Questions SPS FM Pure - A-Level Maths

SPS SPS FM Pure 2021 June Q1

A curve is defined by the parametric equations

$$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ Find the gradient of the curve at the point where $t = - 2$

SPS SPS FM Pure 2021 June Q2

3 marks

2. The equation $x ^ { 3 } - 3 x + 1 = 0$ has three real roots.

Show that one of the roots lies between -2 and -1
Taking $x _ { 1 } = - 2$ as the first approximation to one of the roots, use the Newton-Raphson method to find $x _ { 2 }$, the second approximation.
[0pt] [3 marks]
Explain why the Newton-Raphson method fails in the case when the first approximation is $x _ { 1 } = - 1$

SPS SPS FM Pure 2021 June Q3

3. 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

SPS SPS FM Pure 2021 June Q5

8 marks

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 $l _ { 2 }$.
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4. Solve the quadratic equation $x ^ { 2 } - 4 x - 1 - 12 i = 0$ writing your solutions in the form $a + b i$.
[0pt] [8]
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5. $\int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x$ can be written in the form $p \ln 2 + q$, where $p$ and $q$ are rational numbers. Find $p$ and $q$.

SPS SPS FM Pure 2021 June Q6

6.

Use the binomial expansion, in ascending powers of $x$, to show that $$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$ where $k$ is a rational constant to be found. A student attempts to substitute $x = 1$ into both sides of this equation to find an approximate value for $\sqrt { 3 }$.
State, giving a reason, if the expansion is valid for this value of $x$.

SPS SPS FM Pure 2021 June Q7

7. (a) Determine a sequence of transformations which maps the graph of $y = \cos \theta$ onto the graph of $y = 3 \cos \theta + 3 \sin \theta$ Fully justify your answer.
(b) Hence or otherwise find the least value and greatest value of $$4 + ( 3 \cos \theta + 3 \sin \theta ) ^ { 2 }$$ Fully justify your answer.
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SPS SPS FM Pure 2021 June Q8

8. Prove by induction that, for $n \in \mathbb { Z } ^ { + }$ $$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7
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SPS SPS FM Pure 2021 June Q10

8 marks

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 $l _ { 2 }$.
[0pt] [BLANK PAGE]
4. Solve the quadratic equation $x ^ { 2 } - 4 x - 1 - 12 i = 0$ writing your solutions in the form $a + b i$.
[0pt] [8]
[0pt] [BLANK PAGE]
5. $\int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x$ can be written in the form $p \ln 2 + q$, where $p$ and $q$ are rational numbers. Find $p$ and $q$.
6.
(a) Use the binomial expansion, in ascending powers of $x$, to show that $$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$ where $k$ is a rational constant to be found. A student attempts to substitute $x = 1$ into both sides of this equation to find an approximate value for $\sqrt { 3 }$.
(b) State, giving a reason, if the expansion is valid for this value of $x$.
7. (a) Determine a sequence of transformations which maps the graph of $y = \cos \theta$ onto the graph of $y = 3 \cos \theta + 3 \sin \theta$ Fully justify your answer.
(b) Hence or otherwise find the least value and greatest value of $$4 + ( 3 \cos \theta + 3 \sin \theta ) ^ { 2 }$$ Fully justify your answer.
[0pt] [BLANK PAGE]
8. Prove by induction that, for $n \in \mathbb { Z } ^ { + }$ $$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7
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9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b5fb91-3bc6-4167-afb9-91879ebbfc96-16_595_593_157_822} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the graph $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ (a) State the range of f
(b) Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has two distinct roots, (c) state the set of possible values for $k$.
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10. A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation $x ^ { 2 } + 2 x y + 2 y ^ { 2 } = 10$, where $x$ and $y$ are measured in metres. The $x$ and $y$ axes are horizontal and vertical respectively.
\includegraphics[max width=\textwidth, alt={}, center]{15b5fb91-3bc6-4167-afb9-91879ebbfc96-18_224_478_667_804} Find the maximum vertical height above the platform of the sculpture.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 June Q11

1 marks

11.

Given that $u = 2 ^ { x }$, write down an expression for $\frac { \mathrm { d } u } { \mathrm {~d} x }$
[0pt] [1 mark]
Find the exact value of $\int _ { 0 } ^ { 1 } 2 ^ { x } \sqrt { 3 + 2 ^ { x } } \mathrm {~d} x$ Fully justify your answer.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 June Q12

12. The cubic equation
IGNORE
has roots $\alpha , \beta$, and $\psi$
Without solving the cubic equation, $$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$

determine the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$
find a cubic equation that has roots $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$, giving your answer in $x ^ { 3 } + a x ^ { 2 } - b x + c = 0$, where $a , b$ and $c$ are integers to be determined.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 June Q13

13. $$\mathbf { A } = \left( \begin{array} { c c } 2 & a
a - 4 & b \end{array} \right)$$ where $a$ and $b$ are non-zero constants.
Given that the matrix $\mathbf { A }$ is self-inverse,

determine the value of $b$ and the possible values for $a$. The matrix $\mathbf { A }$ represents a linear transformation $M$.
Using the smaller value of $a$ from part (a),
show that the invariant points of the linear transformation $M$ form a line, stating the equation of this line.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 June Q14

14. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{15b5fb91-3bc6-4167-afb9-91879ebbfc96-26_545_1029_164_571} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { 4 \sin 2 x } { \mathrm { e } ^ { \sqrt { 2 } x - 1 } } , \quad 0 \leqslant x \leqslant \pi$$ The curve has a maximum turning point at $P$ and a minimum turning point at $Q$ as shown in Figure 5.

Show that the $x$ coordinates of point $P$ and point $Q$ are solutions of the equation $$\tan 2 x = \sqrt { 2 }$$
Using your answer to part (a), find the $x$-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2 f ( x )$$ [BLANK PAGE]

SPS SPS FM Pure 2021 June Q15

7 marks

15. The height $x$ metres, of a column of water in a fountain display satisfies the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 8 \sin 2 t } { 3 \sqrt { x } }$, where $t$ is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height.
Express your answer in the form $x = \mathrm { f } ( t )$
[0pt] [7 marks]
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SPS SPS FM Pure 2021 June Q16

16. Given that there are two distinct complex numbers $z$ that satisfy $$\{ z : | z - 3 - 5 \mathrm { i } | = 2 r \} \cap \left\{ z : \arg ( z - 2 ) = \frac { 3 \pi } { 4 } \right\}$$ determine the exact range of values for the real constant $r$.
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SPS SPS FM Pure 2021 May Q1

Points $A , B$ and $C$ have coordinates $( 0,1 , - 4 ) , ( 1,1 , - 2 )$ and $( 3,2,5 )$ respectively.
1. Find the vector product $\overrightarrow { A B } \times \overrightarrow { A C }$.
2. Hence find the equation of the plane $A B C$ in the form $a x + b y + c z = d$.
  [0pt] [BLANK PAGE]
3. The equation of the curve shown on the graph is, in polar coordinates, $r = 3 \sin 2 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
  \includegraphics[max width=\textwidth, alt={}, center]{25055c9c-2d29-476e-887a-a10699814b85-04_505_704_348_292}
4. The greatest value of $r$ on the curve occurs at the point $P$.
  1. Show that $\theta = \frac { 1 } { 4 } \pi$ at the point $P$.
  2. Find the value of $r$ at the point $P$.
  3. Mark the point $P$ on a copy of the graph.
5. In this question you must show detailed reasoning.
Find the exact area of the region enclosed by the curve.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q3

3. You are given the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & 0 & 1
0 & - 1 & 0 \end{array} \right)$.

Find $\mathbf { A } ^ { 4 }$.
Describe the transformation that $\mathbf { A }$ represents. The matrix $\mathbf { B }$ represents a reflection in the plane $x = 0$.
Write down the matrix B. The point $P$ has coordinates $( 2,3,4 )$. The point $P ^ { \prime }$ is the image of $P$ under the transformation represented by $\mathbf { B }$.
Find the coordinates of $P ^ { \prime }$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q4

4. Using the formulae for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$, show that $\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045$.
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SPS SPS FM Pure 2021 May Q5

5. Prove by induction that, for all positive integers $n , 7 ^ { n } + 3 ^ { n - 1 }$ is a multiple of 4 .
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q6

6. $$\mathbf { A } = \left( \begin{array} { r r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is a constant. }$$

Show that the matrix $\mathbf { A }$ is non-singular for all values of $k$. A transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is represented by the matrix $\mathbf { A }$.
The point $P$ has position vector $\binom { a } { 2 a }$ relative to an origin $O$.
The point $Q$ has position vector $\binom { 7 } { - 3 }$ relative to $O$.
Given that the point $P$ is mapped onto the point $Q$ under $T$,
determine the value of $a$ and the value of $k$. Given that, for a different value of $k , T$ maps the line $y = 2 x$ onto itself,
determine this value of $k$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q7

7. Given that $y = \arcsin x , - 1 \leq x < 1$,

show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$. Given that $f ( x ) = \frac { 3 x + 2 } { \sqrt { 4 - x ^ { 2 } } }$,
show that the mean value of $f ( x )$ over the interval $[ 0 , \sqrt { } 2 ]$, is $$\frac { \pi \sqrt { } 2 } { 4 } + A \sqrt { } 2 - A$$ where $A$ is a constant to be determined.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q8

8.

Using the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$
In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3$$ Give your answer in the form $\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }$ where $a , b$ and $c$ are integers.
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SPS SPS FM Pure 2021 May Q9

9.

Using the Maclaurin series for $\ln ( 1 + x )$, find the first four terms in the series expansion for $\ln \left( 1 + 3 x ^ { 2 } \right)$.
Find the range of $x$ for which the expansion is valid.
Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots$$ [BLANK PAGE]

SPS SPS FM Pure 2021 May Q10

10. A particular radioactive substance decays over time.
A scientist models the amount of substance, $x$ grams, at time $t$ hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t$$

Solve the differential equation to find the general solution for $x$ in terms of $t$. Initially there was 10 g of the substance.
Find the particular solution of the differential equation.
Find to 6 significant figures the amount of substance that would be predicted by the model at
(a) 6 hours,
(b) 6.25 hours.
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SPS SPS FM Pure 2021 May Q1

In this question you must show detailed reasoning.
1. By using partial fractions show that $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } + 3 r + 2 } = \frac { 1 } { 2 } - \frac { 1 } { n + 2 }$.
2. Hence determine the value of $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } + 3 r + 2 }$.
  [0pt] [BLANK PAGE]
A plane $\Pi$ has the equation $\mathbf { r } . \left( \begin{array} { r } 3
6
- 2 \end{array} \right) = 15 . C$ is the point $( 4 , - 5,1 )$. Find the shortest distance between $\Pi$ and $C$.
Lines $l _ { 1 }$ and $l _ { 2 }$ have the following equations.
$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)$
$l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
1 \end{array} \right)$
Find, in exact form, the distance between $l _ { 1 }$ and $l _ { 2 }$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2021 May Q3

3. In this question you must show detailed reasoning. Show that $$\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } d x = 1$$ [BLANK PAGE]

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