OCR C2 (Core Mathematics 2)

1. Evaluate

$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$

2. The diagram shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\). Find

1. the length \(B C\),
2. the area of triangle \(A B C\).

3. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point (8, 7). Given that $$f ^ { \prime } ( x ) = 4 x ^ { \frac { 1 } { 3 } } - 5$$ find \(\mathrm { f } ( x )\).

4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\). Give your answers to 1 decimal place.

5. (i) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (ii) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$

1. (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
   (b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
   1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
   2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)
   3. The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.
      

1. Show that the common difference is 5 .
2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
   Given that the sums of the first \(n\) terms of these two sequences are equal,
3. find the value of \(n\).

8. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,

1. find an expression for \(b\) in terms of \(a\). Given also that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\),
2. find the values of \(a\) and \(b\),
3. fully factorise \(\mathrm { p } ( x )\).

9. \includegraphics[max width=\textwidth, alt={}, center]{33f9663f-26bb-445e-af6e-ca5ca927f7dd-3_638_757_1064_493} The diagram shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).

1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).