Questions - SPS SPS FM - A-Level Maths

SPS SPS FM 2020 May Q14

4 marks

14. Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.

Runner	A	B	C	D	E	F	G	H	I
100 metre time (seconds)	13.2	11.6	10.9	12.3	14.7	13.1	11.7	13.6	12.4
Push-ups achieved	32	42	22	36	41	27	37	38	33

Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to examine whether there is any association between time taken for the run and number of push-ups achieved. [4]
Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient. State, with a reason, which test is more appropriate for these data.

SPS SPS FM 2020 May Q15

15. A researcher at a large company thinks that there may be some relationship between the numbers of working days lost due to illness per year and the ages of the workers in the company. The researcher selects a random sample of 190 workers. The ages of the workers and numbers of days lost for a period of 1 year are summarised below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Working days lost
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$\mathbf { 0 }$ to 4	5 to 9	10 or more
\multirow{3}{*}{Age}	Under 35	31	27	4
\cline { 2 - 5 }	35 to 50	28	32	8
\cline { 2 - 5 }	Over 50	16	28	16

Carry out a test at the $1 \%$ significance level to investigate whether the researcher's belief appears to be true. Your working should include a table showing the contributions of each cell to the test statistic.
For the 'Over 50 ' age group, comment briefly on how the working days lost compare with what would be expected if there were no association.

SPS SPS FM 2019 Q2

Find the coefficient of the $x ^ { 4 }$ term in $( 2 - 3 x ) ^ { 6 }$.
A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = 3 n - 1$, for $n \geq 1$.
1. Find the values of $u _ { 1 } , u _ { 2 } , u _ { 3 }$.
2. Find
$$\sum _ { n = 1 } ^ { 40 } u _ { n }$$

SPS SPS FM 2019 Q4

Show that

$$\log _ { a } \left( x ^ { 10 } \right) - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )$$

SPS SPS FM 2019 Q5

Solve the following inequalities giving your answer in set notation:
1. $\quad | 4 x + 3 | < | x - 8 |$
2. $\quad \frac { x } { x ^ { 2 } + 1 } < \frac { 1 } { 2 }$
3. If $a$ and $b$ are odd integers such that 4 is a factor of ( $a - b$ ), prove by contradiction that 4 cannot be a factor of $( a + b )$.
\includegraphics[max width=\textwidth, alt={}]{e41e06f6-7d0a-496a-aa1e-b2dcd787d72c-2_568_608_1749_683}
The diagram shows a circle which passes through the points $A ( 2,9 )$ and $B ( 10,3 )$. $A B$ is a diameter of the circle.
The tangent to the circle at the point $B$ cuts the $x$-axis at $C$. Find the exact coordinates of $C$.
Find the exact area of the triangle formed by $B , C$ and the centre of the circle

SPS SPS FM 2019 Q8

8. Sketch the curve $y = 2 ^ { 2 x + 3 }$, stating the coordinates of any points of intersection with the axes. The point $P$ on the curve $y = 3 ^ { 3 x + 2 }$ has $y$-coordinate equal to 180 .
Use logarithms to find the $x$-coordinate of $P$ correct to 3 significant figures. The curves $y = 2 ^ { 2 x + 3 }$ and $y = 3 ^ { 3 x + 2 }$ intersect at the point $Q$.
Show that the $x$-coordinate of $Q$ can be written as $$x = \frac { 3 \log _ { 3 } 2 - 2 } { 3 - 2 \log _ { 3 } 2 }$$

SPS SPS FM 2019 Q9

(a) Given that $u _ { n + 1 } = 5 u _ { n } + 4 , u _ { 1 } = 4$, prove by induction that $u _ { n } = 5 ^ { n } - 1$.
(b) For all positive integers, $n \geq 2$, prove by induction that

$$\sum _ { r = 2 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n ( n - 1 ) ( n + 1 ) ( 3 n + 2 )$$

SPS SPS FM 2019 Q10

Show that, for any value of the real constant $b$, the equation $x ^ { 3 } - ( b + 1 ) x + b = 0$ has $x = 1$ as a solution.

Find all values of $b$ for which this equation has exactly two real solutions \section*{11. In the question you must show detailed reasoning} Given that the coefficients of $x , x ^ { 2 }$ and $x ^ { 4 }$ in the expansion of $( 1 + k x ) ^ { n }$ are the consecutive terms of a geometric series, where $n \geq 4$ and $k$ is a positive constant

Show that $k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }$
For the case when $k = \frac { 7 } { 5 }$, find the value of $n$.
Given that $= \frac { 7 } { 5 } , n$ is a positive integer, and that the first term of the geometric series is the coefficient of $x$, find the number of terms required for the sum of the geometric series to exceed $1.12 \times 10 ^ { 12 }$. \section*{12. In the question you must show detailed reasoning} Given that $\log _ { a } x = \frac { \log _ { b } x } { \log _ { b } a }$ show that the sum of the infinite series, where $n = 0,1,2 \ldots$, $$\log _ { 2 } e - \log _ { 4 } e + \log _ { 16 } e - \cdots + ( - 1 ) ^ { n } \log _ { 2 ^ { 2 } } n e + \cdots$$ is $$\frac { 1 } { \ln ( 2 \sqrt { 2 } ) }$$ \section*{Advanced GCE (H245)} \section*{Further Mathematics A} \section*{Formulae Booklet} \section*{Pure Mathematics} \section*{Arithmetic series} $S _ { n } = \frac { 1 } { 2 } n ( a + l ) = \frac { 1 } { 2 } n \{ 2 a + ( n - 1 ) d \}$ \section*{Geometric series} $S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$
$S _ { \infty } = \frac { a } { 1 - r }$ for $| r | < 1$ \section*{Binomial series} $( a + b ) ^ { n } = a ^ { n } + { } ^ { n } \mathrm { C } _ { 1 } a ^ { n - 1 } b + { } ^ { n } \mathrm { C } _ { 2 } a ^ { n - 2 } b ^ { 2 } + \ldots + { } ^ { n } \mathrm { C } _ { r } a ^ { n - r } b ^ { r } + \ldots + b ^ { n } \quad ( n \in \mathbb { N } )$,
where ${ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }$ $$( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )$$

SPS SPS FM 2020 December Q1

Solve $2 \sin x = \tan x$ exactly, where $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.
Let $a , b$ satisfy $0 < a < b$.
i. Find, in terms of $a$ and $b$, the value of

$$\int _ { a } ^ { b } \frac { 81 } { x ^ { 4 } } d x$$ ii. Explaining clearly any limiting processes used, find the value of $a$, given that $$\int _ { a } ^ { \infty } \frac { 81 } { x ^ { 4 } } d x = \frac { 216 } { 125 }$$

SPS SPS FM 2020 December Q3

i. Sketch the graph of $y = | 3 x - 1 |$.
ii. Hence, solve $5 x + 3 < | 3 x - 1 |$.
The following diagram shows the curve $y = a \sin ( b ( x + c ) ) + d$, where $a , b , c$ and $d$ are all positive constants and $x$ is measured in radians. The curve has a maximum point at $( 1,3.5 )$ and a minimum point at $( 2,0.5 )$.
\includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-05_826_1109_269_532}
i) Write down the value of $a$ and the value of $d$.
ii) Find the value of $b$.
iii) Find the smallest possible value of $c$, given that $c > 0$.
The $2 \times 2$ matrix $\mathbf { A }$ represents a rotation by $90 ^ { \circ }$ anticlockwise about the origin. The $2 \times 2$ matrix $\mathbf { B }$ represents a reflection in the line $y = - x$.
The matrix $\mathbf { B }$ is given by

$$\mathbf { B } = \left( \begin{array} { c c } 0 & - 1
- 1 & 0 \end{array} \right)$$ i. Write down the matrix representing $\mathbf { A }$.
ii. The $2 \times 2$ matrix $\mathbf { C }$ represents a rotation by $90 ^ { \circ }$ anticlockwise about the origin, followed by a reflection in the line $y = - x$. Compute the matrix $\mathbf { C }$ and describe geometrically the single transformation represented by $\mathbf { C }$.

SPS SPS FM 2020 December Q6

6. Given that $z$ is the complex number $x + i y$ and satisfies $$| z | + z = 6 - 2 i$$ find the value of $x$ and the value of $y$.

SPS SPS FM 2020 December Q7

7. The diagram below shows part of a curve C with equation $y = 1 + 3 x - \frac { 1 } { 2 } x ^ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-08_709_898_214_603}
i. The curve crosses the $y$ axis at the point A . The straight line L is normal to the curve at A and meets the curve again at B . Find the equation of L and the $x$ coordinate of the point B .
ii. The region $R$ is bounded by the curve $C$ and the line $L$. Find the exact area of $R$.

SPS SPS FM 2020 December Q8

8.
a) The $2 \times 2$ matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { l l } 7 & 3
2 & 1 \end{array} \right) .$$ The $2 \times 2$ matrix $\mathbf { B }$ satisfies $$\mathbf { B } \mathbf { A } ^ { 2 } = \mathbf { A } .$$ Find the matrix $\mathbf { B }$.
b) The $2 \times 2$ matrix $\mathbf { C }$ is given by $$\mathbf { C } = \left( \begin{array} { l l } - 2 & 4
- 1 & 2 \end{array} \right) .$$ By considering $\mathbf { C } ^ { 2 }$, show that the matrices $\mathbf { I } - \mathbf { C }$ and $\mathbf { I } + \mathbf { C }$ are inverse to each other.

SPS SPS FM 2020 December Q9

9. Sketch on an Argand diagram the locus of all points that satisfy $| z + 4 - 4 i | = 2 \sqrt { 2 }$ and hence find $\theta , \phi \in ( - \pi , \pi ]$ such that $\theta \leq \arg z \leq \phi$.

SPS SPS FM 2020 December Q10

10. The $2 \times 2$ matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left( \begin{array} { c c } 0 & 0.25
0.36 & 0 \end{array} \right)$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by $\mathbf { M }$.

SPS SPS FM 2020 December Q11

11. In the triangle $P Q R , P Q = 6 , P R = k , P \hat { Q } R = 30 ^ { \circ }$.
i. For the case $k = 4$, find the two possible values of $Q R$ exactly.
ii. Determine the value(s) of $k$ for which the conditions above define a unique triangle.

SPS SPS FM 2020 December Q12

12. Consider the binomial expansion of $\left( 1 + \frac { x } { 5 } \right) ^ { n }$ in ascending powers of $x$, where $n$ is a positive integer.
i. Write down the first four terms of the expansion, giving the coefficients as polynomials in $n$. The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
ii. Show that $n ^ { 3 } - 33 n ^ { 2 } + 182 n = 0$.
iii. Hence find the possible values of $n$ and the corresponding values of the common difference.

SPS SPS FM 2020 December Q13

13. A series is given by $$\sum _ { r = 1 } ^ { k } 9 ^ { r - 1 }$$ i. Write down a formula for the sum of this series.
ii. Prove by induction that $P ( n ) = 9 ^ { n } - 8 n - 1$ is divisible by 64 if $n$ is a positive integer greater than 1 . Spare space for extra working. Spare space for extra working.

SPS SPS FM 2020 October Q1

i. Find the binomial expansion of $( 2 + x ) ^ { 5 }$, simplifying the terms.
ii. Hence find the coefficient of $y ^ { 3 }$ in the expansion of $\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }$.
Let $a = \log _ { 2 } x , b = \log _ { 2 } y$ and $c = \log _ { 2 } z$.

Express $\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)$ in terms of $a , b$ and $c$.

SPS SPS FM 2020 October Q3

3. i. Give full details of a sequence of two transformations needed to transform the graph $y = | x |$ to the graph of $y = | 2 ( x + 3 ) |$.
ii. Solve $| x | > | 2 ( x + 3 ) |$, giving your answer in set notation.

SPS SPS FM 2020 October Q4

4. Prove by induction that, for $n \geq 1 , \sum _ { r = 1 } ^ { n } r ( 3 r + 1 ) = n ( n + 1 ) ^ { 2 }$.

SPS SPS FM 2020 October Q5

5. The diagram shows triangle $A B C$, with $A B = x \mathrm {~cm} , A C = ( x + 2 ) \mathrm { cm } , B C = 2 \sqrt { 7 } \mathrm {~cm}$ and angle $C A B = 60 ^ { \circ }$.
i. Find the value of $x$.
ii. Find the area of triangle $A B C$, giving your answer in an exact form as simply as possible.

SPS SPS FM 2020 October Q6

6. Prove by contradiction that $\sqrt { 7 }$ is irrational.

SPS SPS FM 2020 October Q7

7. A curve has equation $y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }$.
i. Find $\frac { d y } { d x }$.
ii. Hence sketch the gradient function for the curve.
iii. Find the equation of the tangent to the curve $y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }$ at $x = 4$.

SPS SPS FM 2020 October Q8

8. The equation of a circle is $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
i. Find the centre and radius of the circle.
ii. Find the coordinates of any points where the line $y = 2 x - 3$ meets the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
iii. State what can be deduced from the answer to part ii. about the line $y = 2 x - 3$ and the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
iv. The point $A ( - 1,5 )$ lies on the circumference of the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$. Given that $A B$ is a diameter of the circle, find the coordinates of $B$.

Questions — SPS SPS FM (245 questions)

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