Questions — SPS (1106 questions)

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SPS SPS FM Pure 2023 November Q9
9. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio \(r\) and first term \(a\).
Given \(r \neq 1\) and \(a \neq 0\)
  1. prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that \(S _ { 10 }\) is four times \(S _ { 5 }\)
  2. find the exact value of \(r\).
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SPS SPS FM Pure 2023 November Q10
10. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
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SPS SPS SM Pure 2022 November Q1
  1. Do not use a calculator for this question
    a)
Find \(a\), given that \(a ^ { 3 } = 64 x ^ { 12 } y ^ { 3 }\).
b)
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
SPS SPS SM Pure 2022 November Q2
2.
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
SPS SPS SM Pure 2022 November Q3
3. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where k is a constant, has 2 different real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0$$
  2. Hence find the set of possible values of \(k\).
SPS SPS SM Pure 2022 November Q4
4. Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
  1. \(\quad \log _ { 2 } ( 16 x )\),
  2. \(\quad \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\)
    (3)
  3. Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
    (Total 9 marks)
SPS SPS SM Pure 2022 November Q5
5. An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 29 terms is 1102.
  1. Show that \(a + 14 d = 38\).
  2. The sum of the second term and the seventh term is 13 . Find the value of \(a\) and the value of \(d\).
SPS SPS SM Pure 2022 November Q6
6. A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(\quad u _ { n } = 2 n + 5\), for \(n \geq 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. State what type of sequence it is.
  3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).
SPS SPS SM Pure 2022 November Q7
7. There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012-2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10 ^ { b t }\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log _ { 10 } y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\).
  2. Complete the values of \(\log _ { 10 } y\) in the table, draw the graph of \(\log _ { 10 } y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model.
    t12345678910
    \(\log _ { 10 } y\)1.511.581.982.68
SPS SPS SM Pure 2022 November Q8
8.
  1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\).
    \(A B\) is the diameter of a circle.
  2. Find the radius of the circle.
  3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).
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 \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).
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    \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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SPS SPS FM 2023 February Q5
5. (a) Expand \(\sqrt { 1 + 2 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(b) Hence expand \(\frac { \sqrt { 1 + 2 x } } { 1 + 9 x ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(c) Determine the range of values of \(x\) tor which the expansion in part (b) is valid.
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