 June Q7
7. 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 \(P\).
  2. Give a geometric interpretation of the locus of \(P\).
WJEC Further Unit 1 2018 June Q8
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 \(T\).
  2. Find the equation of the line of fixed points of \(T\).
  3. Find \(T ^ { 2 }\) and hence write down \(T ^ { - 1 }\).
WJEC Further Unit 1 2018 June Q9
9. 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. The vector equation of the line \(L _ { 2 }\) is given by \(\mathbf { r } = 2 \mathbf { i } - 4 \mathbf { j } + \mu ( 4 \mathbf { j } + 7 \mathbf { k } )\).
  2. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
  3. Find a vector in the direction of the common perpendicular to \(L _ { 1 }\) and \(L _ { 2 }\).
WJEC Further Unit 1 2019 June Q1
  1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 7
    - 2 & 0 \end{array} \right)\), \(\mathbf { B } = \left( \begin{array} { l l } 5 & 1
    0 & 4 \end{array} \right)\).
The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).
WJEC Further Unit 1 2019 June Q2
2. The position vectors of the points \(A , B , C , D\) are given by
\(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\),
\(\mathbf { b } = 4 \mathbf { j } + 5 \mathbf { k }\),
\(\mathbf { c } = 7 \mathbf { i } - 3 \mathbf { k }\),
\(\mathbf { d } = - 3 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }\),
respectively.
  1. Find the vector equations of the lines \(A B\) and \(C D\).
  2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.
WJEC Further Unit 1 2019 June Q3
3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.
  1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
  2. Find the complex number \(\frac { z } { w }\).
WJEC Further Unit 1 2019 June Q4
4. Prove, by mathematical induction, that \(9 ^ { n } + 15\) is a multiple of 8 for all positive integers \(n\).
WJEC Further Unit 1 2019 June Q5
5. Given that \(x = - \frac { 1 } { 2 }\) and \(x = - 3\) are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.
WJEC Further Unit 1 2019 June Q6
6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.
WJEC Further Unit 1 2019 June Q7
7. (a) Find an expression for \(\sum _ { r = 1 } ^ { 2 m } ( r + 2 ) ^ { 2 }\) in the form \(\frac { 1 } { 3 } m \left( a m ^ { 2 } + b m + c \right)\), where \(a , b , c\) are integers whose values are to be determined.
(b) Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).
WJEC Further Unit 1 2019 June Q8
8. The plane \(\Pi\) contains the three points \(A ( 3,5,6 ) , B ( 5 , - 1,7 )\) and \(C ( - 1,7,0 )\). Find the vector equation of the plane \(\Pi\) in the form r.n \(= d\).
Express this equation in Cartesian form.
WJEC Further Unit 1 2019 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 $$w = z ^ { 2 } - 1$$
  1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
  2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).
WJEC Further Unit 1 2019 June Q10
10. The quadratic equation \(p x ^ { 2 } + q x + r = 0\) has roots \(\alpha\) and \(\beta\), where \(p , q , r\) are non-zero constants.
  1. A cubic equation is formed with roots \(\alpha , \beta , \alpha + \beta\). Find the cubic equation with coefficients expressed in terms of \(p , q , r\).
  2. Another quadratic equation \(p x ^ { 2 } - q x - r = 0\) has roots \(2 \alpha\) and \(\gamma\). Show that \(\beta = - 2 \gamma\).
WJEC Further Unit 1 2022 June Q1
  1. The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).
    1. (i) Find the modulus and argument of \(z w\).
      (ii) Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
    2. The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
    3. The complex conjugate of \(v\) is denoted by \(\bar { v }\).
    Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.
WJEC Further Unit 1 2022 June Q2
2. (a) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left( \begin{array} { c c } 3 & 4
- 1 & - 2 \end{array} \right) , \quad \mathbf { B } = \binom { - 11 } { 7 }$$ Given that \(\mathbf { A X } = \mathbf { B }\), find the matrix \(\mathbf { X }\).
(b) (i) Find the \(2 \times 2\) matrix, \(\mathbf { T }\), which represents a reflection in the line \(y = - 2 x\).
(ii) The images of the points \(C ( 2,7 )\) and \(D ( 3,1 )\), under \(\mathbf { T }\), are \(E\) and \(F\) respectively. Find the coordinates of the midpoint of \(E F\).
WJEC Further Unit 1 2022 June Q3
3. The vector equation of the line \(L\) is given by $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } + \lambda ( 4 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) .$$ The Cartesian equation of the plane \(\Pi\) is given by $$3 x + 8 y - 9 z = 0$$ Find the Cartesian coordinates of the point of intersection of \(L\) and \(\Pi\).
WJEC Further Unit 1 2022 June Q4
4. The positive integer \(N\) is such that \(1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + N ^ { 2 } = ( 3 N - 2 ) ^ { 2 }\). Write down and simplify an equation satisfied by \(N\). Hence find the possible values of \(N\).
WJEC Further Unit 1 2022 June Q5
5. (a) The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 3 + 2 i | = | z - 3 |$$ find the equation of the locus of \(P\).
(b) Give a geometric interpretation of the locus of \(P\).
WJEC Further Unit 1 2022 June Q6
6. The roots of the cubic equation $$2 x ^ { 3 } + p x ^ { 2 } - 126 x + q = 0$$ form a geometric progression with common ratio - 3 .
Find the possible values of \(p\) and \(q\), giving your answers in surd form.
WJEC Further Unit 1 2022 June Q7
7. The vector equations of the lines \(L _ { 1 } , L _ { 2 } , L _ { 3 }\) are given by $$\begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } + n \mathbf { j } + \mathbf { k } )
& \mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } )
& \mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } + v ( p \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \end{aligned}$$ respectively, where \(n\) and \(p\) are constants.
The line \(L _ { 1 }\) is perpendicular to the line \(L _ { 2 }\). The line \(L _ { 1 }\) is also perpendicular to the line \(L _ { 3 }\).
  1. Show that the value of \(n\) is - 3 and find the value of \(p\).
  2. Find the acute angle between the lines \(L _ { 2 }\) and \(L _ { 3 }\).
WJEC Further Unit 1 2022 June Q8
8. The point \(( x , y , z )\) is rotated through \(60 ^ { \circ }\) anticlockwise around the \(z\)-axis. After rotation, the value of the \(x\)-coordinate is equal to the value of the \(y\)-coordinate.
Show that \(y = ( a + \sqrt { b } ) x\), where \(a\), \(b\) are integers whose values are to be determined.