WJEC Unit 3 (Unit 3) 2019 June

a) Express \(\frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } }\) in terms of partial fractions. b) Find \(\int \frac { 9 } { ( x - 1 ) ( x + 2 ) ^ { 2 } } \mathrm {~d} x\).

Expand \(\frac { 4 - x } { \sqrt { 1 + 2 x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). State the range of values of \(x\) for which the expansion is valid.

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\). The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).

a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$

a) Differentiate each of the following functions with respect to \(x\). i) \(x ^ { 5 } \ln x\) ii) \(\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }\) iii) \(( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }\) b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
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The function \(f ( x )\) is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain \(x \geqslant 1\).
a) Find an expression for \(f ^ { - 1 } ( x )\). State the domain for \(f ^ { - 1 }\) and sketch both \(f ( x )\) and \(f ^ { - 1 } ( x )\) on the same diagram.
b) Explain why the function \(f f ( x )\) cannot be formed.
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A chord \(A B\) subtends an angle \(\theta\) radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio \(1 : 2\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that \(\sin \theta = \theta - \frac { 2 \pi } { 3 }\).
b) i) Show that \(\theta\) lies between \(2 \cdot 6\) and \(2 \cdot 7\).
ii) Starting with \(\theta _ { 0 } = 2 \cdot 6\), use the Newton-Raphson Method to find the value of \(\theta\) correct to three decimal places. \section*{TURN OVER} Wildflowers grow on the grass verge by the side of a motorway. The area populated by wildflowers at time \(t\) years is \(A \mathrm {~m} ^ { 2 }\). The rate of increase of \(A\) is directly proportional to \(A\).
a) Write down a differential equation that is satisfied by \(A\).
b) At time \(t = 0\), the area populated by wildflowers is \(0.2 \mathrm {~m} ^ { 2 }\). One year later, the area has increased to \(1.48 \mathrm {~m} ^ { 2 }\). Find an expression for \(A\) in terms of \(t\) in the form \(p q ^ { t }\), where \(p\) and \(q\) are rational numbers to be determined.

a) Find \(\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x\). b) Find \(\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x\).
c) Find \(\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x\).
d) Use the substitution \(u = 2 \cos x + 1\) to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$

Use proof by contradiction to show that \(\sqrt { 6 }\) is irrational.