Questions SPS FM (245 questions)

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SPS SPS FM 2022 October Q1
  1. a) Rationalise the denominator for \(\frac { \sqrt { 8 } + 2 } { 5 - \sqrt { 2 } }\)
    b) Solve
$$( \sqrt { 2 } ) ^ { x + 1 } = \frac { 1 } { 4 ^ { 4 - 3 x } }$$ [BLANK PAGE]
SPS SPS FM 2022 October Q2
2. Given that $$f ( x ) = \ln x , x > 0$$ Sketch on separate axes the graphs of
i) \(y = f ( x )\)
ii) \(\quad y = f ( x - 4 )\) Show on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.
SPS SPS FM 2022 October Q3
3. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
a) Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\) The sum of the first n terms of the series is greater than 300 .
b) Calculate the smallest possible value of n
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SPS SPS FM 2022 October Q4
4. Let \(f ( x )\) be given by: $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$ a) Use the factor theorem to show that ( \(x + 3\) ) is a factor of \(f ( x )\)
b) Factorise \(f ( x )\) into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation \(f ( x ) = 0\), showing detailed reasoning with your working
c) Hence write down the one solution to the equation $$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$ in the form \(\ln ( a + \sqrt { b } )\)
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SPS SPS FM 2022 October Q5
5. Solve, for \(0 < \theta < 360 ^ { \circ }\),
a) \(5 \cos ( \theta + 30 ) = 3\)
b) \(\cos ^ { 2 } ( x ) + 4 \sin ^ { 2 } ( x ) + 4 \sin ( x ) = 0\) Give each non-exact solution to one decimal place.
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SPS SPS FM 2022 October Q6
6. The curve \(C\) has the equation \(y = 6 x ^ { 2 } + 2 \sqrt { x }\). Find the equation of the normal of the curve at the point where \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a x + b y = k\) where \(a , b\) and \(k\) are positive integers. For this question, show detailed reasoning with your working
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SPS SPS FM 2022 October Q7
7. A sequence of positive integers is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = u _ { n } + n ( 3 n + 1 ) , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that $$u _ { n } = n ^ { 2 } ( n - 1 ) + 1 , \quad n \in \mathbb { Z } ^ { + }$$ [BLANK PAGE]
SPS SPS FM 2022 October Q8
8. Given that \(k\) is a positive constant,
a) sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis
b) find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$ [BLANK PAGE]
SPS SPS FM 2022 October Q9
9. a) Write the following as a single logarithm $$3 \log ( x ) - \frac { 1 } { 2 } \log ( y ) + 2$$ b) Solve \(2 ^ { x } e ^ { 3 x + 1 } = 10\) Giving your answer to (b) in the form \(\frac { \ln a + b } { \ln c + d }\), where \(a , b , c\) and \(d\) are integers.
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SPS SPS FM 2022 October Q10
10. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.
a) Show that \(n k ( n - 1 ) = 252\)
b) Find the value of \(k\) and the value of \(n\).
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SPS SPS FM 2022 October Q11
11.
a) Without using a calculator, show that \(5 > 3 \sqrt { 2 }\)
b) Two circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 5 y = \frac { 39 } { 4 } \text { and } x ^ { 2 } + y ^ { 2 } + 2 x - y = \frac { 3 } { 4 }$$ respectively. Show that \(C _ { 2 }\) lies completely inside \(C _ { 1 }\)
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SPS SPS FM 2023 January Q1
1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a
4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
  2. \(\mathbf { A B }\).
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SPS SPS FM 2023 January Q2
2. The transformations \(\mathrm { R } , \mathrm { S }\) and T are defined as follows.
R : reflection in the \(x\)-axis
S : stretch in the \(x\)-direction with scale factor 3
T: translation in the positive \(x\)-direction by 4 units
  1. The curve \(y = \ln x\) is transformed by R followed by T . Find the equation of the resulting curve.
  2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }\). You should make clear the order of the transformations.
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SPS SPS FM 2023 January Q3
3. Express \(\frac { x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( x - 2 ) }\) in partial fractions.
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SPS SPS FM 2023 January Q4
4. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.
SPS SPS FM 2023 January Q5
5.
  1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).
    [0pt] [BLANK PAGE] \section*{6.} The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it.
    \includegraphics[max width=\textwidth, alt={}, center]{d193321f-0471-48cd-b954-4a7330777491-14_424_849_287_520} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
    (a) How many selections of seven cards are possible?
    (b) Find the probability that the seven cards include exactly one card showing the letter A .
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SPS SPS FM 2023 January Q7
7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = ( - 9 \mathbf { i } + 10 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
l _ { 2 } : & \mathbf { r } = ( 3 \mathbf { i } + \mathbf { j } + 17 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{array}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\) has position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\).
  3. Show that \(A\) lies on \(l _ { 1 }\).
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SPS SPS FM 2023 January Q8
8. $$\mathrm { f } ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\)
  1. show all the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram,
  2. find the value of \(p\) and the value of \(q\).
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SPS SPS FM 2023 January Q9
9. Please remember to show detailed reasoning in your answer
\includegraphics[max width=\textwidth, alt={}, center]{d193321f-0471-48cd-b954-4a7330777491-20_467_817_239_639} The diagram shows the curve with equation \(y = ( 2 x - 3 ) ^ { 2 }\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis.
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SPS SPS FM 2023 January Q10
10. The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { c c } - 4 & - 4 \sqrt { 3 }
4 \sqrt { 3 } & - 4 \end{array} \right)$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\) A square \(S\) has vertices at the points with coordinates \(( 0,0 ) , ( a , - a ) , ( 2 a , 0 )\) and \(( a , a )\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
  2. Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
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SPS SPS FM 2023 January Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d193321f-0471-48cd-b954-4a7330777491-24_568_801_264_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows an Argand diagram.
The set \(P\), of points that lie within the shaded region including its boundaries, is defined by $$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$ where \(a , b , c\) and \(d\) are integers.
  1. Write down the values of \(a , b , c\) and \(d\). The set \(Q\) is defined by $$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
  2. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form.
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SPS SPS FM 2023 February Q1
1. Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).
Use A and B to disprove the proposition: "Matrix multiplication is commutative".
SPS SPS FM 2023 February Q2
2. 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 2023 February Q3
3. Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
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SPS SPS FM 2023 February Q4
4. (a) You are given that the matrix \(\left( \begin{array} { c c } 2 & 1
- 1 & 0 \end{array} \right)\) represents a transformation \(T\).
You are given that the line with equation \(y = k x\) is invariant under T. Determine the value of \(k\).
(b) Determine whether the line with equation \(y = k x\) in part above is a line of invariant points under T.
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