OCR C2 (Core Mathematics 2) 2011 January

1

1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).

2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence \(S\) is.
3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-02_510_791_991_678} The diagram shows the curve \(y = \sqrt { x - 3 }\).

1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 5\). Give your answer correct to 3 significant figures.
2. State, with a reason, whether this approximation is an underestimate or an overestimate.

4

1. Use logarithms to solve the equation \(5 ^ { x - 1 } = 120\), giving your answer correct to 3 significant figures.
2. Solve the equation \(\log _ { 2 } x + 2 \log _ { 2 } 3 = \log _ { 2 } ( x + 5 )\).

5 In a geometric progression, the sum to infinity is four times the first term.

1. Show that the common ratio is \(\frac { 3 } { 4 }\).
2. Given that the third term is 9 , find the first term.
3. Find the sum of the first twenty terms.

6

1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
2. 1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
   2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).

7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(3 \tan 2 x = 1\)
2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)

8
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 5 cm . Angle \(A O B\) is \(\theta\) radians. The area of triangle \(A O B\) is \(8 \mathrm {~cm} ^ { 2 }\).

1. Given that the angle \(\theta\) is obtuse, find \(\theta\). The shaded segment in the diagram is bounded by the chord \(A B\) and the arc \(A B\).
2. Find the area of the segment, giving your answer correct to 3 significant figures.
3. Find the perimeter of the segment, giving your answer correct to 3 significant figures.

9
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-04_584_785_255_680} The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + 9 x ^ { 2 } + 10 x - 3\).

1. Verify that the curve crosses the \(x\)-axis at ( 3,0 ) and hence state a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Hence find the other two points of intersection of the curve with the \(x\)-axis.
4. The region enclosed by the curve and the \(x\)-axis is shaded in the diagram. Use integration to find the total area of this region.