SPS SPS FM Pure (SPS FM Pure) 2024 February

1. The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
2. Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
3. It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation

$$z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80 = 0$$ where \(p\) and \(r\) are real numbers.

1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
2. Find the value of \(p\) and the value of \(r\).
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4. Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series $$( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots$$ where \(n\) is a positive integer. Give your answer in fully factorised form.
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5. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers \(n \geq 1\), $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$ [6 marks]
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6.

1. Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x\) is an improper integral.
2. Prove that $$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.
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7. In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\).
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).
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8. A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2
\lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant.

1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
   [0pt] [BLANK PAGE] \section*{9. In this question you must show detailed reasoning.} The complex number \(- 4 + \mathrm { i } \sqrt { 48 }\) is denoted by \(z\).
3. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
4. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
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10. The diagram shows the polar curve \(C _ { 1 }\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C _ { 2 }\) with equation \(r = 1 + \cos 2 \theta\)
\includegraphics[max width=\textwidth, alt={}, center]{44ff7962-1982-46f0-aa02-be7127485bde-18_405_959_331_995}

1. On the diagram above, complete the sketch of \(C _ { 2 }\)
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2. Show that the area of the region shaded in the diagram is equal to $$k \pi + m \alpha - \sin 2 \alpha + q \sin 4 \alpha$$ where \(\alpha = \sin ^ { - 1 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \right)\), and \(k , m\) and \(q\) are rational numbers.
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11. Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3
( k - 1 ) x + ( 3 - k ) y + 2 z & = 1
7 x - 3 y + 4 z & = 2 \end{aligned}$$

1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
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12. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = \frac { x ^ { 3 } } { \sqrt { 4 - 2 x - x ^ { 2 } } }$$ where \(0 < x < \sqrt { 5 } - 1\)
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13. In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh \frac { 1 } { 2 } x ^ { 2 }\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm.
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14. Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
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15. $$y = \cosh ^ { n } x \quad n \geqslant 5$$

1. 1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
   2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer.
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