Questions — SPS SPS FM (245 questions)

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SPS SPS FM 2020 September Q12
12. Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5739a9ae-d1ed-4c9d-a912-587ece5e9627-21_547_517_447_733} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find, in modulus-argument form, the complex number represented by the point P .
  2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    • \(3 + 5 \mathrm { i }\)
    • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\)
      lie within this region.
SPS SPS FM 2022 October Q1
1.
a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 5 }$$ giving each term in its simplest form.
b) Hence write down the first 3 terms, in ascending powers of \(y\), of the binomial expansion of $$\left( 2 + 3 y ^ { \frac { 3 } { 2 } } \right) ^ { 5 }$$
SPS SPS FM 2022 October Q2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ 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 FM 2022 October Q3
3.
  1. Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
  2. Sketch the graph of \(y = \frac { 3 } { x }\) in the space provided and write down the equations of any asymptotes.
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q4
4. Prove, from first principles, that if \(f ( x ) = 2 x ^ { 2 } - 5 x + 2\) then \(f ^ { \prime } ( x ) = 4 x - 5\).
(3)
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q5
5. (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 FM 2022 October Q6
6. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm}\), \(C D = 8 \mathrm {~cm} , A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).
  1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
  2. Find the size of \(\angle B C D\) in degrees.
  3. Find the area of quadrilateral \(A B C D\).
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q7
2 marks
7. (i) Solve the inequality \(| 2 x + 1 | \leqslant | x - 3 |\).
(ii) Given that \(x\) satisfies the inequality \(| 2 x + 1 | \leqslant | x - 3 |\), find the greatest possible value of \(| x + 2 |\).
[0pt] [2]
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SPS SPS FM 2022 October Q8
8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),
  1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
  2. find the value of \(a\),
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q9
9. Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
[0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q10
10. 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 .
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SPS SPS FM 2022 October Q1
  1. a) Solve the inequality:
$$\frac { x - 9 } { 2012 } + \frac { x - 8 } { 2013 } + \frac { x - 7 } { 2014 } + \frac { x - 6 } { 2015 } + \frac { x - 5 } { 2016 } \leq \frac { x - 2012 } { 9 } + \frac { x - 2013 } { 8 } + \frac { x - 2014 } { 7 } + \frac { x - 2015 } { 6 } + \frac { x - 2016 } { 5 }$$ b) Find all ( \(x , y , z\) ) such that: $$\frac { 1 } { x } + \frac { 1 } { y + z } = \frac { 1 } { 3 } , \quad \frac { 1 } { y } + \frac { 1 } { z + x } = \frac { 1 } { 5 } , \quad \frac { 1 } { z } + \frac { 1 } { x + y } = \frac { 1 } { 7 }$$ [Question 1 - Continued]
[0pt] [Question 1 - Continued]
SPS SPS FM 2022 October Q2
2. A function is defined by: $$f ( x ) = \sqrt { \frac { 1 - x } { 1 + x } } , x \in \mathbb { R } , | x | < 1$$ a) P and Q are points on the curve with \(x\)-coordinates \(x\) and \(x + h\) respectively. Find the gradient of the line segment PQ . Simplify your answer to a single fraction.
b) Use differentiation from first principles to show that: $$f ^ { \prime } ( x ) = - \frac { 1 } { ( 1 + x ) \sqrt { 1 - x ^ { 2 } } }$$ c) Sketch the curve on the axes provided over the page, showing clearly the behaviour of the curve near \(x = 0\) and \(x = \pm 1\).
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
\includegraphics[max width=\textwidth, alt={}, center]{5023b2d9-ed3d-4a4a-b0c6-f529550b2e3e-09_1731_1566_913_260}
SPS SPS FM 2022 October Q3
3. If for some \(x , y \in \mathbb { R }\) we have \(| x + y | + | x - y | = 2\), find the maximal value of \(x ^ { 2 } - 6 x + y ^ { 2 }\).
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]
SPS SPS FM 2022 October Q4
4. A sequence is defined by \(u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }\) for \(n \geq 1\).
a) In the case where \(r = \frac { 6 } { 5 }\) find the smallest value of \(n\) such that \(u _ { n } > 10 ^ { 50 }\). A convergent sequence is defined by \(v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }\) for \(n \geq 1\).
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of \(r\), giving your answer in exact form.
c) Evaluate the infinite product: $$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$ [Question 4 - Continued]
[0pt] [Question 4 - Continued]
[0pt] [Question 4 - Continued]
SPS SPS FM 2022 October Q5
5. A cotangent function \(\cot x\) is defined as \(\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }\).
a) If \(- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }\) and \(\cot \alpha = - \frac { 12 } { 5 }\), find the exact value of \(\sin \alpha\) and \(\cos \alpha\).
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle.
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
SPS SPS FM 2022 October Q6
6. A function is defined by: $$f ( x ) = \frac { a x + b } { c x + d } , x \in \mathbb { R } , x \neq - \frac { d } { c }$$ a) Find and simplify an expression for \(f ^ { - 1 } ( x )\), stating the domain. A function is defined by: $$g ( x ) = \frac { x - 6 } { x - 4 } , x \in \mathbb { R } , x \neq 4$$ b) Find \(g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\), stating an appropriate domain for each function.
c) Find \(g ^ { - 1 } ( x ) , g ^ { - 2 } ( x )\) and \(g ^ { - 3 } ( x )\), stating an appropriate domain for each function. NB: \(g ^ { - n } ( x ) = g ^ { - 1 } \left( g ^ { - 1 } \left( \cdots \left( g ^ { - 1 } ( x ) \right) \cdots \right) \right)\) with \(n\) copies of \(g ^ { - 1 }\).
d) State the range of \(g ( x ) , g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\). A function is defined (over an appropriate domain) by \(h ( x ) = g ( x ) + g ^ { - 1 } ( x )\).
e) Solve the inequality \(h ( x ) \geq 4\).
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SPS SPS FM 2022 November Q2
  1. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
    (b) Find the value of \(x\) such that
$$\frac { 1 + x } { x } = \sqrt { 3 }$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
[0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8940254-0663-413e-a802-71519742cfcc-06_597_977_130_351} \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 FM 2022 November Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8940254-0663-413e-a802-71519742cfcc-08_721_982_114_347} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
  2. Find an equation for \(l\).
  3. Find the coordinates of the point where \(l\) intersects \(C\) again.
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q5
5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$ [BLANK PAGE]
SPS SPS FM 2022 November Q6
6. A sequence of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(\left\{ \begin{array} { c } u _ { 1 } = 1
u _ { n + 1 } = 3 u _ { n } + 2 \end{array} ( n \geq 1 ) \right.\)
Prove by induction that \(u _ { n } = 2 \left( 3 ^ { n - 1 } \right) - 1\).
(4)
[0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q7
7. (a) Sketch on the same diagram in the space provided the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$ [BLANK PAGE]
SPS SPS FM 2022 November Q8
8. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.
  1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on circle \(C\),
  2. find the coordinates of the centre of \(C\),
  3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y = k$$ where \(a , b\) and \(k\) are integers to be found.
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SPS SPS FM 2022 January Q1
1. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathbf { A } ^ { - 1 }\).
    [0pt] [BLANK PAGE]