Questions — WJEC (325 questions)

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WJEC Further Unit 4 2023 June Q5
  1. (a) Write down and simplify the Maclaurin series for \(\sin 2 x\) as far as the term in \(x ^ { 5 }\).
    (b) Using your answer to part (a), determine the Maclaurin series for \(\cos ^ { 2 } x\) as far as the term in \(x ^ { 4 }\).
  2. (a) Show that \(\tan \theta\) may be expressed as \(\frac { 2 t } { 1 - t ^ { 2 } }\), where \(t = \tan \left( \frac { \theta } { 2 } \right)\).
The diagram below shows a sketch of the curve \(C\) with polar equation $$r = \cos \left( \frac { \theta } { 2 } \right) , \quad \text { where } - \pi < \theta \leqslant \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{1d01b3d3-8a45-4c64-9d2f-ced042d8fba3-4_680_887_1171_580}
(b) Show that the \(\theta\)-coordinate of the points at which the tangent to \(C\) is perpendicular to the initial line satisfies the equation $$\tan \theta = - \frac { 1 } { 2 } \tan \left( \frac { \theta } { 2 } \right) .$$ (c) Hence, find the polar coordinates of the points on \(C\) where the tangent is perpendicular to the initial line.
(d) Calculate the area of the region enclosed by the curve \(C\) and the initial line for \(0 \leqslant \theta \leqslant \pi\).
WJEC Further Unit 4 2023 June Q7
7. Find the cube roots of the complex number \(z = 11 - 2 \mathrm { i }\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and correct to three decimal places.
WJEC Further Unit 4 2023 June Q8
8. The function \(f\) is defined by $$f ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 3 } }$$
  1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places.
  2. The region \(R\) is bounded by the curve \(y = f ( x )\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\). Find the exact value of the volume of the solid generated when \(R\) is rotated through four right angles about the \(x\)-axis.
WJEC Further Unit 4 2023 June Q9
9. Consider the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 5 y = ( x + 1 ) ^ { 2 } , \quad x > - 1$$ Given that \(y = \frac { 1 } { 4 }\) when \(x = 1\), find the value of \(y\) when \(x = 0\).
WJEC Further Unit 4 2023 June Q10
10. (a) By writing \(y = \sin ^ { - 1 } ( 2 x + 5 )\) as \(\sin y = 2 x + 5\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - ( 2 x + 5 ) ^ { 2 } } }\).
(b) Deduce the range of values of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sin ^ { - 1 } ( 2 x + 5 ) \right)\) is valid.
WJEC Further Unit 4 2023 June Q11
11. Evaluate the integrals
  1. \(\int _ { - 2 } ^ { 0 } \mathrm { e } ^ { 2 x } \sinh x \mathrm {~d} x\),
  2. \(\int _ { \frac { 3 } { 2 } } ^ { 3 } \frac { 5 } { ( x - 1 ) \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).
WJEC Further Unit 4 2023 June Q12
12. Find the general solution of the equation $$\cos 4 \theta + \cos 2 \theta = \cos \theta$$
WJEC Further Unit 4 2023 June Q13
  1. Two species of insects, \(X\) and \(Y\), co-exist on an island. The populations of the species at time \(t\) years are \(x\) and \(y\) respectively, where \(x\) and \(y\) are measured in millions. The situation can be modelled by the differential equations
$$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 3 x + 10 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = x + 6 y \end{aligned}$$
    1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 9 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 8 x = 0\).
    2. Find the general solution for \(x\) in terms of \(t\).
  2. Find the corresponding general solution for \(y\).
  3. When \(t = 0 , \frac { \mathrm {~d} x } { \mathrm {~d} t } = 5\) and the population of species \(Y\) is 4 times the population of species \(X\). Find the particular solution for \(x\) in terms of \(t\).
WJEC Further Unit 4 2024 June Q1
  1. (a) Express the three cube roots of \(5 + 10 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
    …………………………………………………………………………………………………………………………………………………..
    (b) The three cube roots of \(5 + 10 \mathrm { i }\) are plotted in an Argand diagram. The points are joined by straight lines to form a triangle. Find the area of this triangle, giving your answer correct to two significant figures.
  2. The function \(f\) is defined by \(f ( x ) = \cosh \left( \frac { x } { 2 } \right)\).
    (a) State the Maclaurin series expansion for \(\cosh \left( \frac { x } { 2 } \right)\) up to and including the term in \(x ^ { 4 }\).
Another function \(g\) is defined by \(g ( x ) = x ^ { 2 } - 2\). The diagram below shows parts of the graphs of \(y = f ( x )\) and \(y = g ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-04_894_940_1471_552}
(b) The two graphs intersect at the point \(A\), as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of \(A\), giving your answer correct to two decimal places.
(c) Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis.
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WJEC Further Unit 4 2024 June Q3
  1. Given the differential equation
$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = 4 \cos ^ { 3 } x \sin x + 5$$ and \(y = 3 \sqrt { 2 }\) when \(x = \frac { \pi } { 4 }\), find an equation for \(y\) in terms of \(x\).
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WJEC Further Unit 4 2024 June Q4
5 marks
4. (a) Given that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(16 \cos ^ { 4 } \theta\) in the form \(a \cos 4 \theta + b \cos 2 \theta + c\), where \(a , b , c\) are integers whose values are to be determined. [5]
The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4 \cos ^ { 2 } \theta , \quad \text { where } \frac { \pi } { 6 } \leqslant \theta \leqslant \frac { 5 \pi } { 6 }$$
\includegraphics[max width=\textwidth, alt={}]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-11_346_241_580_612}
Initial line
(b) Calculate the area of the region enclosed by the curve \(C\).
(c) Find the exact polar coordinates of the points on \(C\) at which the tangent is perpendicular to the initial line.
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WJEC Further Unit 4 2024 June Q5
  1. Find each of the following integrals.
    1. \(\quad \int \frac { 3 - x } { x \left( x ^ { 2 } + 1 \right) } \mathrm { d } x\)
    2. \(\quad \int \frac { \sinh 2 x } { \sqrt { \cosh ^ { 4 } x - 9 \cosh ^ { 2 } x } } \mathrm {~d} x\)
    3. The matrix \(\mathbf { M }\) is defined by
    $$\mathbf { M } = \left( \begin{array} { c c c } 12 & 30 & 8
    18 & 25 & 20
    19 & 50 & 16 \end{array} \right)$$
  2. Given that \(\operatorname { det } \mathbf { M } = - 1040\), give a geometrical interpretation of the solution to the following equation. $$\left( \begin{array} { c c c } 12 & 30 & 8
    18 & 25 & 20
    19 & 50 & 16 \end{array} \right) \left( \begin{array} { l } x
    y
    z \end{array} \right) = \left( \begin{array} { l } 2668
    3402
    4581 \end{array} \right)$$
  3. Three hotels \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels. The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.
    Types of room
    HotelSingleDoubleFamilyTotal revenue
    A12308£2,668
    B182520£3,402
    C195016£4,581
    Find the price per night of each type of room.
WJEC Further Unit 4 2024 June Q7
7. (a) A curve \(C\) is defined by the equation \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) for \(- 3 \leqslant x \leqslant 1\).
  1. Find the mean value of \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) between \(x = - 3\) and \(x = 1\).
  2. The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines \(x = - 3\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis.
    (b) Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { - 8 \mathrm { e } ^ { - 2 x } } { 4 \mathrm { e } ^ { - 2 x } - 5 } \mathrm {~d} x$$ giving your answer correct to three decimal places.
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WJEC Further Unit 4 2024 June Q8
  1. (a) By writing \(y = \sinh ^ { - 1 } ( 4 x + 3 )\) as \(\sinh y = 4 x + 3\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { 16 x ^ { 2 } + 24 x + 10 } }\).
    (b) Show that the graph of \(\mathrm { e } ^ { - 3 x } y = \sinh 2 x\) has only one stationary point.
\section*{
\includegraphics[max width=\textwidth, alt={}]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-23_56_1600_436_274}
}
WJEC Further Unit 4 2024 June Q9
9. Find the general solution of the equation $$\sin 6 \theta + 2 \cos ^ { 2 } \theta = 3 \cos 2 \theta - \sin 2 \theta + 1 .$$
WJEC Further Unit 4 2024 June Q10
10. The following simultaneous equations are to be solved. $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 4 x + 2 y + 6 \mathrm { e } ^ { 3 t }
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 6 x + 8 y + 15 \mathrm { e } ^ { 3 t } \end{aligned}$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0\).
  2. Given that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 9\) and \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 10\) when \(t = 0\), find the particular solution for \(x\) in terms of \(t\). Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 4 Specimen Q1
  1. (a) Evaluate the integral
$$\int _ { 0 } ^ { \infty } \frac { \mathrm { d } x } { ( 1 + x ) ^ { 5 } }$$ (b) By putting \(u = \ln x\), determine whether or not the following integral has a finite value. $$\int _ { 2 } ^ { \infty } \frac { \mathrm { d } x } { x \ln x }$$
WJEC Further Unit 4 Specimen Q2
  1. Evaluate the integral
$$\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { 2 x ^ { 2 } + 4 x + 6 } }$$
WJEC Further Unit 4 Specimen Q3
  1. The curve \(C\) has polar equation \(r = 3 ( 2 + \cos \theta ) , 0 \leq \theta \leq \pi\). Determine the area enclosed between \(C\) and the initial line. Give your answer in the form \(\frac { a } { b } \pi\), where \(a\) and \(b\) are positive integers whose values are to be found.
  2. Find the three cube roots of the complex number \(2 + 3 \mathrm { i }\), giving your answers in Cartesian form.
  3. Find all the roots of the equation
$$\cos \theta + \cos 3 \theta + \cos 5 \theta = 0$$ lying in the interval \([ 0 , \pi ]\). Give all the roots in radians in terms of \(\pi\).
WJEC Further Unit 4 Specimen Q6
6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left[ \begin{array} { l l l } 2 & 1 & 3
1 & 3 & 2
3 & 2 & 5 \end{array} \right]$$
  1. Find
    1. the adjugate matrix of \(\mathbf { M }\),
    2. hence determine the inverse matrix \(\mathbf { M } ^ { - 1 }\).
  2. Use your result to solve the simultaneous equations $$\begin{aligned} & 2 x + y + 3 z = 13
    & x + 3 y + 2 z = 13
    & 3 x + 2 y + 5 z = 22 \end{aligned}$$
WJEC Further Unit 4 Specimen Q7
  1. The function \(f\) is defined by
$$f ( x ) = \frac { 8 x ^ { 2 } + x + 5 } { ( 2 x + 1 ) \left( x ^ { 2 } + 3 \right) }$$
  1. Express \(f ( x )\) in partial fractions.
  2. Hence evaluate $$\int _ { 2 } ^ { 3 } f ( x ) \mathrm { d } x$$ giving your answer correct to three decimal places.
WJEC Further Unit 4 Specimen Q8
8. The curve \(y = 1 + x ^ { 3 }\) is denoted by \(C\).
  1. A bowl is designed by rotating the arc of \(C\) joining the points \(( 0,1 )\) and \(( 2,9 )\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl.
  2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\).
WJEC Further Unit 4 Specimen Q9
9. (a) Use mathematical induction to prove de Moivre's Theorem, namely that $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
(b) (i) Use this result to show that $$\sin 5 \theta = a \sin ^ { 5 } \theta - b \sin ^ { 3 } \theta + c \sin \theta$$ where \(a , b\) and \(c\) are positive integers to be found.
(ii) Hence determine the value of \(\lim _ { \theta \rightarrow 0 } \frac { \sin 5 \theta } { \sin \theta }\)
WJEC Further Unit 4 Specimen Q10
10. Consider the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$
  1. Find an integrating factor for this differential equation.
  2. Solve the differential equation given that \(y = 0\) when \(x = \frac { \pi } { 4 }\), giving your answer in the form \(y = f ( x )\).
WJEC Further Unit 4 Specimen Q11
11. (a) Show that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , \quad \text { where } - 1 < x < 1$$ (b) Given that $$a \cosh x + b \sinh x \equiv \operatorname { rcosh } ( x + \alpha ) , \quad \text { where } a > b > 0$$ show that $$\alpha = \frac { 1 } { 2 } \ln \left( \frac { a + b } { a - b } \right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\).
(c) Hence solve the equation $$5 \cosh x + 4 \sinh x = 10$$ giving your answers correct to three significant figures.