Questions - SPS SPS FM - A-Level Maths

SPS SPS FM 2023 October Q4

4. In this question you must show detailed reasoning. Find the equation of the normal to the curve $y = 4 \sqrt { x } - 3 x + 1$ at the point on the curve where $x = 4$. Give your answer in the form $a x + b y + c = 0$, where $a$, band $c$ are integers.
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Find the binomial expansion of $( 3 + k x ) ^ { 3 }$, simplifying the terms.
It is given that, in the expansion of $( 3 + k x ) ^ { 3 }$, the coefficient of $x ^ { 2 }$ is equal to the constant term. Find the possible values of $k$, giving your answers in an exact form.
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SPS SPS FM 2023 October Q6

6. In this question you must show detailed reasoning. The functions f and g are defined for all real values of $x$ by $$f ( x ) = x ^ { 3 } \text { and } g ( x ) = x ^ { 2 } + 2$$

Write down expressions for
1. $\mathrm { fg } ( x )$,
2. $\mathrm { gf } ( x )$.
Hence find the values of $x$ for which $\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24$.
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SPS SPS FM 2023 October Q7

7. The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form $p + q \sqrt { r }$ where $p , q$ and $r$ are integers.
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SPS SPS FM 2023 October Q8

8. Prove that $2 ^ { 3 n } - 3 ^ { n }$ is divisible by 5 for all integers $n \geq 1$.
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SPS SPS FM 2023 October Q9

9. (a)
\includegraphics[max width=\textwidth, alt={}, center]{9c377549-1fbd-4790-b57e-37ec9707c9d8-20_540_529_157_278} The shape $A B C$ shown in the diagram is a student's design for the sail of a small boat. The curve $A C$ has equation $y = 2 \log _ { 2 } x$ and the curve $B C$ has equation $y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3$. State the $x$-coordinate of point $A$.
(b) Determine the $x$-coordinate of point $B$.
(c) By solving an equation involving logarithms, show that the $x$-coordinate of point $C$ is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units $^ { 2 }$.
(d) Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.
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SPS SPS FM 2024 February Q1

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]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-04_510_894_443_226} Find the volume of the funnel according to the model.
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SPS SPS FM 2024 February Q2

2. The diagram below shows the graphs of $y = | 3 x - 2 |$ and $y = | 2 x + 1 |$.

\includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-06_318_511_187_904} Give the coordinates of the points of intersection of the graphs with the coordinate axes.
Solve the equation $| 2 x + 1 | = | 3 x - 2 |$.
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SPS SPS FM 2024 February Q3

3. Show that $\int _ { 4 } ^ { \infty } x ^ { - \frac { 3 } { 2 } } d x = 1$
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SPS SPS FM 2024 February Q4

4. 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 }

SPS SPS FM 2024 February Q5

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 $/ _ { 1 }$ and $/ _ { 2 }$.
$Q$ is a point on $/ 1$ which is 12 metres away from $P \cdot R$ is the point on $/ 2$ such that $Q R$ is perpendicular to $/ 1$.
Determine the length $Q R$.
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5. (a) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.
State the set of values for which the expansion in part (b) is valid.
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SPS SPS FM 2024 February Q6

6. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $= \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)$.

Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.
Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.
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SPS SPS FM 2024 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 $/ _ { 1 }$ and $/ _ { 2 }$.
$Q$ is a point on $/ 1$ which is 12 metres away from $P \cdot R$ is the point on $/ 2$ such that $Q R$ is perpendicular to $/ 1$.
Determine the length $Q R$.
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5. (a) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.
State the set of values for which the expansion in part (b) is valid.
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6. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $= \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)$.
Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.
Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.
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7. (a) In this question you must show detailed reasoning. Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.
The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z | = | z - 2 i |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
i. Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$, showing any intercepts with the imaginary axis.
ii. Indicate, by shading on your Argand diagram, the region $\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}$.
i. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
ii. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.
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SPS SPS FM 2024 October Q1

Given the function $f ( x ) = x - x ^ { 2 }$, defined for all real values of $x$,
1. Show that $f ^ { \prime } ( x ) = 1 - 2 x$ by differentiating $f ( x )$ from first principles.
2. Find the maximum value of $f ( x )$.
3. Explain why $f ^ { - 1 } ( x )$ does not exist.
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4. The quadratic equation $k x ^ { 2 } + 2 k x + 2 k = 3 x - 1$, where $k$ is a constant, has no real roots.
5. Show that $k$ satisfies the inequality
$$4 k ^ { 2 } + 16 k - 9 > 0$$
Hence find the set of possible values of $k$. Give your answer in set notation.
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SPS SPS FM 2024 October Q3

3. (a) Find and simplify the first three terms in the expansion, in ascending powers of $x$, of $\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }$, where $k$ is a constant.
(b) In the expansion of $( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }$, the constant term is equal to the coefficient of $x ^ { 2 }$. Determine the exact value of $k$, given that $k$ is positive.
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SPS SPS FM 2024 October Q4

4. The curve $y = \sqrt { 2 x - 1 }$ is stretched by scale factor $\frac { 1 } { 4 }$ parallel to the $x$-axis and by scale factor $\frac { 1 } { 2 }$ parallel to the $y$-axis. Find the resulting equation of the curve, giving your answer in the form $\sqrt { a x - b }$ where $a$ and $b$ are rational numbers.
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(a) Show that $( x + 1 )$ is a factor of $\mathrm { f } ( x )$.
(b) Hence find the exact roots of the equation $\mathrm { f } ( x ) = 0$.
(a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form $\mathrm { f } ( x ) = 0$.
(b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
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SPS SPS FM 2024 October Q6

6. The first three terms of a geometric sequence are $$u _ { 1 } = 3 k + 4 \quad u _ { 2 } = 12 - 3 k \quad u _ { 3 } = k + 16$$ where $k$ is a constant. Given that the sequence converges,

Find the value of $k$, giving a reason for your answer.
Find the value of $\sum _ { r = 2 } ^ { \infty } u _ { r }$
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SPS SPS FM 2024 October Q7

7. The diagram shows part of the graph of $y = x ^ { 2 }$. The normal to the curve at the point $A ( 1,1 )$ meets the curve again at $B$. Angle $A O B$ is denoted by $\alpha$.
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-16_506_741_283_217}

Determine the coordinates of $B$.
Hence determine the exact value of $\tan \alpha$.
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SPS SPS FM 2024 October Q8

8. Prove by induction that $11 \times 7 ^ { n } - 13 ^ { n } - 1$ is divisible by 3 , for all integers $n \geqslant 0$.
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SPS SPS FM 2024 October Q9

9. A circle has centre $C$ which lies on the $x$-axis, as shown in the diagram. The line $y = x$ meets the circle at $A$ and $B$. The midpoint of $A B$ is $M$.
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-20_776_730_280_214} The equation of the circle is $x ^ { 2 } - 6 x + y ^ { 2 } + a = 0$, where $a$ is a constant.

In this question you must show detailed reasoning. Find the $x$-coordinate of M and hence show that the area of triangle ABC is $\frac { 3 } { 2 } \sqrt { 9 - 2 a }$.
1. Find the value of $a$ when the area of triangle $A B C$ is zero.
2. Give a geometrical interpretation of the case in part (b)(i).
Give a geometrical interpretation of the case where $a = 5$.
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SPS SPS FM 2024 October Q1

Solve the following simultaneous equations:

$$\begin{aligned} & y = 4 x ^ { 2 } + 2 x - 5
& y = | 4 x + 1 | \end{aligned}$$

SPS SPS FM 2024 October Q2

The graph of $y = f ( x )$ (where $- 2 \leq x \leq 6$ ) has the following features:

A local maximum at $x = 0$.
A local minimum at $x = 2$.
No other turning points.
Three stationary points.

Sketch a possible graph of $y = f ^ { \prime } ( x )$ on the axes provided.
You can ignore the scale for the $y$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-04_1269_1354_826_447}
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SPS SPS FM 2024 October Q3

3. The function $g ( x )$ is defined as follows: $$g ( x ) = e ^ { \sin \left( x ^ { \circ } \right) } , - 90 \leq x \leq 90$$

Find $g ^ { - 1 } ( x )$, stating its domain.
Sketch $y = g ^ { - 1 } ( x )$ on the axes provided below, being sure to label all key points.
\includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-07_1524_1591_459_342}

SPS SPS FM 2024 October Q4

4. The polynomial $P ( x )$ is defined as follows:
$P ( x ) \equiv x ^ { 8 } + 8 x ^ { 7 } + 28 x ^ { 6 } + 56 x ^ { 5 } + 70 x ^ { 4 } + 56 x ^ { 3 } + 28 x ^ { 2 } + 8 x , x \in \mathbb { R }$
By first factorising $P ( x )$ find all of its real roots.
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SPS SPS FM 2024 October Q5

6 marks

5. While working on a logarithms problem on one of the whiteboards in the Maths corridor, Rehman confidently asserts the following:
"For all positive real numbers $x$ and $y$, we know that $\log _ { 2 } x - \log _ { 2 } y \equiv \frac { \log _ { 2 } x } { \log _ { 2 } y }$."
Soufiane, who happens to be passing, knows that this is wrong. He attempts to prove this by counterexample, picking two values of $x$ and $y$ off the top of his head and plugging both sides of the false identity into his calculator. To his dismay, both sides give exactly the same answer, and Rehman smugly walks off uncorrected. What is the range of possible values of $x$ that Soufiane picked?
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SPS SPS FM 2024 October Q6

6. Neil enjoys playing around with sequences in his spare time, and one day he decides to create a new one. He does this by taking a normal arithmetic sequence but repeatedly halving the common difference between the terms. For instance, if the difference between the first two terms is 16 , the difference between the third term and the second term will be 8 and the difference between the fourth term and the third term will be 4 (continuing in this fashion for as long as he likes).

If $u _ { 1 } = a$ and $u _ { 2 } = a + d$, find a closed form expression for $u _ { n }$, where $n \in \mathbb { N }$. "Closed form" means that you can't have a "..." or sigma notation in your final answer.
With $u _ { n }$ defined as above, find a closed form expression for $S _ { n } = \sum _ { k = 1 } ^ { n } u _ { k }$, where $n \in \mathbb { N }$.
With $u _ { n }$ and $S _ { n }$ defined as above, find the values of $a$ and $d$ if $u _ { 7 } = 23$ and $S _ { 7 } = 41$.
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