Edexcel C4 (Core Mathematics 4)

1. (a) Express \(1.5 \sin 2 x + 2 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate.
   (b) Express \(3 \sin x \cos x + 4 \cos ^ { 2 } x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\) and \(c\) are constants to be found.
   (c) Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos ^ { 2 } x\).

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$ (a) Given that \(\mathrm { f } ( x ) = A + \frac { B } { 1 + x } + \frac { C } { 3 - x }\), find the values of the constants \(A , B\) and \(C\). The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
(b) Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p , q\) and \(r\) are rational constants to be found.

3. A student tests the accuracy of the trapezium rule by evaluating \(I\), where $$I = \int _ { 0.5 } ^ { 1.5 } \left( \frac { 3 } { x } + x ^ { 4 } \right) \mathrm { d } x$$

1. Complete the student's table, giving values to 2 decimal places where appropriate.

   \(x\)	0.5	0.75	1	1.25	1.5
   \(\frac { 3 } { x } + x ^ { 4 }\)	6.06	4.32

2. Use the trapezium rule, with all the values from your table, to calculate an estimate for the value of \(I\).
3. Use integration to calculate the exact value of \(I\).
4. Verify that the answer obtained by the trapezium rule is within \(3 \%\) of the exact value.

4. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{a1b078fe-96e3-4d62-bf0d-415294ba022f-4_588_1008_242_566} Figure 1 shows a cross-section \(R\) of a dam. The line \(A C\) is the vertical face of the dam, \(A B\) is the horizontal base and the curve \(B C\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A , B\) and \(C\) have coordinates \(( 0,0 ) , \left( 3 \pi ^ { 2 } , 0 \right)\) and \(( 0,30 )\) respectively. The area of the cross-section is to be calculated. Initially the profile \(B C\) is approximated by a straight line.

1. Find an estimate for the area of the cross-section \(R\) using this approximation.
   (1) The profile \(B C\) is actually described by the parametric equations. $$x = 16 t ^ { 2 } - \pi ^ { 2 } , \quad y = 30 \sin 2 t , \quad \frac { \pi } { 4 } \leq t \leq \frac { \pi } { 2 }$$
2. Find the exact area of the cross-section \(R\).
   (7)
3. Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a).
   (2)

5. (a) Prove that, when \(x = \frac { 1 } { 15 }\), the value of \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }\) is exactly equal to \(\sin 60 ^ { \circ }\).
(3)
(b) Expand \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } } , | x | < 0.2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(c) Use your answer to part (b) to find an approximation for \(\sin 60 ^ { \circ }\).
(d) Find the difference between the exact value of \(\sin 60 ^ { \circ }\) and the approximation in part (c).

6. (a) Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } x \sec ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \ln 2 .$$ \begin{figure}[h] \end{figure} The finite region \(R\), bounded by the equation \(y = x ^ { \frac { 1 } { 2 } } \sec x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Find the volume of the solid of revolution generated.
(c) Find the gradient of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \sec x\) at the point where \(x = \frac { \pi } { 4 }\).

7. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = 2 \mathrm { e } ^ { x } + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3 .

1. Find the equation of the normal to \(C\) at \(M\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N ( n , 0 )\).
2. Show that \(n = 14\). The point \(P ( \ln 4,13 )\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(P N\), as shown in Fig. 3.
3. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found.

8. Referred to an origin \(O\), the points \(A , B\) and \(C\) have position vectors ( \(9 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ), \(( 6 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k } )\) and \(( 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k } )\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). Given that \(C\) lies on \(l\),
2. find the value of \(p\) and the value of \(q\),
3. calculate, in degrees, the acute angle between \(O C\) and \(A B\). The point \(D\) lies on \(A B\) and is such that \(O D\) is perpendicular to \(A B\).
4. Find the position vector of \(D\).