OCR C2 (Core Mathematics 2) 2014 June

1
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_426_1244_280_413} The diagram shows triangle \(A B C\), with \(A B = 8 \mathrm {~cm}\), angle \(B A C = 65 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The point \(D\) is on \(A C\) such that \(A D = 10 \mathrm {~cm}\).

1. Find the area of triangle \(A B D\).
2. Find the length of \(B D\).
3. Find the length of \(B C\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).

1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 12 cm . The angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians.

1. Find the exact length of the \(\operatorname { arc } A B\).
2. Find the exact area of the shaded segment enclosed by the arc \(A B\) and the chord \(A B\).

4

1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5 Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).

6

1. Find the binomial expansion of \(\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x\).

7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.

8

1. The first term of a geometric progression is 50 and the common ratio is 0.8 . Use logarithms to find the smallest value of \(k\) such that the value of the \(k\) th term is less than 0.15 .
2. In a different geometric progression, the second term is - 3 and the sum to infinity is 4 . Show that there is only one possible value of the common ratio and hence find the first term. \section*{Question 9 begins on page 4.}

9
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-4_387_624_287_717} The diagram shows part of the curve \(y = - 3 + 2 \sqrt { x + 4 }\). The point \(P ( 5,3 )\) lies on the curve. Region \(A\) is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 5\). Region \(B\) is bounded by the curve, the \(y\)-axis and the line \(y = 3\).

1. Use the trapezium rule, with 2 strips each of width 2.5 , to find an approximate value for the area of region \(A\), giving your answer correct to 3 significant figures.
2. Use your answer to part (i) to deduce an approximate value for the area of region \(B\).
3. By first writing the equation of the curve in the form \(x = \mathrm { f } ( y )\), use integration to show that the exact area of region \(B\) is \(\frac { 14 } { 3 }\). \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}