Questions — SPS (1106 questions)

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SPS SPS FM 2024 October Q2
  1. The graph of \(y = f ( x )\) (where \(- 2 \leq x \leq 6\) ) has the following features:
  • A local maximum at \(x = 0\).
  • A local minimum at \(x = 2\).
  • No other turning points.
  • Three stationary points.
Sketch a possible graph of \(y = f ^ { \prime } ( x )\) on the axes provided.
You can ignore the scale for the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-04_1269_1354_826_447}
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SPS SPS FM 2024 October Q3
3. The function \(g ( x )\) is defined as follows: $$g ( x ) = e ^ { \sin \left( x ^ { \circ } \right) } , - 90 \leq x \leq 90$$
  1. Find \(g ^ { - 1 } ( x )\), stating its domain.
  2. Sketch \(y = g ^ { - 1 } ( x )\) on the axes provided below, being sure to label all key points.
    \includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-07_1524_1591_459_342}
SPS SPS FM 2024 October Q4
4. The polynomial \(P ( x )\) is defined as follows:
\(P ( x ) \equiv x ^ { 8 } + 8 x ^ { 7 } + 28 x ^ { 6 } + 56 x ^ { 5 } + 70 x ^ { 4 } + 56 x ^ { 3 } + 28 x ^ { 2 } + 8 x , x \in \mathbb { R }\)
By first factorising \(P ( x )\) find all of its real roots.
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SPS SPS FM 2024 October Q5
6 marks
5. While working on a logarithms problem on one of the whiteboards in the Maths corridor, Rehman confidently asserts the following:
"For all positive real numbers \(x\) and \(y\), we know that \(\log _ { 2 } x - \log _ { 2 } y \equiv \frac { \log _ { 2 } x } { \log _ { 2 } y }\)."
Soufiane, who happens to be passing, knows that this is wrong. He attempts to prove this by counterexample, picking two values of \(x\) and \(y\) off the top of his head and plugging both sides of the false identity into his calculator. To his dismay, both sides give exactly the same answer, and Rehman smugly walks off uncorrected. What is the range of possible values of \(x\) that Soufiane picked?
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SPS SPS FM 2024 October Q6
6. Neil enjoys playing around with sequences in his spare time, and one day he decides to create a new one. He does this by taking a normal arithmetic sequence but repeatedly halving the common difference between the terms. For instance, if the difference between the first two terms is 16 , the difference between the third term and the second term will be 8 and the difference between the fourth term and the third term will be 4 (continuing in this fashion for as long as he likes).
  1. If \(u _ { 1 } = a\) and \(u _ { 2 } = a + d\), find a closed form expression for \(u _ { n }\), where \(n \in \mathbb { N }\). "Closed form" means that you can't have a "..." or sigma notation in your final answer.
  2. With \(u _ { n }\) defined as above, find a closed form expression for \(S _ { n } = \sum _ { k = 1 } ^ { n } u _ { k }\), where \(n \in \mathbb { N }\).
  3. With \(u _ { n }\) and \(S _ { n }\) defined as above, find the values of \(a\) and \(d\) if \(u _ { 7 } = 23\) and \(S _ { 7 } = 41\).
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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 FM 2024 October Q1
  1. The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(f ( x )\).
Use differentiation from first principles to determine the value of \(f ^ { \prime } ( 2 )\).
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SPS SPS FM 2024 October Q2
2. 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 FM 2024 October Q3
3. (i) Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
(ii) In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
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SPS SPS FM 2024 October Q4
4. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
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SPS SPS FM 2024 October Q5
5. A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
    1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    2. Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
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SPS SPS FM 2024 October Q6
6. 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 FM 2024 October Q7
7. In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).
\(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
Calculate the possible values of the angle \(A D B\).
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SPS SPS FM 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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SPS SPS FM 2024 October Q9
9. Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
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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. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\).