OCR C2 (Core Mathematics 2) 2010 January

1

1. Show that the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ can be expressed in the form $$2 \cos ^ { 2 } x + 5 \cos x - 3 = 0$$
2. Hence solve the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 4\). The curve passes through the distinct points ( 2,5 ) and ( \(p , 5\) ).

1. Find the equation of the curve.
2. Find the value of \(p\).

3

1. Find and simplify the first four terms in the expansion of \(( 2 - x ) ^ { 7 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(w ^ { 6 }\) in the expansion of \(\left( 2 - \frac { 1 } { 4 } w ^ { 2 } \right) ^ { 7 }\).

4

1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for $$\int _ { 3 } ^ { 5 } \log _ { 10 } ( 2 + x ) d x$$ giving your answer correct to 3 significant figures.
2. Use your answer to part (i) to deduce an approximate value for \(\int _ { 3 } ^ { 5 } \log _ { 10 } \sqrt { 2 + x } \mathrm {~d} x\), showing your method clearly.

5
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653} The diagram shows parts of the curves \(y = x ^ { 2 } + 1\) and \(y = 11 - \frac { 9 } { x ^ { 2 } }\), which intersect at \(( 1,2 )\) and \(( 3,10 )\). Use integration to find the exact area of the shaded region enclosed between the two curves.

6 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 3\) ) is a factor of \(\mathrm { f } ( x )\) and that, when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 35 .

1. Find the values of \(a\) and \(b\).
2. Using these values of \(a\) and \(b\), divide \(\mathrm { f } ( x )\) by ( \(x + 3\) ).

7
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_469_673_1720_737} The diagram shows triangle \(A B C\), with \(A B = 10 \mathrm {~cm} , B C = 13 \mathrm {~cm}\) and \(C A = 14 \mathrm {~cm} . E\) and \(F\) are points on \(A B\) and \(A C\) respectively such that \(A E = A F = 4 \mathrm {~cm}\). The sector \(A E F\) of a circle with centre \(A\) is removed to leave the shaded region \(E B C F\).

1. Show that angle \(C A B\) is 1.10 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region \(E B C F\).
3. Find the area of the shaded region \(E B C F\).

8 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$

1. Show that \(u _ { 5 } = 20\).
2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
3. State what type of sequence it is.
4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).

9

1. Sketch the curve \(y = 6 \times 5 ^ { x }\), stating the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 9 ^ { x }\) has \(y\)-coordinate equal to 150 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
3. The curves \(y = 6 \times 5 ^ { x }\) and \(y = 9 ^ { x }\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as \(x = \frac { 1 + \log _ { 3 } 2 } { 2 - \log _ { 3 } 5 }\).