SPS SPS FM (SPS FM) 2020 October

1. i. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
   ii. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).
2. Let \(a = \log _ { 2 } x , b = \log _ { 2 } y\) and \(c = \log _ { 2 } z\).

Express \(\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)\) in terms of \(a , b\) and \(c\).

3. i. Give full details of a sequence of two transformations needed to transform the graph \(y = | x |\) to the graph of \(y = | 2 ( x + 3 ) |\).
ii. Solve \(| x | > | 2 ( x + 3 ) |\), giving your answer in set notation.

4. Prove by induction that, for \(n \geq 1 , \sum _ { r = 1 } ^ { n } r ( 3 r + 1 ) = n ( n + 1 ) ^ { 2 }\).

5. The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = ( x + 2 ) \mathrm { cm } , B C = 2 \sqrt { 7 } \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\).
i. Find the value of \(x\).
ii. Find the area of triangle \(A B C\), giving your answer in an exact form as simply as possible.

6. Prove by contradiction that \(\sqrt { 7 }\) is irrational.

7. A curve has equation \(y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }\).
i. Find \(\frac { d y } { d x }\).
ii. Hence sketch the gradient function for the curve.
iii. Find the equation of the tangent to the curve \(y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }\) at \(x = 4\).

8. The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
i. Find the centre and radius of the circle.
ii. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
iii. State what can be deduced from the answer to part ii. about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
iv. The point \(A ( - 1,5 )\) lies on the circumference of the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\). Given that \(A B\) is a diameter of the circle, find the coordinates of \(B\).

9. In this question you must show detailed reasoning. A sequence \(t _ { 1 } , t _ { 2 } , t _ { 3 } \ldots\) is defined by \(t _ { n } = 5 - 2 n\).
Use an algebraic method to find the smallest value of \(N\) such that $$\sum _ { n = 1 } ^ { \infty } 2 ^ { t _ { n } } - \sum _ { n = 1 } ^ { N } 2 ^ { t _ { n } } < 10 ^ { - 8 }$$