Questions — SPS (1106 questions)

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SPS SPS FM 2022 November Q4
4. The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\) has roots \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
  2. Show that \(c = - \frac { 4 } { 9 }\) and find the values of \(a\) and \(b\).
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SPS SPS FM 2022 November Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-12_595_1579_194_274} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curves in Figure 1 have equations $$y = 6 \cosh ( x ) \text { and } y = 9 - 2 \sinh ( x )$$
  1. Find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shaded in Figure 1.
  2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln ( b ) + c\), where \(a , b\) and \(c\) are integers.
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SPS SPS FM 2022 November Q6
6. It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } x\).
  1. Show that \(\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 3 }\).
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SPS SPS FM 2022 November Q7
7.
\(f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q\), where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35. Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).
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SPS SPS FM 2022 November Q8
8. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations
\(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\)
\(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-18_346_840_210_1107} The curves meet at the pole \(O\), and at the points \(A\) and \(B\).
  1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\).
  2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { 3 } } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
  3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
    Given that the length of the line \(A B\) is 4.5 cm ,
  4. calculate the area of this badge, giving your answer to three significant figures.
    (Total 16 marks)
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SPS SPS FM 2024 October Q1
  1. (a) (i) Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
    (ii) Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
    (b) In this question you must show detailed reasoning.
Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).
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SPS SPS FM 2024 October Q2
2. (a) Sketch the curve with equation $$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$ [BLANK PAGE]
SPS SPS FM 2024 October Q3
3. (a) Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
(b) In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).
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SPS SPS FM 2024 October Q4
4. The functions f and g are defined for all real values of \(x\) by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 6 x\) and \(\mathrm { g } ( x ) = 3 x + 2\).
  1. Find the range of f.
  2. Give a reason why f has no inverse.
  3. Given that \(\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )\), where \(a\) is a constant, determine the value of \(a\).
  4. Determine the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\). Give your answer in set notation.
    [0pt] [BLANK PAGE] \section*{5. In this question you must show detailed reasoning} Find the equation of the normal to the curve \(y = \frac { x ^ { 2 } - 32 } { \sqrt { x } }\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
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SPS SPS FM 2024 October Q6
6. Given that the equation $$2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3 ,$$ only has one solution, find the value of \(x\).
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SPS SPS FM 2024 October Q7
7. In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).
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SPS SPS FM 2024 October Q8
8. Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geq 1\).
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SPS SPS FM 2024 October Q9
9. a) Factorise \(8 x y - 4 x + 6 y - 3\) into the form \(( a x + b ) ( c y + d )\) where \(a , b , c\) and \(d\) are integers
b) Hence, or otherwise, solve $$8 \sin \left( x ^ { 2 } \right) \cos \left( e ^ { \frac { x } { 3 } } \right) - 4 \sin \left( x ^ { 2 } \right) + 6 \cos \left( e ^ { \frac { x } { 3 } } \right) - 3 = 0$$ where \(0 ^ { \circ } < x < 19 ^ { \circ }\), giving your answers to 1 decimal place.
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SPS SPS SM 2023 October Q1
1 In this question you must show detailed reasoning.
Find the smallest positive integers m and n such that \(\left( \frac { 64 } { 49 } \right) ^ { - \frac { 3 } { 2 } } = \frac { m } { n }\)
SPS SPS SM 2023 October Q2
2 In this question you must show detailed reasoning. Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{808e5492-febe-434f-91b8-9b2888b17fcb-04_704_912_178_694} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that
  • \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression in \(x\)
  • \(C\) cuts the \(x\)-axis at 0 and 6
  • \(l\) cuts the \(y\)-axis at 60 and intersects \(C\) at the point \(( 10,80 )\)
    use inequalities to define the region \(R\) shown shaded in Figure 3.
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SPS SPS SM 2023 October Q4
  1. In this question you must show detailed reasoning.
A curve has equation $$y = 2 x ^ { 2 } + p x + 1$$ A line has equation $$y = 5 x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points.
Give your answer in exact form using set notation.
(6)
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SPS SPS SM 2023 October Q5
5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$
  1. Show that \(u _ { 5 } = 20\).
  2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
  3. State what type of sequence it is.
  4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).
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SPS SPS SM 2023 October Q6
6. In part (ii) of this question you must show detailed reasoning.
  1. Use logarithms to solve the equation \(8 ^ { 2 x + 1 } = 24\), giving your answer to 3 decimal places.
    (2)
  2. Find the values of \(y\) such that $$\log _ { 2 } ( 11 y - 3 ) - \log _ { 2 } 3 - 2 \log _ { 2 } y = 1 , \quad y > \frac { 3 } { 11 }$$ [BLANK PAGE]
SPS SPS SM 2023 October Q7
7. (a) Sketch the curve with equation $$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$ [BLANK PAGE] \section*{8. In this question you must show detailed reasoning.} The curve \(C _ { 1 }\) has equation \(y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }\)
The curve \(C _ { 2 }\) has equation \(y = x ^ { 2 } - 12 x + 14\)
(a) Verify that when \(x = 1\) the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect. The curves also intersect when \(x = k\).
Given that \(k < 0\)
(b) use algebra to find the exact value of \(k\).
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SPS SPS SM 2023 October Q9
9. The first term of a geometric progression is 10 and the common ratio is 0.8 .
  1. Find the fourth term.
  2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
  3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002$$ and use logarithms to find the smallest possible value of \(N\).
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SPS SPS SM 2023 October Q10
10. In this question you must show detailed reasoning.
A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 12 = 0\). Two tangents to this circle pass through the point \(( 0,1 )\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same.
Find the angle between these two tangents.
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SPS SPS FM 2023 October Q1
  1. This question requires detailed reasoning.
Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
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SPS SPS FM 2023 October Q2
2. 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 2023 October Q3
3.
  1. Sketch the curve \(y = - \frac { 1 } { x }\).
  2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
  3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).
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SPS SPS FM 2023 October Q4
4. In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), band \(c\) are integers.
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  1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
  2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.
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