Questions — SPS SPS FM Pure (237 questions)

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SPS SPS FM Pure 2023 June Q4
4. You are given that \(\boldsymbol { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 }
\sqrt { 3 } & 1 \end{array} \right)\).
  1. Show that \(\boldsymbol { M }\) is non-singular. The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\boldsymbol { M }\). Given that the area of hexagon \(R\) is 5 square units,
  2. find the area of hexagon \(S\). The matrix \(\boldsymbol { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
  3. Find the value of \(k\).
  4. Find the value of \(\theta\).
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SPS SPS FM Pure 2023 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-12_611_334_171_902} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }\), show that \(\angle B A C = 105.9 ^ { \circ }\) to one decimal place.
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SPS SPS FM Pure 2023 June Q6
6. A spherical balloon is inflated so that its volume increases at a rate of \(10 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm .
[0pt] [For a sphere of radius \(r\), surface area \(= 4 \pi r ^ { 2 }\) and volume \(= \frac { 4 } { 3 } \pi r ^ { 3 }\) ].
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SPS SPS FM Pure 2023 June Q7
7. Prove that for all \(n \in \mathbb { N }\) $$\left( \begin{array} { c c } 3 & 4 i
i & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 n + 1 & 4 n i
n i & 1 - 2 n \end{array} \right)$$ [BLANK PAGE]
SPS SPS FM Pure 2023 June Q8
8. (i) Shade on an Argand diagram the set of points $$\{ z \in \mathbb { C } : | z - 4 \mathrm { i } | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 \mathrm { i } ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number \(w\) satisfies \(| w - 4 i | = 3\).
(ii) Find the maximum value of \(\arg w\) in the interval \(( - \pi , \pi ]\). Give your answer in radians correct to 2 decimal places.
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SPS SPS FM Pure 2023 June Q9
9. (i) Use the binomial expansion to show that \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } } \approx 1 + x + \frac { 3 } { 2 } x ^ { 2 }\) for sufficiently small values of \(x\).
(ii) For what values of \(x\) is the expansion valid?
(iii) Find the expansion of \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
(iv) Use \(x = \frac { 1 } { 20 }\) in your answer to part (iii) to find an approximate value for \(\sqrt { 11 }\).
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SPS SPS FM Pure 2023 June Q10
10. The complex number \(z\) is given by \(z = k + 3 i\), where \(k\) is a negative real number. Given that \(z + \frac { 12 } { z }\) is real, find \(k\) and express \(z\) in exact modulus-argument form.
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SPS SPS FM Pure 2023 June Q11
11. In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3 t \text { and } y = 3 t - 4 \cos t - \sin 2 t , \text { for } 0 \leqslant t \leqslant \pi .$$ Show that the gradient of the curve is always negative.
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SPS SPS FM Pure 2023 June Q12
12.
\includegraphics[max width=\textwidth, alt={}, center]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-26_689_1203_182_447} The figure shows part of the graph of \(y = ( x - 3 ) \sqrt { \ln x }\). The portion of the graph below the \(x\)-axis is rotated by \(2 \pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\).
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SPS SPS FM Pure 2023 June Q13
13. (i) Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + y ) ( 1 - x ) ,$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\), where f is a function to be determined.
(ii) By considering the sign of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) near \(( 1,1 )\), or otherwise, show that this point is a maximum point on the curve \(y = \mathrm { f } ( x )\).
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SPS SPS FM Pure 2023 June Q14
14. A curve \(C\) has equation $$x ^ { 3 } + y ^ { 3 } = 3 x y + 48$$ Prove that \(C\) has two stationary points and find their coordinates.
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SPS SPS FM Pure 2023 June Q15
15. In this question you must use detailed reasoning.
  1. Show that \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x = \ln ( \sqrt { a } + b )\), where \(a\) and \(b\) are integers to be determined.
  2. Show that \(\frac { \pi } { \frac { \pi } { 4 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x\) is undefined, explaining your reasoning clearly.
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SPS SPS FM Pure 2023 October Q1
  1. A curve is described by the equation
$$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). A point \(Q\) lies on the curve.
    The tangent to the curve at \(Q\) is parallel to the \(y\)-axis.
    Given that the \(x\) coordinate of \(Q\) is negative,
  2. use your answer to part (a) to find the coordinates of \(Q\).
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SPS SPS FM Pure 2023 October Q2
2. Given that \(x \geqslant 2\), find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ [BLANK PAGE]
SPS SPS FM Pure 2023 October Q3
3. Liquid is pouring into a large vertical circular cylinder at a constant rate of \(1600 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is \(4000 \mathrm {~cm} ^ { 2 }\).
  1. Show that at time \(t\) seconds, the height \(h \mathrm {~cm}\) of liquid in the cylinder satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - k \sqrt { } h , \text { where } k \text { is a positive constant. }$$ When \(h = 25\), water is leaking out of the hole at \(400 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
  2. Show that \(k = 0.02\)
  3. Separate the variables of the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.4 - 0.02 \sqrt { } h$$ to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by $$\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { h } } \mathrm {~d} h$$ Using the substitution \(h = ( 20 - x ) ^ { 2 }\), or otherwise,
  4. find the exact value of \(\int _ { 0 } ^ { 100 } \frac { 50 } { 20 - \sqrt { h } } \mathrm {~d} h\).
  5. Hence find the time taken to fill the cylinder from empty to a height of 100 cm , giving your answer in minutes and seconds to the nearest second.
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SPS SPS FM Pure 2023 October Q4
4. The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
  2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).
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SPS SPS FM Pure 2023 October Q5
5. (i) Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$ (ii) Show that \(( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1\).
(iii) Use the results in parts (i) and (ii) and the standard result for \(\sum _ { r = 1 } ^ { n } r\) to show that $$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$ [BLANK PAGE]
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SPS SPS FM Pure 2023 September Q1
1. Show that \(\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = 1\).
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SPS SPS FM Pure 2023 September Q2
2. (i) Sketch the graph of \(y = | 2 x - 7 a |\), where \(a\) is a positive constant. State the coordinates of the points where the graph meets each axis.
(ii) Solve the inequality \(| 2 x - 7 a | < 4 a\).
(iii) Deduce the largest integer \(N\) satisfying the inequality \(| 2 \ln N - 10.5 | < 6\).
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SPS SPS FM Pure 2023 September Q3
3. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. \(\sin \frac { 1 } { 2 } x = 0.8\)
  2. \(\sin x = 3 \cos x\)
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SPS SPS FM Pure 2023 September Q4
4. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geq 9\).
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SPS SPS FM Pure 2023 September Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-12_423_743_123_731} The diagram shows the curve \(y = \mathrm { f } ( x )\), where f is the function defined for all real values of \(x\) by $$f ( x ) = 3 + 4 e ^ { - x }$$
  1. State the range of f.
  2. Find an expression for \(f ^ { - 1 } ( x )\), and state the domain and range of \(f ^ { - 1 }\).
  3. The straight line \(y = x\) meets the curve \(y = \mathrm { f } ( x )\) at the point \(P\). By using an iterative process based on the equation \(x = \mathrm { f } ( x )\), with a starting value of 3 , find the coordinates of the point \(P\). Show all your working and give each coordinate correct to 3 decimal places.
  4. How is the point \(P\) related to the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) ?
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SPS SPS FM Pure 2023 September Q6
6. The matrix \(\left( \begin{array} { l l } 1 & 3
0 & 1 \end{array} \right)\) represents a transformation \(P\).
  1. Describe fully the transformation \(P\). The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1
    - 1 & 0 \end{array} \right)\).
  2. Given that \(M\) represents transformation \(Q\) followed by transformation \(P\), find the matrix that represents the transformation Q and describe fully the transformation Q .
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SPS SPS FM Pure 2023 September Q7
7. The complex number \(2 - \mathrm { i }\) is denoted by \(z\).
  1. Find \(| z |\) and arg \(z\).
  2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).
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SPS SPS FM Pure 2023 September Q8
8. A curve has equation \(y = 2 + \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The region \(R\) is bounded by the curve and by the straight lines \(x = 0 , x = 4\) and \(y = 0\). Find the exact volume of the solid obtained when \(R\) is rotated completely about the x-axis.
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