Questions - SPS - A-Level Maths

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 Statistics 2023 April Q1

21 marks

$\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c$,
if $X$ and $Y$ are independent then $\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )$.

\section*{Discrete distributions} $X$ is a random variable taking values $x _ { i }$ in a discrete distribution with $\mathrm { P } \left( X = x _ { i } \right) = p _ { i }$
Expectation: $\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }$
Variance: $\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }$ Greg and Nilaya play a game with these dice.
Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
The probability of Greg winning the game is $\frac { 1 } { 6 }$
(b) Find the value of $a$ and the value of $b$ Show your working clearly. The random variable $X = 2 W - 5$
Given that $\mathrm { E } ( X ) = 2.6$
(c) find the exact value of $\operatorname { Var } ( X )$ END OF EXAMINATION

SPS SPS FM Pure 2023 June Q1

You are given that $g f ( x ) = | 3 x - 1 |$ for $x \in \mathbb { R }$.
1. Given that $f ( x ) = 3 x - 1$, express $g ( x )$ in terms of $x$.
2. State the range of $g f ( x )$.
3. Solve the inequality $| 3 x - 1 | > 1$.
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\section*{2. In this question you must show detailed reasoning.}
Express $8 \cos x + 5 \sin x$ in the form $R \cos ( x - \alpha )$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
Hence solve the equation $8 \cos x + 5 \sin x = 6$ for $0 \leqslant x < 2 \pi$, giving your answers correct to 4 decimal places.
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SPS SPS FM Pure 2023 June Q3

3. You are given that $f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30$, for $x > 2.5$.

Show that $f ( x ) = 0$ has a root $\alpha$ in the interval [3.5, 4]. A student takes 4 as the first approximation to $\alpha$.
Given $f ( 4 ) = 3.099$ and $f ^ { \prime } ( 4 ) = 16.67$ to 4 significant figures,
apply the Newton-Raphson procedure once to obtain a second approximation for $\alpha$, giving your answer to 3 significant figures.
Show that $\alpha$ is the only root of $f ( x ) = 0$.
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SPS SPS FM Pure 2023 June Q4

4. You are given that $\boldsymbol { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 }
\sqrt { 3 } & 1 \end{array} \right)$.

Show that $\boldsymbol { M }$ is non-singular. The hexagon $R$ is transformed to the hexagon $S$ by the transformation represented by the matrix $\boldsymbol { M }$. Given that the area of hexagon $R$ is 5 square units,
find the area of hexagon $S$. The matrix $\boldsymbol { M }$ represents an enlargement, with centre $( 0,0 )$ and scale factor $k$, where $k > 0$, followed by a rotation anti-clockwise through an angle $\theta$ about $( 0,0 )$.
Find the value of $k$.
Find the value of $\theta$.
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SPS SPS FM Pure 2023 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-12_611_334_171_902} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a triangle $A B C$.
Given $\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }$, show that $\angle B A C = 105.9 ^ { \circ }$ to one decimal place.
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SPS SPS FM Pure 2023 June Q6

6. A spherical balloon is inflated so that its volume increases at a rate of $10 \mathrm {~cm} ^ { 3 }$ per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm .
[0pt] [For a sphere of radius $r$, surface area $= 4 \pi r ^ { 2 }$ and volume $= \frac { 4 } { 3 } \pi r ^ { 3 }$ ].
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SPS SPS FM Pure 2023 June Q7

7. Prove that for all $n \in \mathbb { N }$ $$\left( \begin{array} { c c } 3 & 4 i
i & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 n + 1 & 4 n i
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SPS SPS FM Pure 2023 June Q8

8. (i) Shade on an Argand diagram the set of points $$\{ z \in \mathbb { C } : | z - 4 \mathrm { i } | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 \mathrm { i } ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number $w$ satisfies $| w - 4 i | = 3$.
(ii) Find the maximum value of $\arg w$ in the interval $( - \pi , \pi ]$. Give your answer in radians correct to 2 decimal places.
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SPS SPS FM Pure 2023 June Q9

9. (i) Use the binomial expansion to show that $( 1 - 2 x ) ^ { - \frac { 1 } { 2 } } \approx 1 + x + \frac { 3 } { 2 } x ^ { 2 }$ for sufficiently small values of $x$.
(ii) For what values of $x$ is the expansion valid?
(iii) Find the expansion of $\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }$ in ascending powers of $x$ as far as the term in $x ^ { 2 }$.
(iv) Use $x = \frac { 1 } { 20 }$ in your answer to part (iii) to find an approximate value for $\sqrt { 11 }$.
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SPS SPS FM Pure 2023 June Q10

10. The complex number $z$ is given by $z = k + 3 i$, where $k$ is a negative real number. Given that $z + \frac { 12 } { z }$ is real, find $k$ and express $z$ in exact modulus-argument form.
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SPS SPS FM Pure 2023 June Q11

11. In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3 t \text { and } y = 3 t - 4 \cos t - \sin 2 t , \text { for } 0 \leqslant t \leqslant \pi .$$ Show that the gradient of the curve is always negative.
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SPS SPS FM Pure 2023 June Q12

12.
\includegraphics[max width=\textwidth, alt={}, center]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-26_689_1203_182_447} The figure shows part of the graph of $y = ( x - 3 ) \sqrt { \ln x }$. The portion of the graph below the $x$-axis is rotated by $2 \pi$ radians around the $x$-axis to form a solid of revolution, $S$. Determine the exact volume of $S$.
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SPS SPS FM Pure 2023 June Q13

13. (i) Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + y ) ( 1 - x ) ,$$ given that $y = 1$ when $x = 1$. Give your answer in the form $y = \mathrm { f } ( x )$, where f is a function to be determined.
(ii) By considering the sign of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ near $( 1,1 )$, or otherwise, show that this point is a maximum point on the curve $y = \mathrm { f } ( x )$.
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SPS SPS FM Pure 2023 June Q14

14. A curve $C$ has equation $$x ^ { 3 } + y ^ { 3 } = 3 x y + 48$$ Prove that $C$ has two stationary points and find their coordinates.
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SPS SPS FM Pure 2023 June Q15

15. In this question you must use detailed reasoning.

Show that $\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x = \ln ( \sqrt { a } + b )$, where $a$ and $b$ are integers to be determined.
Show that $\frac { \pi } { \frac { \pi } { 4 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x$ is undefined, explaining your reasoning clearly.
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SPS SPS SM Pure 2023 June Q1

1. Find $$\int \left( x ^ { 4 } - 6 x ^ { 2 } + 7 \right) \mathrm { d } x$$ giving your answer in simplest form. Curve C has equation $$y = x ^ { 3 } - 7 x ^ { 2 } + 5 x + 4$$ The point $P ( 2 , - 6 )$ lies on $C$ Find the equation of the tangent to $C$ at $P$
Give your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found.

SPS SPS SM Pure 2023 June Q3

3. Express in partial fractions, $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) }$$

SPS SPS SM Pure 2023 June Q4

4. Relative to a fixed origin $O$,

the point $A$ has position vector $5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$
the point $B$ has position vector $7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$
the point $C$ has position vector $4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }$
1. Find $| \overrightarrow { A B } |$ giving your answer as a simplified surd.

Given that $A B C D$ is a parallelogram,

find the position vector of the point $D$. The point $E$ is positioned such that

$A C E$ is a straight line
$A C : C E = 2 : 1$
Find the coordinates of the point $E$.

SPS SPS SM Pure 2023 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-10_684_689_260_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of the curve with equation $y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }$ for $x \geq 0$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis, the $x$-axis, and the line with equation $x = 2$ The table below shows corresponding values of $x$ and $y$ for $y = e ^ { \frac { 1 } { 5 } x ^ { 2 } }$

$x$	0	0.5	1	1.5	2
$y$	1	$e ^ { 0.05 }$	$e ^ { 0.2 }$	$e ^ { 0.45 }$	$e ^ { 0.8 }$

Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for the area of $R$, giving your answer to 2 decimal places.
Use your answer to part (a) to deduce an estimate for
1. $\quad \int _ { 0 } ^ { 2 } \left( 4 + e ^ { \frac { 1 } { 5 } x ^ { 2 } } \right) d x$
2. $\quad \int _ { 1 } ^ { 3 } e ^ { \frac { 1 } { 5 } ( x - 1 ) ^ { 2 } } d x$
  giving your answers to 2 decimal places.

SPS SPS SM Pure 2023 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-12_622_1196_251_495} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The shape $A O C B A$, shown in Figure 2, consists of a sector $A O B$ of a circle centre $O$ joined to a triangle $B O C$. The points $A , O$ and $C$ lie on a straight line with $A O = 7.5 \mathrm {~cm}$ and $O C = 8.5 \mathrm {~cm}$.
The size of angle $A O B$ is 1.2 radians.
Find, in cm, the perimeter of the shape $A O C B A$, giving your answer to one decimal place.

\(x\)	0	0.5	1	1.5	2
\(y\)	1	\(e ^ { 0.05 }\)	\(e ^ { 0.2 }\)	\(e ^ { 0.45 }\)	\(e ^ { 0.8 }\)

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