SPS SPS FM (SPS FM) 2020 December

1. Solve \(2 \sin x = \tan x\) exactly, where \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
2. Let \(a , b\) satisfy \(0 < a < b\).
   i. Find, in terms of \(a\) and \(b\), the value of

$$\int _ { a } ^ { b } \frac { 81 } { x ^ { 4 } } d x$$ ii. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int _ { a } ^ { \infty } \frac { 81 } { x ^ { 4 } } d x = \frac { 216 } { 125 }$$

1. i. Sketch the graph of \(y = | 3 x - 1 |\).
   ii. Hence, solve \(5 x + 3 < | 3 x - 1 |\).
2. The following diagram shows the curve \(y = a \sin ( b ( x + c ) ) + d\), where \(a , b , c\) and \(d\) are all positive constants and \(x\) is measured in radians. The curve has a maximum point at \(( 1,3.5 )\) and a minimum point at \(( 2,0.5 )\).
   \includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-05_826_1109_269_532}
   i) Write down the value of \(a\) and the value of \(d\).
   ii) Find the value of \(b\).
   iii) Find the smallest possible value of \(c\), given that \(c > 0\).
3. The \(2 \times 2\) matrix \(\mathbf { A }\) represents a rotation by \(90 ^ { \circ }\) anticlockwise about the origin. The \(2 \times 2\) matrix \(\mathbf { B }\) represents a reflection in the line \(y = - x\).
   The matrix \(\mathbf { B }\) is given by

$$\mathbf { B } = \left( \begin{array} { c c } 0 & - 1
- 1 & 0 \end{array} \right)$$ i. Write down the matrix representing \(\mathbf { A }\).
ii. The \(2 \times 2\) matrix \(\mathbf { C }\) represents a rotation by \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = - x\). Compute the matrix \(\mathbf { C }\) and describe geometrically the single transformation represented by \(\mathbf { C }\).

6. Given that \(z\) is the complex number \(x + i y\) and satisfies $$| z | + z = 6 - 2 i$$ find the value of \(x\) and the value of \(y\).

7. The diagram below shows part of a curve C with equation \(y = 1 + 3 x - \frac { 1 } { 2 } x ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-08_709_898_214_603}
i. The curve crosses the \(y\) axis at the point A . The straight line L is normal to the curve at A and meets the curve again at B . Find the equation of L and the \(x\) coordinate of the point B .
ii. The region \(R\) is bounded by the curve \(C\) and the line \(L\). Find the exact area of \(R\).

8.
a) The \(2 \times 2\) matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l } 7 & 3
2 & 1 \end{array} \right) .$$ The \(2 \times 2\) matrix \(\mathbf { B }\) satisfies $$\mathbf { B } \mathbf { A } ^ { 2 } = \mathbf { A } .$$ Find the matrix \(\mathbf { B }\).
b) The \(2 \times 2\) matrix \(\mathbf { C }\) is given by $$\mathbf { C } = \left( \begin{array} { l l } - 2 & 4
- 1 & 2 \end{array} \right) .$$ By considering \(\mathbf { C } ^ { 2 }\), show that the matrices \(\mathbf { I } - \mathbf { C }\) and \(\mathbf { I } + \mathbf { C }\) are inverse to each other.

9. Sketch on an Argand diagram the locus of all points that satisfy \(| z + 4 - 4 i | = 2 \sqrt { 2 }\) and hence find \(\theta , \phi \in ( - \pi , \pi ]\) such that \(\theta \leq \arg z \leq \phi\).

10. The \(2 \times 2\) matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c } 0 & 0.25
0.36 & 0 \end{array} \right)$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by \(\mathbf { M }\).

11. In the triangle \(P Q R , P Q = 6 , P R = k , P \hat { Q } R = 30 ^ { \circ }\).
i. For the case \(k = 4\), find the two possible values of \(Q R\) exactly.
ii. Determine the value(s) of \(k\) for which the conditions above define a unique triangle.

12. Consider the binomial expansion of \(\left( 1 + \frac { x } { 5 } \right) ^ { n }\) in ascending powers of \(x\), where \(n\) is a positive integer.
i. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
ii. Show that \(n ^ { 3 } - 33 n ^ { 2 } + 182 n = 0\).
iii. Hence find the possible values of \(n\) and the corresponding values of the common difference.

13. A series is given by $$\sum _ { r = 1 } ^ { k } 9 ^ { r - 1 }$$ i. Write down a formula for the sum of this series.
ii. Prove by induction that \(P ( n ) = 9 ^ { n } - 8 n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1 . Spare space for extra working. Spare space for extra working.