OCR C2 (Core Mathematics 2)

1. Giving your answers in terms of \(\pi\), solve the equation

$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).

2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for

1. \(\quad \log _ { 2 } 45\),
2. \(\log _ { 2 } 0.3\)

3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),

1. find and simplify the first three terms,
2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).

4. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
2. find the coordinates of \(Q\) and \(R\).

5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$

1. Find \(\mathrm { f } ( x )\).
2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).

6. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.

1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).

7. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for

1. \(3 ^ { x + 1 }\),
2. \(3 ^ { 2 x - 1 }\).
   (b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$

1. (i) Given that

$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.

9. The second and fifth terms of a geometric series are - 48 and 6 respectively.

1. Find the first term and the common ratio of the series.
2. Find the sum to infinity of the series.
3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).