OCR C2 (Core Mathematics 2) 2011 June

1 The diagram shows triangle \(A B C\), with \(A B = 9 \mathrm {~cm} , A C = 17 \mathrm {~cm}\) and angle \(B A C = 40 ^ { \circ }\).

1. Find the length of \(B C\).
2. Find the area of triangle \(A B C\).
3. \(D\) is the point on \(A C\) such that angle \(B D A = 63 ^ { \circ }\). Find the length of \(B D\).

2

1. Find \(\int \left( 6 x ^ { \frac { 1 } { 2 } } - 1 \right) \mathrm { d } x\).
2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 1\) and which passes through the point \(( 4,17 )\).

3
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-2_515_501_1439_822} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius 8 cm . The perimeter of the sector is 23.2 cm .

1. Find angle \(A O B\) in radians.
2. Find the area of the sector.

4
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-3_588_1136_255_502} The diagram shows the curve \(y = - 1 + \sqrt { x + 4 }\) and the line \(y = 3\).

1. Show that \(y = - 1 + \sqrt { x + 4 }\) can be rearranged as \(x = y ^ { 2 } + 2 y - 3\).
2. Hence find by integration the exact area of the shaded region enclosed between the curve, the \(y\)-axis and the line \(y = 3\).

5 The first four terms in the binomial expansion of \(( 3 + k x ) ^ { 5 }\), in ascending powers of \(x\), can be written as \(a + b x + c x ^ { 2 } + d x ^ { 3 }\).

1. State the value of \(a\).
2. Given that \(b = c\), find the value of \(k\).
3. Hence find the value of \(d\).

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).

1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.

7

1. The first term of a geometric progression is 7 and the common ratio is - 2 .
   1. Find the ninth term.
   2. Find the sum of the first 15 terms.
2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).

8
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_417_931_255_607} The diagram shows the curve \(y = 2 ^ { x } - 3\).

1. Describe the geometrical transformation that transforms the curve \(y = 2 ^ { x }\) to the curve \(y = 2 ^ { x } - 3\).
2. State the \(y\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(y\)-axis.
3. Find the \(x\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(x\)-axis, giving your answer in the form \(\log _ { a } b\).
4. The curve \(y = 2 ^ { x } - 3\) passes through the point ( \(p , 62\) ). Use logarithms to find the value of \(p\), correct to 3 significant figures.
5. Use the trapezium rule, with 2 strips each of width 0.5 , to find an estimate for \(\int _ { 3 } ^ { 4 } \left( 2 ^ { x } - 3 \right) \mathrm { d } x\). Give your answer correct to 3 significant figures.

9

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_362_979_1505_625} The diagram shows part of the curve \(y = \cos 2 x\), where \(x\) is in radians. The point \(A\) is the minimum point of this part of the curve.
   1. State the period of \(y = \cos 2 x\).
   2. State the coordinates of \(A\).
   3. Solve the inequality \(\cos 2 x \leqslant 0.5\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
2. Solve the equation \(\cos 2 x = \sqrt { 3 } \sin 2 x\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.