Questions — SPS SPS SM (145 questions)

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SPS SPS SM 2022 October Q10
10. A circle \(C\) with radius \(r\)
  • lies only in the 1st quadrant
  • touches the \(x\)-axis and touches the \(y\)-axis
The line \(l\) has equation \(2 x + y = 12\)
  1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
  2. find the two possible values of \(r\), giving your answers as fully simplified surds.
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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 SM 2024 October Q1
1. A is inversely proportional to B . B is inversely proportional to the square of C . When A is \(2 , \mathrm { C }\) is 8 . Find C when A is 12 .
SPS SPS SM 2024 October Q2
2.
  1. Write \(3 x ^ { 2 } + 24 x + 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants to be determined. The finite region \(R\) is enclosed by the curve \(y = 3 x ^ { 2 } + 24 x + 5\) and the \(x\)-axis.
  2. State the inequalities that define \(R\), including its boundaries.
SPS SPS SM 2024 October Q3
3. The 11th term of an arithmetic progression is 1 . The sum of the first 10 terms is 120 . Find the 4th term.
SPS SPS SM 2024 October Q4
4. The quadratic equation \(k x ^ { 2 } + 2 k x + 2 k = 3 x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4 k ^ { 2 } + 16 k - 9 > 0 .$$
  2. Hence find the set of possible values of \(k\). Give your answer in set notation.
SPS SPS SM 2024 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4db8f2e8-e4f8-4463-bf1e-c24413c34d6f-08_680_808_173_688} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The line \(l _ { 1 }\) has equation \(y = \frac { 3 } { 5 } x + 6\)
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(B ( 8,0 )\), as shown in the sketch in Figure 4.
  1. Show that an equation for line \(l _ { 2 }\) is $$5 x + 3 y = 40$$ Given that
    • lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\)
    • line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\)
    • find the exact area of triangle \(A B C\), giving your answer as a fully simplified fraction in the form \(\frac { p } { q }\)
SPS SPS SM 2024 October Q6
6. In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$
  1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
  2. Find the time when the mass is 30 grams.
  3. Sketch the graph of \(m\) against \(t\).
SPS SPS SM 2024 October Q7
7. Express \(\frac { a ^ { \frac { 7 } { 2 } } - a ^ { \frac { 5 } { 2 } } } { a ^ { \frac { 3 } { 2 } } - a }\) in the form \(a ^ { m } + \sqrt { a ^ { n } }\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1 .
SPS SPS SM 2024 October Q8
8. A circle, \(C\), has equation \(x ^ { 2 } - 6 x + y ^ { 2 } = 16\).
A second circle, \(D\), has the following properties:
  • The line through the centres of circle \(C\) and circle \(D\) has gradient 1 .
  • Circle \(D\) touches circle \(C\) at exactly one point.
  • The centre of circle \(D\) lies in the first quadrant.
  • Circle \(D\) has the same radius as circle \(C\).
Find the coordinates of the centre of circle \(D\). \section*{9. In this question you must show detailed reasoning.} The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
  1. (a) Show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
    (b) Hence find the exact roots of the equation \(\mathrm { f } ( x ) = 0\).
  2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( x ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
SPS SPS SM 2024 October Q10
10. The first three terms of a geometric sequence are $$u _ { 1 } = 3 k + 4 \quad u _ { 2 } = 12 - 3 k \quad u _ { 3 } = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of k , giving a reason for your answer.
  2. Find the value of \(\sum _ { r = 2 } ^ { \infty } u _ { r }\)
SPS SPS SM 2024 October Q1
  1. The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(\mathrm { vms } ^ { - 1 }\).
When \(v = 3.6 , P = 50\). Determine the wind speed that will give a power output of 225 watts.
SPS SPS SM 2024 October Q2
2. Solve the inequalities
  1. \(3 - 8 x > 4\),
  2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\).
SPS SPS SM 2024 October Q3
3. The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where p is a constant. Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).
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SPS SPS SM 2024 October Q4
4. The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
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SPS SPS SM 2024 October Q6
6. The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).
\(t\)051025
\(m\)200160
  1. Find the values missing from the table.
  2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.
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SPS SPS SM 2024 October Q7
7. A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0
& \log _ { 3 } x - 2 = 0
& \log _ { 3 } x = 2
& x = 8 \end{aligned}$$
  1. Identify the two mistakes that the student has made.
  2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
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SPS SPS SM 2024 October Q8
8. In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac { 5 } { 3 x - 4 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 2 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 3 } + \ldots$$ is convergent.
  1. Find the set of possible values of \(x\), giving your answer in set notation.
  2. Given that the sum to infinity of the series is \(\frac { 2 } { 3 }\), find the value of \(x\).
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