Questions Further Paper 2

Browse by module

All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4

Browse by board

AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

CAIE Further Paper 2 2021 November Q4

By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
Use a similar method to find, in terms of $N$, a lower bound for $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }$.

Use a vector product to show that the area of triangle $A B C$ is $\frac { 5 \sqrt { 11 } } { 2 }$
[0pt] [4 marks]
7
The points $A , B$ and $C$ lie in a plane.
Find a vector equation of the plane in the form r.n $= k$
7
Hence find the exact distance of the plane from the origin.

AQA Further Paper 2 2019 June Q8

The line $y = m x$ is a tangent to $P _ { 2 }$
Prove that $m = \pm \sqrt { \frac { a } { b } }$
Solutions using differentiation will be given no marks.
8
The line $y = \sqrt { \frac { a } { b } } x$ meets $P _ { 2 }$ at the point $D$.
The finite region $R$ is bounded by the $x$-axis, $P _ { 2 }$ and a line through $D$ perpendicular to the $x$-axis. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid.
Find, in terms of $a$ and $b$, the volume of this solid.
Fully justify your answer.
Find the eigenvalues and corresponding eigenvectors of the matrix

AQA Further Paper 2 2019 June Q9

4 marks

Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { 5 } & \frac { 2 } { 5 }
\frac { - 3 } { 5 } & \frac { 13 } { 10 } \end{array} \right]$$ 9
Find matrices $\mathbf { U }$ and $\mathbf { D }$ such that $\mathbf { D }$ is a diagonal matrix and $\mathbf { M } = \mathbf { U D U } ^ { - 1 }$
9
Given that $\mathbf { M } ^ { n } \rightarrow \mathbf { L }$ as $n \rightarrow \infty$, find the matrix $\mathbf { L }$.
[0pt] [4 marks]
9
The transformation represented by $\mathbf { L }$ maps all points onto a line. Find the equation of this line.
\begin{center} \begin{tabular}{ | l | }

AQA Further Paper 2 2019 June Q10

7 marks

10
- $\begin{array} { c } \text { Prove by induction that } \mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n \text { is divisible by } 6 \text { for all integers } n \geq 1
\text { [7 marks] }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - } \end{array}$
-
\end{tabular} \end{center}

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-13_2488_1716_219_153}

AQA Further Paper 2 2019 June Q11

11 The line $L _ { 1 }$ has equation $$\frac { x - 2 } { 3 } = \frac { y + 4 } { 8 } = \frac { 4 z - 5 } { 5 }$$ The line $L _ { 2 }$ has equation $$\left( \mathbf { r } - \left[ \begin{array} { c } - 2
0
3 \end{array} \right] \right) \times \left[ \begin{array} { l } 2
1
3 \end{array} \right] = \mathbf { 0 }$$ Find the shortest distance between the two lines, giving your answer to three significant figures.

AQA Further Paper 2 2019 June Q12

10 marks

12 Abel and Bonnie are trying to solve this mathematical problem: $$\begin{gathered} z = 2 - 3 \mathrm { i } \text { is a root of the equation }
2 z ^ { 3 } + m z ^ { 2 } + p z + 91 = 0 \end{gathered}$$ Find the value of $m$ and the value of $p$. Abel says he has solved the problem.
Bonnie says there is not enough information to solve the problem.
12

Abel's solution begins as follows: Since $z = 2 - 3 \mathrm { i }$ is a root of the equation, $z = 2 + 3 \mathrm { i }$ is another root. State one extra piece of information about $m$ and $p$ which could be added to the problem to make the beginning of Abel's solution correct.

12	Prove that Bonnie is right.

13(a) Explain why $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$ is an improper integral.

[1 mark]

13(b) Evaluate $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$

Show the limiting process.

[9 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-18_97_150_215_1884}

AQA Further Paper 2 2019 June Q14

Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where $a , b$ and $c$ are integers.
Question 14 continues on the next page 14
Show that, for any number $k$ greater than $\frac { 12 } { 5 }$, if the difference between $\frac { 5 } { 12 }$ and $S _ { n }$ is less than $\frac { 1 } { k }$, then $$n > \frac { k - 5 + \sqrt { k ^ { 2 } + 1 } } { 2 }$$

AQA Further Paper 2 2019 June Q15

9 marks

Find the value of $r$. 15
Show that $\mu = 3$ 15
Solve the coupled differential equations to find both $x$ and $y$ in terms of $t$.
[0pt] [9 marks]

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-27_2493_1721_214_150}
Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Further Paper 2 2020 June Q1

1 Three of the four expressions below are equivalent to each other.
Which of the four expressions is not equivalent to any of the others? Circle your answer.
$\mathbf { a } \times ( \mathbf { a } + \mathbf { b } )$
$( \mathbf { a } + \mathbf { b } ) \times \mathbf { b }$
$( \mathbf { a } - \mathbf { b } ) \times \mathbf { b }$
$\mathbf { a } \times ( \mathbf { a } - \mathbf { b } )$

AQA Further Paper 2 2020 June Q2

2 Given that arg $( a + b \mathrm { i } ) = \varphi$, where $a$ and $b$ are positive real numbers and $0 < \varphi < \frac { \pi } { 2 }$, three of the following four statements are correct. Which statement is not correct? Tick $( \checkmark )$ one box. $$\begin{aligned} & \arg ( - a - b \mathrm { i } ) = \pi - \varphi
& \arg ( a - b \mathrm { i } ) = - \varphi
& \arg ( b + a \mathrm { i } ) = \frac { \pi } { 2 } - \varphi
& \arg ( b - a \mathrm { i } ) = \varphi - \frac { \pi } { 2 } \end{aligned}$$

AQA Further Paper 2 2020 June Q3

3 Find the gradient of the tangent to the curve $$y = \sin ^ { - 1 } x$$ at the point where $x = \frac { 1 } { 5 }$
Circle your answer.

$\frac { 5 \sqrt { 6 } } { 12 }$

$\frac { 2 \sqrt { 6 } } { 5 }$

$\frac { 4 \sqrt { 3 } } { 25 }$

$\frac { 25 } { 24 }$

AQA Further Paper 2 2020 June Q4

4 The matrices A and B are defined as follows: $$\begin{aligned} & \mathbf { A } = \left[ \begin{array} { l l } x + 1 & 2
x + 2 & - 3 \end{array} \right]
& \mathbf { B } = \left[ \begin{array} { c c } x - 4 & x - 2
0 & - 2 \end{array} \right] \end{aligned}$$ Show that there is a value of $x$ for which $\mathbf { A B } = k \mathbf { I }$, where $\mathbf { I }$ is the $2 \times 2$ identity matrix and $k$ is an integer to be found.

AQA Further Paper 2 2020 June Q5

5 Solve the inequality $$\frac { 2 x + 3 } { x - 1 } \leq x + 5$$

AQA Further Paper 2 2020 June Q6

6 Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers.

AQA Further Paper 2 2020 June Q7

7 The diagram shows part of the graph of $y = \cos ^ { - 1 } x$ The diagram shows part of the graph of $y = \cos ^ { - 1 } x$
\includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-07_689_958_358_539} The finite region enclosed by the graph of $y = \cos ^ { - 1 } x$, the $y$-axis, the $x$-axis and the line $x = 0.8$ is rotated by $2 \pi$ radians about the $x$-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places.

AQA Further Paper 2 2020 June Q8

$\quad$ Factorise $\left| \begin{array} { c c c } 2 a + b + x & x + b & x ^ { 2 } + b ^ { 2 }
0 & a & - a ^ { 2 }
a + b & b & b ^ { 2 } \end{array} \right|$ as fully as possible.
8
The matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left[ \begin{array} { c c c } 13 + x & x + 3 & x ^ { 2 } + 9
0 & 5 & - 25
8 & 3 & 9 \end{array} \right]$$ Under the transformation represented by $\mathbf { M }$, a solid of volume $0.625 \mathrm {~m} ^ { 3 }$ becomes a solid of volume $300 \mathrm {~m} ^ { 3 }$ Use your answer to part (a) to find the possible values of $x$.
Use $\mathbf { C }$ to show that $\cos \frac { \pi } { 12 }$ can be written in the form $\frac { \sqrt { \sqrt { m } + n } } { 2 }$, where $m$ and $n$ are integers.

AQA Further Paper 2 2020 June Q10

10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers $n \geq 1$, $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$

AQA Further Paper 2 2020 June Q11

Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$\frac { \sin x } { x } - \cos x$$ is $$( - 1 ) ^ { r + 1 } \frac { 2 r } { ( 2 r + 1 ) ! } x ^ { 2 r }$$ 11
Show that $$\lim _ { x \rightarrow 0 } \left[ \frac { \frac { \sin x } { x } - \cos x } { 1 - \cos x } \right] = \frac { 2 } { 3 }$$

AQA Further Paper 2 2020 June Q12

6 marks

Given that $I = \int _ { a } ^ { b } \mathrm { e } ^ { 2 t } \sin t \mathrm {~d} t$, show that $$I = \left[ q \mathrm { e } ^ { 2 t } \sin t + r \mathrm { e } ^ { 2 t } \cos t \right] _ { a } ^ { b }$$ where $q$ and $r$ are rational numbers to be found.
[0pt] [6 marks]
12

A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v = 5 \mathrm { e } ^ { t } \sin t$$ where $v$ is the velocity at time $t$.
Find the speed of the object when $t = 2 \pi$, giving your answer in exact form.

13	Charlotte is trying to solve this mathematical problem:
	Find the general solution of the differential equation $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 10 \mathrm { e } ^ { - 2 x }$

Charlotte's solution starts as follows:

Particular integral: $y = \lambda \mathrm { e } ^ { - 2 x }$

so $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \lambda \mathrm { e } ^ { - 2 x }$

and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 \lambda \mathrm { e } ^ { - 2 x }$

Questions Further Paper 2 (287 questions)

Browse by module

Browse by board

CAIE Further Paper 2 2021 November Q4

AQA Further Paper 2 2019 June Q1

AQA Further Paper 2 2019 June Q2

AQA Further Paper 2 2019 June Q3

AQA Further Paper 2 2019 June Q4

AQA Further Paper 2 2019 June Q5

AQA Further Paper 2 2019 June Q7

AQA Further Paper 2 2019 June Q8

AQA Further Paper 2 2019 June Q9

AQA Further Paper 2 2019 June Q10

AQA Further Paper 2 2019 June Q11

AQA Further Paper 2 2019 June Q12

AQA Further Paper 2 2019 June Q14

AQA Further Paper 2 2019 June Q15

AQA Further Paper 2 2020 June Q1

AQA Further Paper 2 2020 June Q2

AQA Further Paper 2 2020 June Q3

AQA Further Paper 2 2020 June Q4

AQA Further Paper 2 2020 June Q5

AQA Further Paper 2 2020 June Q6

AQA Further Paper 2 2020 June Q7

AQA Further Paper 2 2020 June Q8

AQA Further Paper 2 2020 June Q10

AQA Further Paper 2 2020 June Q11

AQA Further Paper 2 2020 June Q12