SPS SPS FM Pure (SPS FM Pure) 2021 May

1. Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
   1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
   2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).
      [0pt] [BLANK PAGE]
   3. The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
      \includegraphics[max width=\textwidth, alt={}, center]{25055c9c-2d29-476e-887a-a10699814b85-04_505_704_348_292}
   4. The greatest value of \(r\) on the curve occurs at the point \(P\).
      1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
      2. Find the value of \(r\) at the point \(P\).
      3. Mark the point \(P\) on a copy of the graph.
   5. In this question you must show detailed reasoning.
   Find the exact area of the region enclosed by the curve.
   [0pt] [BLANK PAGE]

3. You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & 0 & 1
0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix B. The point \(P\) has coordinates \(( 2,3,4 )\). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).
   [0pt] [BLANK PAGE]

4. Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).
[0pt] [BLANK PAGE]

5. Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
[0pt] [BLANK PAGE]

6. $$\mathbf { A } = \left( \begin{array} { r r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is a constant. }$$

1. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(k\). A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
   The point \(P\) has position vector \(\binom { a } { 2 a }\) relative to an origin \(O\).
   The point \(Q\) has position vector \(\binom { 7 } { - 3 }\) relative to \(O\).
   Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),
2. determine the value of \(a\) and the value of \(k\). Given that, for a different value of \(k , T\) maps the line \(y = 2 x\) onto itself,
3. determine this value of \(k\).
   [0pt] [BLANK PAGE]

7. Given that \(y = \arcsin x , - 1 \leq x < 1\),

1. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\). Given that \(f ( x ) = \frac { 3 x + 2 } { \sqrt { 4 - x ^ { 2 } } }\),
2. show that the mean value of \(f ( x )\) over the interval \([ 0 , \sqrt { } 2 ]\), is $$\frac { \pi \sqrt { } 2 } { 4 } + A \sqrt { } 2 - A$$ where \(A\) is a constant to be determined.
   [0pt] [BLANK PAGE]

8.

1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$
2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.
   [0pt] [BLANK PAGE]

9.

1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
2. Find the range of \(x\) for which the expansion is valid.
3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots$$ [BLANK PAGE]

10. A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t$$

1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). Initially there was 10 g of the substance.
2. Find the particular solution of the differential equation.
3. Find to 6 significant figures the amount of substance that would be predicted by the model at
   (a) 6 hours,
   (b) 6.25 hours.
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]