WJEC Unit 3 (Unit 3) 2024 June

1. The function \(f\) is given by

$$f ( x ) = \frac { 25 x + 32 } { ( 2 x - 5 ) ( x + 1 ) ( x + 2 ) }$$

1. Express \(f ( x )\) in terms of partial fractions.
2. Show that \(\int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x = - \ln P\), where \(P\) is an integer whose value is to be found. \section*{
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3. Show that the sign of \(f ( x )\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f ( x ) = 0\) in this case.

2. (a) Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 360 ^ { \circ }\) satisfying $$3 \cot \theta + 4 \operatorname { cosec } ^ { 2 } \theta = 5 .$$ (b) By writing \(24 \cos x - 7 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), solve the equation $$24 \cos x - 7 \sin x = 16$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

3. The diagram below shows a badge \(O D C\). The shape \(O A B\) is a sector of a circle centre \(O\) and radius \(r \mathrm {~cm}\). The shape \(O D C\) is a sector of a circle with the same centre \(O\). The length \(A D\) is 5 cm and angle \(A O B\) is \(\frac { \pi } { 5 }\) radians. The area of the shaded region, \(A B C D\), is \(\frac { 13 \pi } { 2 } \mathrm {~cm} ^ { 2 }\).
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1. Determine the value of \(r\).
2. Calculate the perimeter of the shaded region.

4. A function \(f\) is given by \(f ( x ) = | 3 x + 4 |\).

1. Sketch the graph of \(y = f ( x )\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis.
2. On a separate set of axes, sketch the graph of \(y = \frac { 1 } { 2 } f ( x ) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\).
   Clearly label the coordinates of the points \(A ^ { \prime }\) and \(B ^ { \prime }\).

5. Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac { 81 } { x } \geqslant 18\).
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1. (a) Differentiate \(\cos x\) from first principles.
   (b) Differentiate \(\mathrm { e } ^ { 3 x } \sin 4 x\) with respect to \(x\).
   (c) Find \(\int x ^ { 2 } \sin 2 x \mathrm {~d} x\).
   
2. Showing all your working, evaluate
   (a) \(\quad \sum _ { r = 3 } ^ { 50 } ( 4 r + 5 )\),
   (b) \(\quad \sum _ { r = 2 } ^ { \infty } \left( 540 \times \left( \frac { 1 } { 3 } \right) ^ { r } \right)\).
   
3. The function \(f\) is defined by

$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x - 1$$ (a) Show that the equation \(f ( x ) = 0\) has a root in the interval \([ 0,1 ]\).
(b) Using the Newton-Raphson method with \(x _ { 0 } = 0 \cdot 8\),
(i) write down in full the decimal value of \(x _ { 1 }\) as given in your calculator,
(ii) determine the value of this root correct to six decimal places.
Explain why the Newton-Raphson method does not work if \(x _ { 0 } = \frac { 1 } { 3 }\).

9. The diagram below shows a sketch of the curve \(C _ { 1 }\) with equation \(y = - x ^ { 2 } + \pi x + 1\) and a sketch of the curve \(C _ { 2 }\) with equation \(y = \cos 2 x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{e4a4ea5b-7278-4735-925c-265a556ad679-18_817_1173_577_431} Calculate the area of the shaded region enclosed by \(C _ { 1 } , C _ { 2 }\) and the \(x\)-axis. Give your answer in terms of \(\pi\).

10. The function \(f\) has domain \([ 4 , \infty )\) and is defined by $$f ( x ) = \frac { 2 ( 3 x + 1 ) } { x ^ { 2 } - 2 x - 3 } + \frac { x } { x + 1 }$$

1. Show that \(f ( x ) = \frac { x + 2 } { x - 3 }\).
   10. The function \(f\) has domain \([ 4 , \infty )\) and is defined by $$f ( x ) = \frac { 2 ( 3 x + 1 ) } { x ^ { 2 } - 2 x - 3 } + \frac { x } { x + 1 } .$$
2. Show that \(f ( x ) = \frac { x + 2 } { x - 3 }\). [4]
3. Determine the range of \(f ( x )\).
4. Find an expression for \(f ^ { - 1 } ( x )\) and write down the domain and range of \(f ^ { - 1 }\).
5. Find the value of \(x\) when \(f ( x ) = f ^ { - 1 } ( x )\).

11. A curve is defined parametrically by $$x = 2 \theta + \sin 2 \theta , \quad y = 1 + \cos 2 \theta$$

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(- \tan \theta\).
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { \pi } { 4 }\).

12. (a) Given that \(\theta\) is small, show that \(2 \cos \theta + \sin \theta - 1 \approx 1 + \theta - \theta ^ { 2 }\).
(b) Hence, when \(\theta\) is small, show that $$\frac { 1 } { 2 \cos \theta + \sin \theta - 1 } \approx 1 + a \theta + b \theta ^ { 2 }$$ where \(a , b\) are constants to be found.
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14. (a) Given that \(y = \frac { 1 + \ln x } { x }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - \ln x } { x ^ { 2 } }\).
(b) Hence, solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ^ { 2 } t } { \ln x } ,$$ given that \(t = 3\) when \(x = 1\).
Give your answer in the form \(t ^ { 2 } = g ( x )\), where \(g\) is a function of \(x\).

15. Robert wants to deposit \(\pounds P\) into a savings account. He has a choice of two accounts.

• Account \(A\) offers an annual compound interest rate of \(1 \%\).
• Account \(B\) offers an interest rate of \(5 \%\) for the first year and an annual compound interest rate of \(0.6 \%\) for each subsequent year.

After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\).
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