Questions — WJEC Further Unit 4 (51 questions)

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WJEC Further Unit 4 2019 June Q1
  1. A complex number is defined by \(z = 3 + 4 \mathrm { i }\).
    1. Express \(z\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi \leqslant \theta \leqslant \pi\).
      1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
      2. Write down the geometrical name of the triangle.
    3. (a) Show that \(3 \sin x + 4 \cos x - 2\) can be written as \(\frac { 6 t + 2 - 6 t ^ { 2 } } { 1 + t ^ { 2 } }\), where \(t = \tan \left( \frac { x } { 2 } \right)\).
    4. Hence, find the general solution of the equation \(3 \sin x + 4 \cos x - 2 = 3\).
    5. (a) Determine whether or not the following set of equations
    $$\left( \begin{array} { r r r } 2 & - 7 & 2
    0 & 3 & - 2
    - 7 & 8 & 4 \end{array} \right) \left( \begin{array} { l } x
    y
    z \end{array} \right) = \left( \begin{array} { l } a
    b
    c \end{array} \right)$$ has a unique solution, where \(a , b , c\) are constants.
  2. Solve the set of equations $$\begin{aligned} x + 8 y - 6 z & = 5
    2 x + 4 y + 6 z & = - 3
    - 5 x - 4 y + 9 z & = - 7 \end{aligned}$$ Show all your working.
WJEC Further Unit 4 2019 June Q4
4. (a) Given that \(y = \cot ^ { - 1 } x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { x ^ { 2 } + 1 }\).
(b) Express \(\frac { 6 x ^ { 2 } - 10 x - 9 } { ( 2 x + 3 ) \left( x ^ { 2 } + 1 \right) }\) in terms of partial fractions.
(c) Hence find \(\int \frac { 6 x ^ { 2 } - 8 x - 6 } { ( 2 x + 3 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x\).
(d) Explain why \(\int _ { - 2 } ^ { 5 } \frac { 6 x ^ { 2 } - 8 x - 6 } { ( 2 x + 3 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x\) cannot be evaluated.
WJEC Further Unit 4 2019 June Q5
5. (a) Show that \(\sin \theta - \sin 3 \theta\) can be expressed in the form \(a \cos b \theta \sin \theta\), where \(a , b\) are integers whose values are to be determined.
(b) Find the mean value of \(y = 2 \cos 2 \theta \sin \theta + 7\) between \(\theta = 1\) and \(\theta = 3\), giving your answer correct to two decimal places.
WJEC Further Unit 4 2019 June Q6
6. Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 7 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 0$$ where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 8\) when \(x = 0\).
WJEC Further Unit 4 2019 June Q7
7. (a) Write down the Maclaurin series expansion for \(\ln ( 1 - x )\) as far as the term in \(x ^ { 3 }\).
(b) Show that \(- 2 \ln \left( \frac { 1 - x } { ( 1 + x ) ^ { 2 } } \right)\) can be expressed in the form \(a x + b x ^ { 2 } + c x ^ { 3 } + \ldots\), where \(a , b , c\) are integers whose values are to be determined.
WJEC Further Unit 4 2019 June Q8
8. The curve \(C\) has polar equation $$r = \sin 2 \theta , \quad \text { where } \quad 0 < \theta \leqslant \frac { \pi } { 2 }$$
  1. Find the polar coordinates of the point on \(C\) at which the tangent is parallel to the initial line. Give your answers correct to three decimal places.
  2. Write the coordinates of this point in Cartesian form.
WJEC Further Unit 4 2019 June Q9
9. (a) Given that \(y = \sin ^ { - 1 } ( \cos \theta )\), where \(0 \leqslant \theta \leqslant \pi\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = k\), where the value of \(k\) is to be determined.
(b) Find the value of the gradient of the curve \(y = x ^ { 3 } \tan ^ { - 1 } 4 x\) when \(x = \frac { \pi } { 2 }\).
(c) Find the equation of the normal to the curve \(y = \tanh ^ { - 1 } ( 1 - x )\) when \(x = 1 \cdot 7\).
WJEC Further Unit 4 2019 June Q10
10. Given the differential equation $$\sec x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \operatorname { cosec } x = 2$$ and \(x = \frac { \pi } { 2 }\) when \(y = 3\), find the value of \(y\) when \(x = \frac { \pi } { 4 }\).
WJEC Further Unit 4 2019 June Q11
11. (a) Find the area of the region enclosed by the curve \(y = x \sinh x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).
(b) The region \(R\) is bounded by the curve \(y = \cosh 2 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis.
(c) Using your answer to part (b), find the total volume of the solid generated by rotating the region bounded by the curve \(y = \cosh 2 x\) and the lines \(x = - 1\) and \(x = 1\).
WJEC Further Unit 4 2019 June Q12
12. (a) Evaluate \(\int _ { 3 } ^ { 4 } \frac { 1 } { \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x\), giving your answer correct to three decimal places.
(b) Given that \(\int _ { 1 } ^ { 2 } \frac { k } { 9 - x ^ { 2 } } \mathrm {~d} x = \ln \frac { 25 } { 4 }\), find the value of \(k\).
(c) Show that \(\int \frac { ( \cosh x - \sinh x ) ^ { 3 } } { \cosh ^ { 2 } x + \sinh ^ { 2 } x - \sinh 2 x } \mathrm {~d} x\) can be expressed as \(- \mathrm { e } ^ { - x } + c\), where \(c\) is a constant.
WJEC Further Unit 4 2022 June Q1
  1. A function \(f\) has domain \(( - \infty , \infty )\) and is defined by \(f ( x ) = \cosh ^ { 3 } x - 3 \cosh x\).
    1. Show that the graph of \(y = f ( x )\) has only one stationary point.
    2. Find the nature of this stationary point.
    3. State the largest possible range of \(f ( x )\).
    4. When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3 \sqrt { 3 } \mathrm { i }\) lie on a circle. Find the equation of this circle.
    5. (a) By putting \(t = \tan \left( \frac { \theta } { 2 } \right)\), show that the equation
    $$4 \sin \theta + 5 \cos \theta = 3$$ can be written in the form $$4 t ^ { 2 } - 4 t - 1 = 0$$
  2. Hence find the general solution of the equation $$4 \sin \theta + 5 \cos \theta = 3$$
WJEC Further Unit 4 2022 June Q4
  1. The region \(R\) is bounded by the curve \(x = \sin y\), the \(y\)-axis and the lines \(y = 1 , y = 3\). Find the volume of the solid generated when \(R\) is rotated through four right angles about the \(y\)-axis. Give your answer correct to two decimal places.
  2. (a) Determine the number of solutions of the equations
$$\begin{array} { r } x + 2 y = 3
2 x - 5 y + 3 z = 8
6 y - 2 z = 0 \end{array}$$ (b) Give a geometric interpretation of your answer in part (a).
WJEC Further Unit 4 2022 June Q6
6. Solve the equation $$\cos 2 \theta - \cos 4 \theta = \sin 3 \theta \quad \text { for } \quad 0 \leqslant \theta \leqslant \pi$$
WJEC Further Unit 4 2022 June Q7
  1. (a) Express \(4 x ^ { 2 } + 10 x - 24\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found.
    (b) Hence evaluate the integral
$$\int _ { 3 } ^ { 5 } \frac { 6 } { \sqrt { 4 x ^ { 2 } + 10 x - 24 } } \mathrm {~d} x$$ Give your answer correct to 3 decimal places.
WJEC Further Unit 4 2022 June Q8
8. By writing \(x = \sinh y\), show that \(\sinh ^ { - 1 } x = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
WJEC Further Unit 4 2022 June Q9
9. (a) (i) Expand \(\left( \cos \frac { \theta } { 3 } + i \sin \frac { \theta } { 3 } \right) ^ { 3 }\).
(ii) Hence, by using de Moivre's theorem, show that \(\cos \theta\) can be expressed as $$4 \cos ^ { 3 } \frac { \theta } { 3 } - 3 \cos \frac { \theta } { 3 }$$ (b) Hence, or otherwise, find the general solution of the equation \(\frac { \cos \theta } { \cos \frac { \theta } { 3 } } = 1\).
WJEC Further Unit 4 2022 June Q10
10. The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 8 & 0
0 & \lambda & - 2
4 & 0 & \lambda \end{array} \right)$$
  1. Show that there are two values of \(\lambda\) for which \(\mathbf { A }\) is singular.
  2. Given that \(\lambda = 3\),
    1. determine the adjugate matrix of \(\mathbf { A }\),
    2. determine the inverse matrix \(\mathbf { A } ^ { - 1 }\).
WJEC Further Unit 4 2022 June Q11
11. (a) Differentiate each of the following with respect to \(x\).
  1. \(y = \mathrm { e } ^ { 3 x } \sin ^ { - 1 } x\)
  2. \(y = \ln \left( \cosh ^ { 2 } \left( 2 x ^ { 2 } + 7 x \right) \right)\)
    (b) Find the equations of the tangents to the curve \(x = \sinh ^ { - 1 } \left( y ^ { 2 } \right)\) at the points where \(x = 1\).
WJEC Further Unit 4 2022 June Q12
12. Find the solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 8 + 6 x - 2 x ^ { 2 }$$ where \(y = 6\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\) when \(x = 0\).
WJEC Further Unit 4 2022 June Q13
13. The curve \(C\) has polar equation \(r = 2 - \cos \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  1. Sketch the curve \(C\).
    1. Show that the values of \(\theta\) at which the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line satisfy the equation $$2 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0$$
    2. Find the polar coordinates of the points where the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line.
WJEC Further Unit 4 2022 June Q14
14. Evaluate the integral $$\int _ { 2 } ^ { 4 } \frac { 6 x ^ { 2 } + 2 x + 16 } { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } \mathrm {~d} x$$ giving your answer correct to three decimal places.
WJEC Further Unit 4 2023 June Q1
  1. The functions \(f\) and \(g\) have domains \(( - 1 , \infty )\) and \(( 0 , \infty )\) respectively and are defined by
$$f ( x ) = \cosh x , \quad g ( x ) = x ^ { 2 } - 1$$
  1. State the domain and range of \(f g\).
  2. Solve the equation \(f g ( x ) = 3\). Give your answer correct to three decimal places.
WJEC Further Unit 4 2023 June Q2
2. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } \lambda & 1 & 14
- 1 & 2 & 8
- 3 & 2 & \lambda \end{array} \right)\), where \(\lambda\) is a real constant.
  1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(\lambda\). Give your answer in the form \(a \lambda ^ { 2 } + b \lambda + c\), where \(a , b , c\) are integers whose values are to be determined.
  2. Show that \(\mathbf { A }\) is non-singular for all values of \(\lambda\).
WJEC Further Unit 4 2023 June Q3
3. (a) Given that \(z = \cos \theta + \operatorname { isin } \theta\), use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ (b) Express \(32 \cos ^ { 6 } \theta\) in the form \(a \cos 6 \theta + b \cos 4 \theta + c \cos 2 \theta + d\), where \(a , b , c , d\) are integers whose values are to be determined.
WJEC Further Unit 4 2023 June Q4
4. Solve the simultaneous equations $$\begin{aligned} 4 x - 2 y + 3 z & = 8
2 x - 3 y + 8 z & = - 1
2 x + 4 y - z & = 0 \end{aligned}$$