Edexcel C34 (Core Mathematics 3 & 4) 2018 October

1. (a) Write \(\cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < \pi\), the equation

$$\cos 2 \theta + 4 \sin 2 \theta = 1.2$$ giving your answers to 2 decimal places.

2. A curve \(C\) has equation $$x ^ { 3 } - 4 x y + 2 x + 3 y ^ { 2 } - 3 = 0$$ Find an equation of the normal to \(C\) at the point ( \(- 3,2\) ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{c6bde466-61ec-437d-a3b4-84511a98d788-05_108_166_2612_1781}

3. Given
\(\cos \theta ^ { \circ } = p\), where \(p\) is a constant and \(\theta ^ { \circ }\) is acute use standard trigonometric identities to find, in terms of \(p\),

1. \(\sec \theta ^ { \circ }\)
2. \(\sin ( \theta - 90 ) ^ { \circ }\)
3. \(\sin 2 \theta ^ { \circ }\) Write each answer in its simplest form.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = 8 x - x \mathrm { e } ^ { 3 x } , x \geqslant 0\) The curve meets the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).

1. Find the exact \(x\) coordinate of \(A\), giving your answer in its simplest form. The curve has a maximum turning point at the point \(M\).
2. Show, by using calculus, that the \(x\) coordinate of \(M\) is a solution of $$x = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x _ { n } } \right)$$ with \(x _ { 0 } = 0.4\) to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.

5. $$f ( x ) = \frac { 4 x ^ { 2 } + 5 x + 3 } { ( x + 2 ) ( 1 - x ) ^ { 2 } } \equiv \frac { A } { ( x + 2 ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$$

1. Find the values of the constants \(A\), \(B\) and \(C\).
2. 1. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
   2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\), writing your answer in the form \(p + \ln q\), where \(p\) and \(q\) are constants.
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6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\)
(b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$

7. A curve has equation $$y = \ln ( 1 - \cos 2 x ) , \quad x \in \mathbb { R } , 0 < x < \pi$$ Show that

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \cot x\), where \(k\) is a constant to be found. Hence find the exact coordinates of the point on the curve where
2. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { 3 }\)

8. (i) Find \(\int x \sin x d x\)
(ii) (a) Use the substitution \(x = \sec \theta\) to show that
(b) Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { \pi } { 3 } } \tan ^ { 2 } \theta \mathrm {~d} \theta$$ Hence find the exact value of $$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x$$

9. A rare species of mammal is being studied. The population \(P\), \(t\) years after the study started, is modelled by the formula $$P = \frac { 900 \mathrm { e } ^ { \frac { 1 } { 4 } t } } { 3 \mathrm { e } ^ { \frac { 1 } { 4 } t } - 1 } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$ Using the model,

1. calculate the number of mammals at the start of the study,
2. calculate the exact value of \(t\) when \(P = 315\) Give your answer in the form \(a \ln k\), where \(a\) and \(k\) are integers to be determined.
3. 1. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
   2. Hence find the value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 8\), giving your answer to 2 decimal places.

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .

1. State the range of g .
2. State the coordinates of
   1. point \(A\)
   2. point \(B\)
3. Find \(\operatorname { gg } ( x )\) in its simplest form.
4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)

11. Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
4
3 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) passes through the origin and is parallel to \(l _ { 1 }\)

1. Find a vector equation for \(l _ { 2 }\) The point \(A\) and the point \(B\) both lie on \(l _ { 1 }\) with parameters \(\lambda = 0\) and \(\lambda = 3\) respectively.
   Write down
2. 1. the coordinates of \(A\),
   2. the coordinates of \(B\).
3. Find the size of the acute angle between \(O A\) and \(l _ { 1 }\) Give your answer in degrees to one decimal place. The point \(D\) lies on \(l _ { 2 }\) such that \(O A B D\) is a parallelogram.
4. Find the area of \(O A B D\), giving your answer to the nearest whole number.

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 7 t ^ { 2 } - 5 , \quad y = t \left( 9 - t ^ { 2 } \right) , \quad t \in \mathbb { R }$$

1. Find an equation of the tangent to \(C\) at the point where \(t = 1\) Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\), as shown in Figure 3
2. 1. Find the \(x\) coordinate of the point \(A\).
   2. Find the \(x\) coordinate of the point \(B\). The region \(R\), shown shaded in Figure 3, is enclosed by the loop of the curve \(C\).
3. Use integration to find the area of \(R\).

13. The volume of a spherical balloon of radius \(r \mathrm {~m}\) is \(V \mathrm {~m} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) Given that the volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } } , \quad t \geqslant 0$$
2. find an expression in terms of \(r\) and \(t\) for \(\frac { \mathrm { d } r } { \mathrm {~d} t }\) Given that \(V = 1\) when \(t = 0\)
3. solve the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 20 } { V ( 0.05 t + 1 ) ^ { 3 } }$$ giving your answer in the form \(V ^ { 2 } = \mathrm { f } ( t )\).
4. Hence find the radius of the balloon at time \(t = 20\), giving your answer to 3 significant figures.
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