Questions — SPS SPS SM (145 questions)

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SPS SPS SM 2020 October Q7
7. i. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided below.
\includegraphics[max width=\textwidth, alt={}, center]{e1b41613-a703-4eb3-9760-7b47b1dad099-06_849_921_1683_644}
ii. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
iii. Hence, solve the inequality \(\frac { 3 } { x ^ { 2 } } \leq x ^ { 2 } - 2\), giving your answer in interval notation.
SPS SPS SM 2020 October Q9
9. In this question you must show detailed reasoning. Solve the following simultaneous equations: $$\begin{gathered} \left( \log _ { 3 } x \right) ^ { 2 } + \log _ { 3 } \left( y ^ { 2 } \right) = 5
\log _ { 3 } \left( \sqrt { 3 } x y ^ { - 1 } \right) = 2 \end{gathered}$$
SPS SPS SM 2020 October Q10
  1. In this question you must show detailed reasoning.
A sequence \(t _ { 1 } , t _ { 2 } , t _ { 3 } \ldots\) is defined by \(t _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that $$\sum _ { n = 1 } ^ { \infty } t _ { n } - \sum _ { n = 1 } ^ { N } t _ { n } < 10 ^ { - 4 }$$
SPS SPS SM 2022 October Q1
  1. (a) Sketch the curve \(y = 3 ^ { - x }\)
    (b) Solve the inequality \(3 ^ { - x } < 27\)
  2. (a) Complete the square for \(1 - 4 x - x ^ { 2 }\)
    (b) Sketch the curve \(y = 1 - 4 x - x ^ { 2 }\), including the coordinates of any maximum or minimum points and the y intercept only.
    [0pt] [BLANK PAGE]
  3. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010 .
The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
(a) the number of houses built in 1986, the first year of the building programme,
(b) the total number of houses built in the 25 years of the programme.
[0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q4
4. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$ [BLANK PAGE]
SPS SPS SM 2022 October Q5
5. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
  1. Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\).
  2. Find, to 2 decimal places, the difference between the 5th and 6th term.
  3. Calculate the sum of the first 7 terms. The sum of the first \(n\) terms of the series is greater than 300 .
  4. Calculate the smallest possible value of \(n\).
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q6
6. \begin{displayquote} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Solve the equation \end{displayquote} $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
  2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$ [BLANK PAGE]
SPS SPS SM 2022 October Q7
7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 61 }\)
  3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q8
8. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
  2. Find the range of possible values for \(k\).
    [0pt] [BLANK PAGE] \section*{9. In this question you must show detailed reasoning.} The centre of a circle C is the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\).
    Show that L is a tangent to C .
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
SPS SPS SM 2021 November Q1
2 marks
  1. Do not use a calculator for this question
Find the value of \(x\) for which \(\sqrt { 3 } \times 3 ^ { x } = \frac { 1 } { 9 }\)
[0pt] [2 marks]
SPS SPS SM 2021 November Q2
4 marks
2. You must show detailed working in this question Determine whether the line with equation \(2 x + 3 y + 4 = 0\) is parallel to the line through the points with coordinates \(( 9,4 )\) and \(( 3,8 )\).
[0pt] [4 marks]
SPS SPS SM 2021 November Q3
4 marks
3. An arithmetic sequence has first term \(a\) and common difference \(d\).
The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. Show that \(4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }\)
[0pt] [4 marks]
SPS SPS SM 2021 November Q4
2 marks
4. Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
[0pt] [2 marks]
SPS SPS SM 2021 November Q5
5 marks
5.
\(\mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)
Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\)
[0pt] [2 marks]
L
L
L
L
L
L Factorise \(\mathrm { p } ( x )\) completely.
[0pt] [3 marks]
L
L
L
L
L
SPS SPS SM 2021 November Q6
3 marks
6. You are not allowed to use a calculator for this question. Show detailed reasoning. Show that \(\frac { 5 \sqrt { 2 } + 2 } { 3 \sqrt { 2 } + 4 }\) can be expressed in the form \(m + n \sqrt { 2 }\), where \(m\) and \(n\) are integers.
[0pt] [3 marks]
SPS SPS SM 2021 November Q7
4 marks
7. The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
Find the possible values of the constant \(k\)
Fully justify your answer.
[0pt] [4 marks]
SPS SPS SM 2021 November Q8
10 marks
8. The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\).
    1. Write down the radius of the circle.
      [0pt] [1 mark]
  2. (ii) Write down the coordinates of \(C\).
    [0pt] [1 mark]
  3. The point \(P ( 5 , - 1 )\) lies on the circle. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = m x + c\)
    [0pt] [4 marks]
  4. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
    [0pt] [4 marks]
SPS SPS SM 2021 November Q9
4 marks
9. David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.
  1. Using David's model:
    1. state the population of rabbits on the island on 1 January 2016;
  3. (ii) predict the population of rabbits on 1 January 2021.
  4. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
  5. Give one reason why David's model may not be appropriate.
    [0pt] [1 mark]
  6. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where \(t\) is the time in years after 1 January 2016.
    Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
    [0pt] [3 marks]
SPS SPS SM 2022 January Q1
1.
  1. Express \(\frac { 21 } { \sqrt { 7 } }\) in the form \(k \sqrt { 7 }\).
  2. Express \(8 ^ { - \frac { 1 } { 3 } }\) as an exact fraction in its simplest form.
SPS SPS SM 2022 January Q2
2. A curve has equation \(y = 16 x + \frac { 1 } { x ^ { 2 } }\). Find
(A) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(B) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
[0pt] [BLANK PAGE]
SPS SPS SM 2022 January Q3
3. Triangle ABC is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-06_517_652_237_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the perimeter of triangle ABC .
(3)
[0pt] [BLANK PAGE]
SPS SPS SM 2022 January Q4
4. Find $$\int \frac { 2 x ^ { 2 } + 6 x - 5 } { 3 \sqrt { x ^ { 3 } } } d x$$ simplifying your answer.
[0pt] [BLANK PAGE]
SPS SPS SM 2022 January Q5
5. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)
[0pt] [BLANK PAGE]
SPS SPS SM 2022 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-12_570_922_118_374} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
      [0pt] [BLANK PAGE]
SPS SPS SM 2022 January Q7
7. Prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is an irrational number.
[0pt] [BLANK PAGE]