OCR C2 (Core Mathematics 2)

1. Find

$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$

2. The diagram shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).

3.
\includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-1_474_863_1283_520} The diagram shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).

1. Use the trapezium rule with four intervals, each of width 0.25 , to find an estimate for the area of the shaded region.
2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.

4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.

1. Find the value of \(k\). Using this value of \(k\),
2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375 ,
3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$

1. (i) Given that

$$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0$$ (ii) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0$$

6. $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
2. 1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
   2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0$$

1. (i) Given that

$$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$ show that $$y = 2 x + 1$$ (ii) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x
& 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$

1. The first two terms of an arithmetic progression are \(( t - 1 )\) and \(\left( t ^ { 2 } - 5 \right)\) respectively, where \(t\) is a positive constant.
   1. Find and simplify expressions in terms of \(t\) for
      1. the common difference,
      2. the third term.
   Given also that the third term is 19 ,
2. find the value of \(t\),
3. show that the 10th term is 75,
4. find the sum of the first 40 terms.

9.
\includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447} The diagram shows the curves \(y = 2 x ^ { 2 } - 6 x - 3\) and \(y = 9 + 3 x - x ^ { 2 }\).

1. Find the coordinates of the points where the two curves intersect.
2. Find the area of the shaded region bounded by the two curves.