SPS SPS FM Pure (SPS FM Pure) 2024 June

1. The matrix \(\mathbf { M }\) is such that \(\mathbf { M } \left( \begin{array} { r r r } 1 & 0 & k
2 & - 1 & 1 \end{array} \right) = \left( \begin{array} { l l l } 1 & - 2 & 0 \end{array} \right)\).
Find

• the matrix \(\mathbf { M }\),
• the value of the constant \(k\).
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1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
   the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
   and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
   Given that \(A B C D\) is a parallelogram,
   1. find the position vector of point \(D\).
   The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).
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3. (a) Sketch on the Argand diagram below the locus of points satisfying the equation \(| z - 2 | = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-08_1260_1303_260_468}
(b) Given that \(| z - 2 | = 2\) and \(\arg ( z - 2 ) = - \frac { \pi } { 3 }\), express \(z\) in the form \(a + b i\) where \(a , b \in \mathbb { R }\).
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4. Prove by induction that the sum of the first \(n\) cube numbers is \(\frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
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5. (a) The diagram shows the graph of \(y = a \sec ( b x ) + 1\) for \(x \in [ 0 , \pi )\). Find the values of \(a\) and \(b\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-12_818_556_201_897}
(b) The diagram shows the graph of \(y = \arccos ( x + c )\).
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-12_511_766_1667_790}

1. State the value of c .
2. State the coordinates of the point \(P\).
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6. In this question you must show detailed reasoning. Given that $$( 1 + a x ) ^ { n } = 1 + 6 x - 6 x ^ { 2 } + \ldots$$ where \(a\) and \(n\) are constants, find the values of \(a\) and \(n\).
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7. In the quartic equation \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d = 0\), the coefficients \(a , b , c\) and \(d\) are real. Two of the roots of the equation are i and \(2 - \mathrm { i }\). Find the value of \(a , b , c\) and \(d\).
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8. Using the substitution \(x = \mathrm { e } ^ { u }\), find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
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9. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 2.

1. Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
2. Show that the \(x\) coordinate of \(Q\) satisfies $$x = \frac { 8 } { 1 + \ln x }$$
3. Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
4. Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with \(x _ { 1 } = 3.5\) to
   1. find the value of \(x _ { 5 }\) to 4 decimal places,
   2. find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.

10. $$\begin{aligned} & \boldsymbol { v } _ { \mathbf { 1 } } = \left( \begin{array} { c } \sqrt { 17 }
\cos 2 \theta
- 4 \end{array} \right)
& \boldsymbol { v } _ { \mathbf { 2 } } = \left( \begin{array} { c } - \sin 2 \theta
2 \sqrt { 2 }
1 \end{array} \right) \end{aligned}$$ Given that \(\boldsymbol { v } _ { \mathbf { 1 } }\) and \(\boldsymbol { v } _ { \mathbf { 2 } }\) are perpendicular and that \(0 \leq \theta \leq \pi\), find all possible values of \(\theta\). Give your answers to 3 significant figures.
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11. \begin{figure}[h] \end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl.
When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)
Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\)
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12. A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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13. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$

1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).
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14. A theme park ride lasts for 70 seconds. The height above ground, \(H\) metres, of a passenger on the theme park ride is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = \frac { t \sin \left( \frac { \pi t } { 5 } \right) } { 10 H } \quad 0 \leqslant t \leqslant 70$$ where \(t\) seconds is the time from the start of the ride.
Given that the passenger is 5 m above ground at the start of the ride find the height above ground of the passenger 52 seconds after the start of the ride.
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15. Two angles, \(x\) and \(y\), are acute. $$\begin{aligned} \sin x \cos y & = \frac { 1 + \sqrt { 3 } } { 4 }
\cos x \sin y & = \frac { - 1 + \sqrt { 3 } } { 4 } \end{aligned}$$

1. Find the exact value of \(\sin ( x + y )\).
2. Find all possible pairs of values of \(x\) and \(y\), giving your answers in terms of \(\pi\). Fully justify your answer.
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16. $$\begin{gathered} M _ { 1 } = \left( \begin{array} { c c } 2 k - 9 & 5 - k
- k & k - 2 \end{array} \right)
M _ { 2 } = \left( \begin{array} { c c } 5 & 1
2 k - 3 & k - 3 \end{array} \right)
k \in \mathbb { R } \end{gathered}$$ Matrices \(M _ { 1 }\) and \(M _ { 2 }\) represent transformations \(T _ { 1 }\) and \(T _ { 2 }\) respectively.
\(\Delta\) is a triangle in the \(x y\)-plane with vertices at \(( 0,0 ) , ( 4,0 )\) and \(( 3,2 )\).
The image of \(\Delta\) under \(T _ { 1 }\) is \(\Delta _ { 1 }\) and the image of \(\Delta\) under \(T _ { 2 }\) is \(\Delta _ { 2 }\).
The area of \(\Delta _ { 2 }\) is greater than the area of \(\Delta _ { 1 }\).
Find the range of possible values of \(k\).
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