Questions - A-Level Maths

OCR MEI Further Pure Core 2023 June Q14

13 marks Challenging +1.2

14 Three planes have equations $$\begin{aligned} k x - z & = 2 \\ - x + k y + 2 z & = 1 \\ 2 k x + 2 y + 3 z & = 0 \end{aligned}$$ where $k$ is a constant.

By considering a suitable determinant, show that the three planes meet at a point for all values of $k$.
Using a matrix method, find, in terms of $k$, the coordinates of the point of intersection of the planes.

OCR MEI Further Pure Core 2023 June Q16

10 marks Challenging +1.8

16 The point $P ( 4,1,0 )$ is equidistant from the plane $2 x + y + 2 z = 0$ and the line $\frac { x - 3 } { 2 } = \frac { y - 1 } { b } = \frac { z + 5 } { 3 }$, where $b > 0$. Determine the value of $b$.

OCR MEI Further Pure Core 2023 June Q17

24 marks Challenging +1.2

17 Two similar species, X and Y , of a small mammal compete for food and habitat. A model of this competition assumes, in a particular area, the following.

In the absence of the other species, each species would increase at a rate proportional to the number present with the same constant of proportionality in each case.
The competition reduces the rate of increase of each species by an amount proportional to the number of the other species present.

So if the numbers of species X and Y present at time $t$ years are $x$ and $y$ respectively, the model gives the differential equations $\frac { d x } { d t } = k x - a y$ and $\frac { d y } { d t } = k y - b x$,
where $k , a$ and $b$ are positive constants.

1. Show that the general solution for $x$ is $x = A e ^ { ( k + n ) t } + B e ^ { ( k - n ) t }$, where $n = \sqrt { a b }$ and $A$ and $B$ are arbitrary constants.
2. Hence find the general solution for $y$ in terms of $A , B , k , n , a$ and $t$. Observations suggest that suitable values for the model are $k = 0.015 , a = 0.04$ and $b = 0.01$. You should use these values in the rest of this question.
When $t = 0$, the numbers present of species X and Y in this area are $x _ { 0 }$ and $y _ { 0 }$ respectively.
1. Show that $\mathrm { x } = \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } - 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { 0.035 \mathrm { t } } + \frac { 1 } { 2 } \left( \mathrm { x } _ { 0 } + 2 \mathrm { y } _ { 0 } \right) \mathrm { e } ^ { - 0.005 \mathrm { t } }$.
2. Hence show that $y = \frac { 1 } { 4 } \left( x _ { 0 } + 2 y _ { 0 } \right) e ^ { - 0.005 t } - \frac { 1 } { 4 } \left( x _ { 0 } - 2 y _ { 0 } \right) e ^ { 0.035 t }$.
Use initial values $x _ { 0 } = 500$ and $y _ { 0 } = 300$ with the results in part (b) to determine what the model predicts for each of the following questions.
1. What numbers of each species will be present after 25 years?
2. In this question you must show detailed reasoning. When will the numbers of the two species be equal?
3. Does either species ever disappear from the area? Justify your answer.
Different initial values will apply in other areas where the two species compete, but previous studies indicate that one species or the other will eventually dominate in any given area.
1. Identify a relationship between $x _ { 0 }$ and $y _ { 0 }$ where the model does not predict this outcome.
2. Explain what the model predicts in the long term for this exceptional case.

OCR MEI Further Pure Core 2024 June Q1

4 marks Moderate -0.3

1 By expressing $\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 }$ as a single fraction, find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$ in terms of $n$.

OCR MEI Further Pure Core 2024 June Q2

7 marks Moderate -0.8

2 Two complex numbers are given by $u = - 1 + \mathrm { i }$ and $v = - 2 - \mathrm { i }$.

1. Find $\mathrm { u } - \mathrm { v }$ in the form $\mathrm { a } + \mathrm { bi }$, where $a$ and $b$ are real.
2. In this question you must show detailed reasoning. Find $\frac { \mathrm { u } } { \mathrm { v } }$ in the form $\mathrm { a } + \mathrm { bi }$, where $a$ and $b$ are real.
Express $u$ in exact modulus-argument form.

OCR MEI Further Pure Core 2024 June Q3

4 marks Standard +0.3

3 The equation $2 x ^ { 3 } - 2 x ^ { 2 } + 8 x - 15 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Determine the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.

OCR MEI Further Pure Core 2024 June Q4

4 marks Standard +0.8

4 The equation of a curve is $\mathrm { y } = \frac { 1 } { \sqrt { \mathrm {~K} ^ { 2 } + \mathrm { x } ^ { 2 } } }$, where $k$ is a positive constant. The region between the $x$-axis, the $y$-axis and the line $x = k$ is rotated through $2 \pi$ radians about the $x$-axis. Given that the volume of the solid of revolution formed is 1 unit ${ } ^ { 3 }$, find the exact value of $k$.

OCR MEI Further Pure Core 2024 June Q5

6 marks Standard +0.3

5

Given that $\mathbf { u } = \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } a \\ 0 \\ 1 \end{array} \right)$ and $\mathbf { u } \times \mathbf { v } = \left( \begin{array} { l } 1 \\ b \\ 3 \end{array} \right)$, find $a$ and $b$.
Using $\mathbf { u } \times \mathbf { v }$, determine the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$, given that this angle is acute.

OCR MEI Further Pure Core 2024 June Q6

6 marks Moderate -0.8

6 On separate Argand diagrams, sketch the set of points represented by each of the following.

$| z - 1 - 2 i | \leqslant 4$.
$\quad \arg ( z + \mathrm { i } ) = \frac { 1 } { 3 } \pi$.

OCR MEI Further Pure Core 2024 June Q7

5 marks Challenging +1.2

7

Explain why $\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt [ 3 ] { x - 2 } } \mathrm {~d} x$ is an improper integral.
In this question you must show detailed reasoning. Use an appropriate limit argument to evaluate this integral.

OCR MEI Further Pure Core 2024 June Q8

10 marks Standard +0.3

8

Specify fully the transformation T of the plane associated with the matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)$ and $\lambda$ is a non-zero constant.
1. Find detM.
2. Deduce two properties of the transformation T from the value of detM.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)$, where $n$ is a positive integer.
Hence specify fully a single transformation which is equivalent to $n$ applications of the transformation T.

OCR MEI Further Pure Core 2024 June Q9

8 marks Challenging +1.2

9 A curve has polar equation $r = \operatorname { asin } 3 \theta$, for $0 \leqslant \theta \leqslant \pi$, where $a$ is a positive constant.

Sketch the curve. Indicate the parts of the curve where $r$ is negative by using a broken line.
In this question you must show detailed reasoning. Determine the area of one of the loops of the curve.

OCR MEI Further Pure Core 2024 June Q10

6 marks Moderate -0.3

10

Write down the first three terms of the Maclaurin series for $\ln \left( 1 + x ^ { 3 } \right)$.
Use these three terms to show that $\ln ( 1.125 ) \approx \frac { n } { 1536 }$, where $n$ is an integer to be determined.
Charlie uses the same first three terms of the series to approximate $\ln 9$ and gets an answer of 147, correct to 3 significant figures. However, $\ln 9 = 2.20$ correct to 3 significant figures. Explain Charlie's error.

OCR MEI Further Pure Core 2024 June Q11

14 marks Standard +0.3

11 The plane $\Pi$ has equation $2 x - y + 2 z = 4$. The point $P$ has coordinates $( 8,4,5 )$.

Calculate the shortest distance from P to $\Pi$. The line $L$ has equation $\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }$.
Verify that P lies on L .
Find the coordinates of the point of intersection of L and $\Pi$.
Determine the acute angle between L and $\Pi$.
Use the results of parts (b), (c) and (d) to verify your answer to part (a).

OCR MEI Further Pure Core 2024 June Q12

12 marks Challenging +1.2

12 The diagram shows the curve with parametric equations $x = 2 \cosh t + \sinh t , y = \cosh t - 2 \sinh t$. \includegraphics[max width=\textwidth, alt={}, center]{83275e7c-7f5a-4f26-b81d-a041e67ac9a2-5_812_808_1283_246}

The curve crosses the positive $x$-axis at A .
1. Determine the value of the parameter $t$ at A , giving your answer in logarithmic form.
2. Find the $x$-coordinate of A , giving your answer correct to $\mathbf { 3 }$ significant figures.
The point B has parameter $t = 0$. Determine the equation of the tangent to the curve at B .

OCR MEI Further Pure Core 2024 June Q13

10 marks Challenging +1.2

13 The complex number $z$ is defined as $z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }$ where $0 < \theta < \frac { 1 } { 2 } \pi$.
On an Argand diagram, the point O represents the complex number 0 , and the points $P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots$ represent the complex numbers $z , z ^ { 2 } , z ^ { 3 } , \ldots$ respectively.

Write down each of the following.
1. The ratio of the lengths $\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }$
2. The angle $\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }$
1. Show that $\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta$, where $a$ and $b$ are integers to be determined.
2. By considering the sum to infinity of the series $z + z ^ { 2 } + z ^ { 3 } + \ldots$, show that $$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$

OCR MEI Further Pure Core 2024 June Q14

12 marks Standard +0.8

14

Find the general solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } - 2 y = 12 e ^ { - x }$. You are given that $y$ tends to zero as $x$ tends to infinity, and that $\frac { \mathrm { dy } } { \mathrm { dx } } = 0$ when $x = 0$.
Find the exact value of $x$ for which $y = 0$.

OCR MEI Further Pure Core 2024 June Q15

10 marks Standard +0.8

15 Three planes have equations $$\begin{aligned} x + k y + 3 z & = 1 \\ 3 x + 4 y + 2 z & = 3 \\ x + 3 y - z & = - k \end{aligned}$$ where $k$ is a constant.

Show that the planes meet at a point except for one value of $k$, which should be determined.
Show that, when the planes do meet at a point, the $y$-coordinate of this point is independent of $k$.

OCR MEI Further Pure Core 2024 June Q17

20 marks Standard +0.8

17 In an industrial process, a container initially contains 1000 litres of liquid. Liquid is drawn from the bottom of the container at a rate of 5 litres per minute. At the same time, salt is added to the top of the container at a constant rate of 10 grams per minute. After $t$ minutes the mass of salt in the container is $x$ grams, and you are given that $x = 0$ when $t = 0$. In modelling the situation, it is assumed that the salt dissolves instantly and uniformly in the liquid, and that adding the salt does not change the volume of the liquid.

1. Show that the concentration of salt in the liquid after $t$ minutes is $\frac { \mathrm { X } } { 1000 - 5 \mathrm { t } }$ grams per litre.
2. Hence show that the mass of salt in the container is given by the differential equation $$\frac { d x } { d t } + \frac { x } { 200 - t } = 10$$
Show by integration that $\mathrm { x } = 10 ( 200 - \mathrm { t } ) \ln \left( \frac { 200 } { 200 - \mathrm { t } } \right)$.
1. Hence determine the mass of salt in the container when half the liquid is drawn off.
2. Determine also the time at which the mass of salt in the container is greatest.
When the process is run, it is found that the concentration of salt over time is higher than predicted by the model. Suggest a reason for this.

OCR MEI Further Pure Core 2020 November Q1

6 marks Standard +0.8

1 Using standard summation of series formulae, determine the sum of the first $n$ terms of the series $( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots$,
where $n$ is a positive integer. Give your answer in fully factorised form.

OCR MEI Further Pure Core 2020 November Q2

6 marks Standard +0.3

2

The matrices $\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)$ and $\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)$ are such that $\mathbf { M } \mathbf { N } = \mathbf { I }$.
Find $a , b$ and $c$.
State with a reason whether or not $\mathbf { N }$ is the inverse of $\mathbf { M }$.

OCR MEI Further Pure Core 2020 November Q3

4 marks Standard +0.3

3 In this question you must show detailed reasoning.
Find $\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { 1 } { \sqrt { 4 - 9 x ^ { 2 } } } \mathrm {~d} x$, expressing your answer in terms of $\pi$.

OCR MEI Further Pure Core 2020 November Q4

8 marks Standard +0.8

4 The roots of the equation $2 x ^ { 3 } - 5 x + 7 = 0$ are $\alpha , \beta$ and $\gamma$.

Find $\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$.
Find an equation with integer coefficients whose roots are $2 \alpha - 1,2 \beta - 1$ and $2 \gamma - 1$.

OCR MEI Further Pure Core 2020 November Q5

8 marks Standard +0.3

5 Fig. 5 shows the curve with polar equation $r = a ( 3 + 2 \cos \theta )$ for $- \pi \leqslant \theta \leqslant \pi$, where $a$ is a constant. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-3_607_718_351_244} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

Write down the polar coordinates of the points A and B .
Explain why the curve is symmetrical about the initial line.
In this question you must show detailed reasoning. Find in terms of $a$ the exact area of the region enclosed by the curve.

OCR MEI Further Pure Core 2020 November Q6