Edexcel C2 (Core Mathematics 2)

1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ),

1. find the value of \(p\). Given also that \(q = 3\), and \(p\) has the value found in part (a),
2. find the value of the remainder.

2. \begin{figure}[h] \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\).
2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
3. Find, to 3 significant figures, the length of \(P T\).

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\hline \end{tabular} \end{center} 1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ),

1. find the value of \(p\). Given also that \(q = 3\), and \(p\) has the value found in part (a),
2. find the value of the remainder.
   2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}
   \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.
3. Write down the value of \(a\) and the value of \(b\).
4. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
5. Find, to 3 significant figures, the length of \(P T\).
   3. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
6. Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
   4. The first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + p x ) ^ { n }\), are \(1 - 18 x + 36 p ^ { 2 } x ^ { 2 }\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\).
   (7 marks)
   5. Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
7. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
8. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.
   6. Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
9. \(\log _ { 2 } ( 16 x )\),
10. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
11. Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
    7. The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
12. Sketch \(C\).
13. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
14. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi\), $$\cos \left( x + \frac { \pi } { 4 } \right) = 0.5$$ giving your answers in terms of \(\pi\).

8. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).

1. Show that the equation of the curve may be written as $$y = x ( x - 3 ) ^ { 2 } ,$$ and hence write down the coordinates of \(A\).
2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
3. Find the area of \(R\).

9 \begin{figure}[h] \end{figure} Triangle \(A B C\) has \(A B = 9 \mathrm {~cm} , B C 10 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\). A circle, centre \(A\) and radius 3 cm , intersects \(A B\) and \(A C\) at \(P\) and \(Q\) respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, \(\angle B A C = 1.504\) radians. Calculate,
2. the area, in \(\mathrm { cm } ^ { 2 }\), of the sector \(A P Q\),
3. the area, in \(\mathrm { cm } ^ { 2 }\), of the shaded region \(B P Q C\),
4. the perimeter, in cm , of the shaded region \(B P Q C\).