OCR C2 (Core Mathematics 2)

2. \(f ( x ) = x ^ { 3 } + k x - 20\). Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),

1. find the value of the constant \(k\),
2. solve the equation \(\mathrm { f } ( x ) = 0\).

3. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { x } - x ^ { 2 }$$ and that \(y = \frac { 2 } { 3 }\) when \(x = 1\), find the value of \(y\) when \(x = 4\).

4. A geometric progression has third term 36 and fourth term 27. Find

1. the common ratio,
2. the fifth term,
3. the sum to infinity.

5. (i) Solve the equation $$\log _ { 2 } ( 6 - x ) = 3 - \log _ { 2 } x$$ (ii) Find the smallest integer \(n\) such that $$3 ^ { n - 2 } > 8 ^ { 250 }$$

1. \(f ( x ) = \cos 2 x , 0 \leq x \leq \pi\).
   1. Sketch the curve \(y = \mathrm { f } ( x )\).
   2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
   3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).
   4. (i) Find
   $$\int \left( x + 5 + \frac { 3 } { \sqrt { x } } \right) \mathrm { d } x$$
2. Evaluate $$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } d x$$

1. (a) An arithmetic series has a common difference of 7 .

Given that the sum of the first 20 terms of the series is 530 , find

1. the first term of the series,
2. the smallest positive term of the series.
   (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1$$ where \(k\) is a positive constant.
   Given that \(u _ { 2 } = 2 u _ { 1 }\),
3. find the value of \(k\),
4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

9.
\includegraphics[max width=\textwidth, alt={}, center]{61af807c-1f2c-417a-85cf-86f2cf566cb9-3_670_1022_1263_374} The diagram shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).

1. Find the coordinates of the points where the curve and line intersect.
2. Show that the area of the shaded region bounded by the curve and line is given by $$\int _ { - 3 } ^ { 1 } \left( 6 - 4 x - 2 x ^ { 2 } \right) d x$$
3. Hence find the area of the shaded region.