OCR C2 (Core Mathematics 2)

1. Solve the equation

$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of

$$( 1 + x ) ( 1 - x ) ^ { 6 }$$

1. (i) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.

4. The diagram shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

5. (i) Find $$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$ The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and the curve passes through the point \(( 1,1 )\).
(ii) Show that the equation of the curve can be written in the form $$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found.

6. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),

1. find the value of the constant \(p\),
2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.

7. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.

1. Find the common ratio of the series.
2. Show that the first term of the series is \(\log _ { 3 } 2\).
3. Find, to 3 significant figures, the sum of the first six terms of the series.

8. (i) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\tan 2 x = 3$$ (ii) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y$$

9. The diagram shows the curve \(C\) with equation \(y = 3 x - 4 \sqrt { x } + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,

1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2 x - 2\). The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the \(y\)-axis.
2. Find the area of the shaded region.