Edexcel C2 (Core Mathematics 2)

1. Find the remainder when \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6\) is divided by \(( 2 x + 1 )\).
2. Given that \(2 \sin 2 \theta = \cos 2 \theta\),
   1. show that \(\tan 2 \theta = 0.5\).
   2. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }\).
      [0pt] [P1 June 2001 Question 2]
   3. (a) Using the substitution \(u = 2 ^ { x }\), show that the equation \(4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0\) can be written in the form \(u ^ { 2 } - 2 u - 15 = 0\).
   4. Hence solve the equation \(4 ^ { x } - 2 ^ { ( x + 1 ) } - 15 = 0\), giving your answers to 2 d . p.
      [0pt] [P2 November 2002 Question 2]
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1425d933-47e3-4a12-bcab-fd2ca41827e2-2_558_643_1101_609}
   \end{figure} The shape of a badge is a sector \(A B C\) of a circle with centre \(A\) and radius \(A B\), as shown in Fig 1. The triangle \(A B C\) is equilateral and has a perpendicular height 3 cm .

1. Find, in surd form, the length \(A B\).
2. Find, in terms of \(\pi\), the area of the badge.
3. Prove that the perimeter of the badge is \(\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }\).

5. A circle \(C\) has centre \(( 3,4 )\) and radius \(3 \sqrt { } 2\). A straight line \(l\) has equation \(y = x + 3\).

1. Write down an equation of the circle \(C\).
2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds.
3. Find the distance between these two points.

6. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots , u _ { n }\) is defined by the recurrence relation $$u _ { n + 1 } = p u _ { n } + 5 , u _ { 1 } = 2 , \text { where } p \text { is a constant. }$$ Given that \(u _ { 3 } = 8\),

1. show that one possible value of \(p\) is \(\frac { 1 } { 2 }\) and find the other value of \(p\). Using \(p = \frac { 1 } { 2 }\),
2. write down the value of \(\log _ { 2 } p\). Given also that \(\log _ { 2 } q = t\),
3. express \(\log _ { 2 } \left( \frac { p ^ { 3 } } { \sqrt { q } } \right)\) in terms of \(t\).
   [0pt] [P2 November 2002 Question 4]

7. \begin{figure}[h] \end{figure} Fig. 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6 x - x ^ { 2 } - 3\).
The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(R\) is bounded by the line and the curve.
2. Calculate the area of \(R\).

8. $$\mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2 .$$ Given that the coefficient of \(x ^ { 3 }\) is twice the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(\mathrm { f } ( x )\),

1. prove that \(n = 6 k + 2\). Given also that the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal and non-zero,
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\),
3. expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 5 }\). Give each coefficient as an exact fraction in its lowest terms

9. \begin{figure}[h] \end{figure} A rectangular sheet of metal measures 50 cm by 40 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 3.

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the tray is given by \(V = 4 x \left( x ^ { 2 } - 45 x + 500 \right)\).
2. State the range of possible values of \(x\).
3. Find the value of \(x\) for which \(V\) is a maximum.
4. Hence find the maximum value of \(V\).
5. Justify that the value of \(V\) you found in part (d) is a maximum.