Questions - SPS SPS FM - A-Level Maths

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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SPS SPS FM 2023 February Q6

6. (a) The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. Find the probability that all the men are next to each other.
(b) Find the probability that no two men are next to one another.
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SPS SPS FM 2023 February Q7

7. Two lines, $l _ { 1 }$ and $l _ { 2 }$, have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $l _ { 1 }$ and $l _ { 2 }$. $Q$ is a point on $l _ { 1 }$ which is 12 metres away from $P . R$ is the point on $l _ { 2 }$ such that $Q R$ is perpendicular to $l _ { 1 }$.
Determine the length $Q R$.
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SPS SPS FM 2023 February Q8

8. In this question you must show detailed reasoning. The equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169$, has a root $x = 2 + 3 \mathrm { i }$.

Express $\mathrm { f } ( x )$ as a product of two quadratic factors.
Hence write down all the roots of the equation $\mathrm { f } ( x ) = 0$.
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SPS SPS FM 2023 February Q10

10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $l _ { 1 }$ and $l _ { 2 }$. $Q$ is a point on $l _ { 1 }$ which is 12 metres away from $P . R$ is the point on $l _ { 2 }$ such that $Q R$ is perpendicular to $l _ { 1 }$.
Determine the length $Q R$.
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8. In this question you must show detailed reasoning. The equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169$, has a root $x = 2 + 3 \mathrm { i }$.
Express $\mathrm { f } ( x )$ as a product of two quadratic factors.
Hence write down all the roots of the equation $\mathrm { f } ( x ) = 0$.
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9. $O$ is the origin of a coordinate system whose units are cm . The points $A , B , C$ and $D$ have coordinates $( 1,0 ) , ( 1,4 ) , ( 6,9 )$ and $( 0,9 )$ respectively.
The arc $B C$ is part of the curve with equation $x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37$.
The closed shape $O A B C D$ is formed, in turn, from the line segments $O A$ and $A B$, the arc $B C$ and the line segments $C D$ and $D O$ (see diagram).
A funnel can be modelled by rotating $O A B C D$ by $2 \pi$ radians about the $y$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{7126440a-3ac4-4dc5-b39f-09d7d6b5c87b-18_663_1166_541_210} Find the volume of the funnel according to the model.
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10. A transformation is equivalent to a shear parallel to the $x$-axis followed by a shear parallel to the $y$-axis and is represented by the matrix $\left( \begin{array} { c c } 1 & s
t & 0 \end{array} \right)$. Find in terms of $s$ the matrices which represent each of the shears.
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SPS SPS FM 2023 February Q11

11. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which $| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }$ and $\operatorname { Re } ( z ) \geq 9$.
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SPS SPS FM 2022 November Q1

1.
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-05_1031_938_262_529} The Argand diagram above shows a half-line $l$ and a circle $C$. The circle has centre 3 i and passes through the origin.

Write down, in complex number form, the equations of $l$ and $C$.
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Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]

SPS SPS FM 2022 November Q2

2.

Show that $\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }$.
Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
State, giving a brief reason, whether the series $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$ converges.
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SPS SPS FM 2022 November Q3

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of $a$.
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SPS SPS FM 2022 November Q4

4. The cubic equation $3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Write down the values of $\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$. The cubic equation $x ^ { 3 } + a x ^ { 2 } + b x + c = 0$ has roots $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.
Show that $c = - \frac { 4 } { 9 }$ and find the values of $a$ and $b$.
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SPS SPS FM 2022 November Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-12_595_1579_194_274} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The curves in Figure 1 have equations $$y = 6 \cosh ( x ) \text { and } y = 9 - 2 \sinh ( x )$$

Find exact values for the $x$-coordinates of the two points where the curves intersect. The finite region between the two curves is shaded in Figure 1.
Using calculus, find the area of the shaded region, giving your answer in the form $a \ln ( b ) + c$, where $a , b$ and $c$ are integers.
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SPS SPS FM 2022 November Q6

6. It is given that $\mathrm { f } ( x ) = \tanh ^ { - 1 } x$.

Show that $\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }$.
Hence find the Maclaurin series for $\mathrm { f } ( x )$, up to and including the term in $x ^ { 3 }$.
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SPS SPS FM 2022 November Q7

7.
$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$, where $p$ and $q$ are real constants.
The equation $f ( z ) = 0$ has roots $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$.
When plotted on an Argand diagram, the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ form the vertices of a triangle of area 35. Given that $z _ { 1 } = 3$, find the values of $p$ and $q$.
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SPS SPS FM 2022 November Q8

8. The diagram is a sketch of the two curves $C _ { 1 }$ and $C _ { 2 }$ with polar equations
$C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi$
$\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-18_346_840_210_1107} The curves meet at the pole $O$, and at the points $A$ and $B$.

Find, in terms of $a$, the polar coordinates of the points $A$ and $B$.
Show that the length of the line $A B$ is $\frac { 3 \sqrt { 3 } } { 2 } a$. The region inside $C _ { 2 }$ and outside $C _ { 1 }$ is shown shaded in the diagram above.
Find, in terms of $a$, the area of this region. A badge is designed which has the shape of the shaded region.
Given that the length of the line $A B$ is 4.5 cm ,
calculate the area of this badge, giving your answer to three significant figures.
(Total 16 marks)
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SPS SPS FM 2024 October Q1

(a) (i) Show that $\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }$ can be written in the form $\frac { a } { b + c x }$, where $a , b$ and $c$ are constants to be determined.
(ii) Hence solve the equation $\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2$.
(b) In this question you must show detailed reasoning.

Solve the equation $2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0$.
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SPS SPS FM 2024 October Q2

2. (a) Sketch the curve with equation $$y = \frac { k } { x } \quad x \neq 0$$ where $k$ is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$ [BLANK PAGE]

SPS SPS FM 2024 October Q3

3. (a) Find and simplify the first three terms in the expansion of $( 2 - 5 x ) ^ { 5 }$ in ascending powers of $x$.
(b) In the expansion of $( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }$, the coefficient of $x$ is 48 . Find the value of $a$.
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SPS SPS FM 2024 October Q4

4. The functions f and g are defined for all real values of $x$ by $\mathrm { f } ( x ) = 2 x ^ { 2 } + 6 x$ and $\mathrm { g } ( x ) = 3 x + 2$.

Find the range of f.
Give a reason why f has no inverse.
Given that $\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )$, where $a$ is a constant, determine the value of $a$.
Determine the set of values of $x$ for which $\mathrm { f } ( x ) > \mathrm { g } ( x )$. Give your answer in set notation.
[0pt] [BLANK PAGE] \section*{5. In this question you must show detailed reasoning} Find the equation of the normal to the curve $y = \frac { x ^ { 2 } - 32 } { \sqrt { x } }$ at the point on the curve where $x = 4$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
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SPS SPS FM 2024 October Q6

6. Given that the equation $$2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3 ,$$ only has one solution, find the value of $x$.
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SPS SPS FM 2024 October Q7

7. In this question you must show detailed reasoning. A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots$ is defined by $u _ { n } = 25 \times 0.6 ^ { n }$.
Use an algebraic method to find the smallest value of $N$ such that $\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }$.
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SPS SPS FM 2024 October Q8

8. Prove by induction that $2 ^ { n + 1 } + 5 \times 9 ^ { n }$ is divisible by 7 for all integers $n \geq 1$.
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SPS SPS FM 2024 October Q9

9. a) Factorise $8 x y - 4 x + 6 y - 3$ into the form $( a x + b ) ( c y + d )$ where $a , b , c$ and $d$ are integers
b) Hence, or otherwise, solve $$8 \sin \left( x ^ { 2 } \right) \cos \left( e ^ { \frac { x } { 3 } } \right) - 4 \sin \left( x ^ { 2 } \right) + 6 \cos \left( e ^ { \frac { x } { 3 } } \right) - 3 = 0$$ where $0 ^ { \circ } < x < 19 ^ { \circ }$, giving your answers to 1 decimal place.
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SPS SPS FM 2023 October Q1

This question requires detailed reasoning.

Express $\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }$ in the form $a + b \sqrt { 5 }$.
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SPS SPS FM 2023 October Q2

2. Solve each of the following equations, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.

$\sin \frac { 1 } { 2 } x = 0.8$
$\sin x = 3 \cos x$
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SPS SPS FM 2023 October Q3

3.

Sketch the curve $y = - \frac { 1 } { x }$.
The curve $y = - \frac { 1 } { x }$ is translated by 2 units parallel to the $x$-axis in the positive direction. State the equation of the transformed curve.
Describe a transformation that transforms the curve $y = - \frac { 1 } { x }$ to the curve $y = - \frac { 1 } { 3 x }$.
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Questions — SPS SPS FM (245 questions)

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