Questions — SPS SPS SM (145 questions)

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SPS SPS SM 2022 January Q8
8. (a) Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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SPS SPS SM 2022 January Q9
9. $$\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\) in the space provided.
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
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SPS SPS SM 2022 January Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-22_659_970_141_614} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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SPS SPS SM 2022 February Q1
1.
  1. Evaluate \(27 ^ { - \frac { 2 } { 3 } }\).
  2. Express \(5 \sqrt { 5 }\) in the form \(5 ^ { n }\).
  3. Express \(\frac { 1 - \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
SPS SPS SM 2022 February Q2
2.
  1. Solve the equation \(x ^ { 4 } - 10 x ^ { 2 } + 25 = 0\).
  2. Given that \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Hence find the number of stationary points on the curve \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\).
SPS SPS SM 2022 February Q3
3. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
  2. \(\sin 2 x = - \cos 2 x\).
SPS SPS SM 2022 February Q4
4.
  1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
  3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).
SPS SPS SM 2022 February Q5
5. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point \(( 4,5 )\). Find the equation of the curve. \section*{6.} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_408_435_27_1560}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
  2. Use integration to find the area of the shaded region bounded by the line and the curve.
SPS SPS SM 2022 February Q7
7. The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the \(\operatorname { arc } D C\).
  1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
  2. Find the area of the shaded segment.
  3. Find the perimeter of the shaded segment.
    \includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_641_1360_1219_310}
SPS SPS SM 2022 February Q8
8. The diagram shows the graph of \(y = \mathrm { f } ( x )\), where
  1. Evaluate \(\mathrm { ff } ( - 3 )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\). $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
  3. Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). Indicate the coordinates of the points where the graph meets the axes.
    \includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_476_490_2014_1352}
SPS SPS SM 2021 November Q1
  1. Find \(\frac { d y } { d x }\) for the following functions, simplifying your answers as far as possible.
    i) \(y = \cos x - 2 \sin 2 x\)
    ii) \(y = \frac { 1 } { 2 } x ^ { 4 } + 2 x ^ { 4 } \ln x\)
    iii) \(y = \frac { 2 e ^ { 3 x } - 1 } { 3 e ^ { 3 x } - 1 }\)
a. Express \(\frac { 5 x + 7 } { ( x + 3 ) ( x + 1 ) ^ { 2 } }\) in partial fractions. In this question you must show all of your algebraic steps clearly. The function \(f ( x ) = \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) }\) can be written in the form; $$f ( x ) = \frac { - 2 } { x } + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { 1 - 2 x }$$ b. Hence find the exact value of \(\int _ { 2 } ^ { 3 } \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) } d x\)
SPS SPS SM 2021 November Q3
3. In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = ( 1 - 3 x ) ( 3 - x ) ^ { 3 }$$
SPS SPS SM 2021 November Q4
  1. Find the equation of the normal to the curve \(y = 4 \ln ( 2 x - 3 )\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(a x + b y + k = 0\) where \(a > 0\).
i) Write \(\log _ { 16 } y - \log _ { 16 } x\) as a single logarithm.
ii) Solve the simultaneous equations, giving your answers in an exact form. $$\begin{gathered} \log _ { 3 } y = \log _ { 3 } ( 9 - 6 x ) + 1
\log _ { 16 } y - \log _ { 16 } x = \frac { 1 } { 4 } \end{gathered}$$
SPS SPS SM 2021 November Q6
6. a. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$( \cos x + \sin x ) ( \operatorname { cosec } x - \sec x ) \equiv 2 \cot 2 x$$ b. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } \sin \left( 2 x - \frac { \pi } { 6 } \right)$$
SPS SPS SM 2021 November Q7
  1. The diagram below represents the graph of the function \(y = ( 2 x - 5 ) ^ { 4 } - 1\)
    \includegraphics[max width=\textwidth, alt={}, center]{1650b28f-be4e-4600-89ca-67c2d3026c5b-10_784_657_233_694}
    a. Find the intersections of this graph with the \(x\) axis.
    b. Hence find the exact value of the area bounded by the curve and the \(x\) axis.
a. Express \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
b. Hence
i. Solve the equation \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x = 6\) for \(0 \leq x \leq 2 \pi\) Giving your answers in terms of \(\pi\).
c. It can be shown that \(y = 9 \sin 2 x + 4 \cos 2 x\) can be written as \(y = \sqrt { 97 } \sin \left( 2 x + 24.0 ^ { \circ } \right)\)
i. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2 x + 4 \cos 2 x\) to the curve \(y = \sin x\)
ii. Find the exact maximum and minimum values of the function; $$f ( x ) = \frac { 1 } { 11 - 9 \sin 2 x - 4 \cos 2 x }$$
SPS SPS SM 2021 November Q9
9. a.
i. Show that \(\cos ^ { 2 } x \equiv \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos 2 x\)
ii. Hence find \(\int 2 \cos ^ { 2 } 4 x d x\)
b. Find \(\int \sin ^ { 3 } x d x\)
SPS SPS SM 2021 November Q10
10.
a. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\)
b. A curve is defined parametrically by the equations; $$x = \cos \theta \quad y = \left( \frac { \sin \theta } { 2 } \right) \left( \sin \frac { \theta } { 2 } \right)$$ Show that the cartesian equation of the curve can be written as \(y ^ { 2 } = \frac { 1 } { 8 } ( 1 - x ) ^ { 2 } ( 1 + x )\)
SPS SPS SM 2022 October Q1
1 Simplify \(\left( \frac { x ^ { 12 } } { 16 } \right) ^ { - \frac { 3 } { 4 } }\)
SPS SPS SM 2022 October Q2
2 A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = - 3 x ^ { 2 } + 12 x + 8$$
  1. Write \(f ( x )\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The curve \(C\) has a maximum turning point at \(M\).
  2. Find the coordinates of \(M\). Solutions relying on calculator technology are not acceptable. Simplify $$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$ giving your answer in the form \(b \sqrt { } 2 + c\), where \(b\) and \(c\) are integers.
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SPS SPS SM 2022 October Q4
4. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
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SPS SPS SM 2022 October Q5
5. (a) Given that $$y = \log _ { 3 } x$$ find expressions in terms of \(y\) for
  1. \(\log _ { 3 } \left( \frac { x } { 9 } \right)\)
  2. \(\log _ { 3 } \sqrt { x }\) Write each answer in its simplest form.
    (b) Hence or otherwise solve $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) - \log _ { 3 } \sqrt { x } = 2$$ [BLANK PAGE]
SPS SPS SM 2022 October Q6
6. An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54
  1. show that $$a + 4 d = 6$$ Given also that the 8th term is half the 7th term,
  2. find the values of \(a\) and \(d\).
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SPS SPS SM 2022 October Q7
7. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Using algebra, find all solutions of the equation $$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
  2. Hence find all real solutions of $$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$ [BLANK PAGE]
SPS SPS SM 2022 October Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba499f70-f2ee-4eff-b15c-33c3f09297f0-14_622_805_141_660} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = p m ^ { q }$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg .
Figure 2 illustrates the linear relationship between \(\log _ { 10 } h\) and \(\log _ { 10 } m\)
The line meets the vertical \(\log _ { 10 } h\) axis at 2.25 and has a gradient of - 0.235
  1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
  2. Comment on the suitability of the model for this mammal.
  3. With reference to the model, interpret the value of the constant \(p\).
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SPS SPS SM 2022 October Q9
9. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$ where \(k\) is a constant.
Given that
  • the sequence is a periodic sequence of order 3
  • \(a _ { 1 } = 2\)
    1. show that
$$k ^ { 2 } + k - 2 = 0$$
  • For this sequence explain why \(k \neq 1\)
  • Find the value of $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$ [BLANK PAGE]
    •