OCR C2 (Core Mathematics 2)

1. Expand \(( 3 - 2 x ) ^ { 4 }\) in ascending powers of \(x\) and simplify each coefficient.

The diagram shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).

3. (i) Given that $$5 \cos \theta - 2 \sin \theta = 0$$ show that \(\tan \theta = 2.5\)
(ii) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.

4. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
2. \(\quad \log _ { 2 } ( \sqrt { x } )\).
   (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$

5. The diagram shows the sector \(O A B\) of a circle, centre \(O\), in which \(\angle A O B = 2.5\) radians. Given that the perimeter of the sector is 36 cm ,

1. find the length \(O A\),
2. find the perimeter and the area of the shaded segment.

6.
\includegraphics[max width=\textwidth, alt={}, center]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_503_839_1208_557} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).

1. Show that \(a = 8\).
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.

7. (a) Evaluate $$\sum _ { r = 10 } ^ { 30 } ( 7 + 2 r )$$ (b) (i) Write down the formula for the sum of the first \(n\) positive integers.
(ii) Using this formula, find the sum of the integers from 100 to 200 inclusive.
(iii) Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3 .

8. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.

1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
3. find the common ratio of the series,
4. find the sum of the first eight terms of the series.

9. (i) Evaluate $$\int _ { 1 } ^ { 3 } ( 3 - \sqrt { x } ) ^ { 2 } d x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$ where \(k\) is a constant. Given that the curve passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\), show that \(k = 2\) and find an equation for the curve.