SPS SPS FM Pure (SPS FM Pure) 2022 June

1. (a) For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
   1. \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
      (3 marks)
   2. \(\quad \int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
      (3 marks)
      (b) Explain briefly why the integrals in part (a) are improper integrals.
      (1 mark)
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   \section*{2.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-06_592_1027_258_593}
   \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
   (a) \(y = \mathrm { f } ( x + 1 )\),
   (b) \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
   Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
   (c) the value of \(a\) and the value of \(b\),
   (d) the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
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   3. (a) Show on an Argand diagram the locus of points given by $$| z - 10 - 12 i | = 8$$ Set \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg ( z - 10 - 10 i ) \leqslant \frac { \pi } { 2 } \right\} \cap \{ z : | z - 10 - 12 i | \leqslant 8 \}$$ (b) Shade the region defined by \(A\) on your Argand diagram.
   (c) Determine the area of the region defined by \(A\).
   [0pt] [BLANK PAGE]
   4. The curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)
   (a) Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
   Starting with \(x _ { 1 } = 0.3\)
   (b) calculate, giving each answer to 4 decimal places,
2. the value of \(x _ { 2 }\)
3. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
   (c) show that \(\alpha\) is 0.341 to 3 decimal places.
   [0pt] [BLANK PAGE]
   5. The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { c c } 6 & - 2
   1 & 2 \end{array} \right)$$ Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\).
   [0pt] [BLANK PAGE]
   6. The complex number \(w\) is given by $$w = 10 - 5 \mathrm { i }$$ (a) Find \(| w |\).
   (b) Find arg \(w\), giving your answer in radians to 2 decimal places. The complex numbers \(z\) and \(w\) satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$ (c) Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where \(\lambda\) is a real constant,
   (d) find the value of \(\lambda\).
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   7. \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-16_634_1025_191_479}
   Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = x \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
   Use integration by parts to find an exact value for the volume of the solid generated.
   [0pt] [BLANK PAGE]
   8. With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { c } 13
   8
   1 \end{array} \right) + \lambda \left( \begin{array} { r }

3. (a) Show on an Argand diagram the locus of points given by $$| z - 10 - 12 i | = 8$$ Set \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg ( z - 10 - 10 i ) \leqslant \frac { \pi } { 2 } \right\} \cap \{ z : | z - 10 - 12 i | \leqslant 8 \}$$ (b) Shade the region defined by \(A\) on your Argand diagram.
(c) Determine the area of the region defined by \(A\).
[0pt] [BLANK PAGE]

4. The curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)

1. Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
   Starting with \(x _ { 1 } = 0.3\)
2. calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
3. show that \(\alpha\) is 0.341 to 3 decimal places.
   [0pt] [BLANK PAGE]

5. The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { c c } 6 & - 2
1 & 2 \end{array} \right)$$ Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\).
[0pt] [BLANK PAGE]

6. The complex number \(w\) is given by $$w = 10 - 5 \mathrm { i }$$

1. Find \(| w |\).
2. Find arg \(w\), giving your answer in radians to 2 decimal places. The complex numbers \(z\) and \(w\) satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
3. Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where \(\lambda\) is a real constant,
4. find the value of \(\lambda\).
   [0pt] [BLANK PAGE]

7. \section*{Figure 1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = x \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
[0pt] [BLANK PAGE]

8. With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { c } 13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point \(A\) lies on \(l\) and has coordinates \(( 3 , - 2,6 )\).
The point \(P\) has position vector ( \(- \boldsymbol { i } + 2 \boldsymbol { k }\) ) relative to \(O\).
Given that vector \(\overrightarrow { P A }\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle B P A = 45 ^ { \circ }\), find the coordinates of the two possible positions of \(B\).
(5)
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9. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
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10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point \(P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)\) is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

1. Show that the equation of the normal to the curve at \(P\) can be written as \(3 y - x = \frac { 7 \sqrt { 3 } } { 2 }\)
2. Show that the Cartesian equation of the curve may be written as \(a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0\) where \(a\), \(b\) and \(c\) are integers to be found.
   [0pt] [BLANK PAGE]

11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which \(y = 0\) at \(x = \frac { \pi } { 3 }\), giving your answer in the form \(\sec ^ { 2 } x = \mathrm { g } ( y )\).
[0pt] [BLANK PAGE]

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point \(A\) lies on \(l\) and has coordinates \(( 3 , - 2,6 )\).
The point \(P\) has position vector ( \(- \boldsymbol { i } + 2 \boldsymbol { k }\) ) relative to \(O\).
Given that vector \(\overrightarrow { P A }\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle B P A = 45 ^ { \circ }\), find the coordinates of the two possible positions of \(B\).
(5)
[0pt] [BLANK PAGE]
9. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
[0pt] [BLANK PAGE]
10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point \(P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)\) is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

1. Show that the equation of the normal to the curve at \(P\) can be written as \(3 y - x = \frac { 7 \sqrt { 3 } } { 2 }\)
2. Show that the Cartesian equation of the curve may be written as \(a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0\) where \(a\), \(b\) and \(c\) are integers to be found.
   [0pt] [BLANK PAGE]
   11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which \(y = 0\) at \(x = \frac { \pi } { 3 }\), giving your answer in the form \(\sec ^ { 2 } x = \mathrm { g } ( y )\).
   [0pt] [BLANK PAGE]
   12. A linear transformation T of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l c } \lambda & k
   1 & \lambda - k \end{array} \right)\), and \(\lambda\)
   and \(k\) are real constants. and \(k\) are real constants.
3. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\).
   2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T ? For the remainder of this question, take \(k = - 2\).
4. Determine whether there are any lines through the origin that are invariant lines for the transformation T .
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   13. (i) Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
   (ii) Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
   [0pt] [BLANK PAGE]

14. Using an appropriate substitution, or otherwise, show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$ [BLANK PAGE]
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