OCR C2 (Core Mathematics 2) 2010 June

1 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x - 14\), where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Using this value of \(a\), find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ).

2

1. Use the trapezium rule, with 3 strips each of width 3 , to estimate the area of the region bounded by the curve \(y = \sqrt [ 3 ] { 7 + x }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 10\). Give your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate of the area.

3

1. Find and simplify the first four terms in the binomial expansion of \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 + 4 x + 2 x ^ { 2 } \right) \left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{570435e0-5685-4c5b-9ed8-f2bc22bdfb24-02_396_1070_1768_536} The diagram shows two congruent triangles, \(B C D\) and \(B A E\), where \(A B C\) is a straight line. In triangle \(B C D , B D = 8 \mathrm {~cm} , C D = 11 \mathrm {~cm}\) and angle \(C B D = 65 ^ { \circ }\). The points \(E\) and \(D\) are joined by an arc of a circle with centre \(B\) and radius 8 cm .

1. Find angle \(B C D\).
2. (a) Show that angle \(E B D\) is 0.873 radians, correct to 3 significant figures.
   (b) Hence find the area of the shaded segment bounded by the chord \(E D\) and the arc \(E D\), giving your answer correct to 3 significant figures.

6

1. Use integration to find the exact area of the region enclosed by the curve \(y = x ^ { 2 } + 4 x\), the \(x\)-axis and the lines \(x = 3\) and \(x = 5\).
2. Find \(\int ( 2 - 6 \sqrt { y } ) \mathrm { d } y\).
3. Evaluate \(\int _ { 1 } ^ { \infty } \frac { 8 } { x ^ { 3 } } \mathrm {~d} x\).

7

1. Show that \(\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv \tan ^ { 2 } x - 1\).
2. Hence solve the equation $$\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5 - \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

8

1. Use logarithms to solve the equation \(5 ^ { 3 w - 1 } = 4 ^ { 250 }\), giving the value of \(w\) correct to 3 significant figures.
2. Given that \(\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4\), express \(y\) in terms of \(x\).

9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.

1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
2. Given that the geometric progression converges, find the exact value of \(r\).
3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).