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\layout Standard


\latex latex 

\backslash thispagestyle{empty}
\latex default 

\layout Standard


\latex latex 

\backslash vspace*{2cm}
\latex default 

\layout Standard


\align center 

\size huge 

\series bold 
MATHEMATICS WORKBOOK
\size default

\series default 

\layout Standard


\latex latex 

\backslash vfill
\latex default 

\hfill 

\latex latex 

\backslash today
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash newpage
\newline 

\latex default 

\layout Section*

Introduction
\layout Standard

The Mathematical Tripos is designed to be accessible to students who are
 familiar with the content of a typical single mathematics A-level.
 Half of any A-level syllabus is so-called `common core', but the material
 chosen for the other half varies between examining boards and according
 to the choice of modules. 
\layout Standard

The purpose of this workbook is to present a set of material which it would
 be useful to know before starting the Tripos.
 It is not 
\shape slanted 
necessary
\shape default 
 to know all this material, but it would make your first term more comfortable
 if you did know it.
 Rather than present a list of topics, the workbook contains questions on
 each topic that are supposed to be straightforward; by tackling these questions
,
 you will see how much knowledge is expected. 
\layout Standard

Most of you will be familiar with most of the material covered here.
 It is still worth sketching out a solution to each question: it will be
 good revision and anyway you never really know that you have understood
 a problem until you do it.
 The answers are given at the end of the workbook.
 Note that none of the questions requires the use of a calculator. 
\layout Standard

If you find some of the material is unfamiliar, you should to look it up,
 if you have time.
 It is all explained in 
\shape slanted 
Mathematics - The Core Course for A Level
\shape default 
 by L.
 Bostock and S.
 Chandler published by Nelson Thornes, 1981 (ISBN 0859503062), or in 
\shape slanted 
Further Pure Mathematics
\shape default 
, by Bostock, Chandler and C.
 Rourke, published by Nelson Thornes, 1982 (ISBN 0859501035).
 You will also probably find most of it in any standard A-level (or the
 equivalent) text.
 If a particular area is really unfamiliar to you, it would be worth doing
 exercises from a text book to supplement those given here.
 If you have difficulties with some questions, don't worry; you will have
 oppportunities to cover the material when you get to Cambridge. 
\layout Standard

You may also find that you need to get into the swing of mathematics again
 after your long break (especially if you took a gap year).
 The best way to do this is to practise problem-solving.
 One source of problems is past STEP papers (available from OCR Publications,
 PO Box 5050, Annesley, Nottingham NG15 0DL) which will be especially useful
 if you did not take STEP.
 Alternatively, your Director of Studies may supply you with some work of
 this sort. 
\layout Standard

Please e-mail comments or corrections to faculty@maths.cam.ac.uk. 
\latex latex 

\backslash bigbreak
\newline 

\latex default 

\hfill 

\latex latex 

\backslash today
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash newpage
\newline 

\latex default 

\layout Section*

Algebra
\layout Standard

Although computers and even calculators are very good at algebra, all mathematic
ians
 agree that it is important to be able to do routine algebra quickly and
 accurately.
 You should be able to state elementary series expansions including binomial,
 sine and cosine, and ln series. 
\layout Standard


\latex latex 

\backslash nq{Factorization}
\latex default 
Factorize the following polynomials: 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \(  x^2 -3x +2\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\latex latex 

\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  3x^3 -3x^2 -6x\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( x^2 -x -1\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( x^3 -1\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\newline 

\latex latex 
 
\backslash >
\latex default 
(v) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( x^4-3x^3-3x^2 +11x-6\)
\end_inset 

. 
\latex latex 

\backslash et 
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash notes 
\latex default 
In part (iii) you will need the quadratic formula to find the factors
\latex latex 

\backslash ,
\latex default 
; part (iv) has one linear and one quadratic factor
\latex latex 

\backslash ,
\latex default 
; for part (v) you can use the factor theorem. 
\layout Standard


\latex latex 

\backslash nq{More factorization}
\latex default 
Find the values of 
\begin_inset Formula \( x\)
\end_inset 

 for which 
\begin_inset Formula \( x^3<2x^2+3x\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Partial fractions}
\latex default 
Express the following in partial fractions: 
\layout Standard


\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( \ds  \frac{2}{(x+1)(x-1)}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\latex latex 

\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{1} {x^3 +1}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{4x+1}{(x+1)^2(x-2)}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\latex latex 

\backslash >
\latex default 
 (iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac{x^2-7}{(x-2)(x+1)}\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash note 
\latex default 
It is best for these purposes not to use the `cover-up rule'
\latex latex 

\backslash ,
\latex default 
; there are at least two other ways which involve elementary mathematics,
 whereas the cover-up rule works for more sophisticated reasons and to most
 users is simply a recipe (which does not always work). 
\layout Standard


\latex latex 

\backslash nq{Completing the square}
\latex default 
Find the smallest value (for real 
\begin_inset Formula \( x\)
\end_inset 

 and 
\begin_inset Formula \( y\)
\end_inset 

) of: 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
 (i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x^2-2x +6\)
\end_inset 

 ;
\latex latex 

\backslash >
\latex default 
 (ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x^4+2x^2+y^4-2y^2+3\)
\end_inset 

 ;
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \sin^2 x + 4\sin x\)
\end_inset 

 . 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash note 
\latex default 
Of course, you could find the smallest value by calculus, but expressing
 the function as a perfect square plus a remainder term is a surprisingly
 useful technique -- for example, when integrating a function with a quadratic
 denominator. 
\layout Standard


\latex latex 

\backslash nq{Exponentials and logs}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 The exponential function 
\begin_inset Formula \( e^x\)
\end_inset 

 can be defined by the series expansion 
\begin_inset Formula \[ e^x =1 + x +\frac{x^2}{2!}+\frac{x^3}{3!} + \cdots.\]
\end_inset 

 Use this definition to show that 
\begin_inset Formula \( \ds\frac{\d {e}^x}{\d {x}}= e^x.\)
\end_inset 


\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
The natural log function 
\begin_inset Formula \( \ln t\)
\end_inset 

 can be defined (for 
\begin_inset Formula \( t>0\)
\end_inset 

) as the inverse of the exponential function, so that 
\begin_inset Formula \( \ln e^x=x\)
\end_inset 

.
 Set 
\begin_inset Formula \( t=e^x\)
\end_inset 

 and use the relationship 
\begin_inset Formula \( \ds \frac{\d {x}}{\d {t}}
= 1\bigg/\frac{\d {t}}{\d {x}}\)
\end_inset 

 to show that 
\begin_inset Formula \[ \frac{\d {\ln }t}{\d {t}}=\frac1 t.
\]
\end_inset 


\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Assuming that the exponential function has the property 
\begin_inset Formula \( e^se^t=e^{s+t}\)
\end_inset 

, prove that 
\begin_inset Formula \( \ln (xy) = \ln x +\ln y\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 The definition of 
\begin_inset Formula \( a^x\)
\end_inset 

 for any 
\begin_inset Formula \( a\)
\end_inset 

 is 
\begin_inset Formula \( e^{x\ln a}\)
\end_inset 

.
 Prove that 
\begin_inset Formula \( a^xa^y = a^{x+y}\)
\end_inset 

 and 
\begin_inset Formula \( a^x b^x = (ab)^x\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash note 
\latex default 
If you have not thought of defining 
\begin_inset Formula \( a^x\)
\end_inset 

 in this way, it is worth considering how else you could give it a meaning
 when 
\begin_inset Formula \( x\)
\end_inset 

 is not an integer. 
\layout Standard


\latex latex 

\backslash nq{Binomial expansions}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Find the coefficient of 
\begin_inset Formula \( x^k\)
\end_inset 

 (for 
\begin_inset Formula \( 0\leq k\leq 10\)
\end_inset 

) in the binomial expansion of 
\begin_inset Formula \( (2+3x)^{10}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Use the binomial theorem to find the expansion in powers of 
\begin_inset Formula \( x\)
\end_inset 

 up to 
\begin_inset Formula \( x^4\)
\end_inset 

 of 
\begin_inset Formula \( (1+x+x^2)^6\)
\end_inset 

, by writing it in the form 
\begin_inset Formula \( \big(1+(x+x^2)\big)^6\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Use binomial expansions to find the expansion in powers of 
\begin_inset Formula \( x\)
\end_inset 

 up to 
\begin_inset Formula \( x^4\)
\end_inset 

 of 
\begin_inset Formula \( (1-x^3)^6(1-x)^{-6}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 Find the first four terms in the binomial expansion of 
\begin_inset Formula \( (2+x)^\half\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Series expansions of elementary functions}
\latex default 
Using only the series expansions 
\begin_inset Formula \( \sin x =x -x^3/3! +x^5/5!+\cdots\)
\end_inset 

, 
\begin_inset Formula \( \cos x =1-x^2/2!+x^4/4! +\cdots\)
\end_inset 

, 
\begin_inset Formula \( \e^x=1+x+x^2/2!+x^3/3!+\cdots\)
\end_inset 

 and 
\begin_inset Formula \( \ln (1+x)=x-x^2/2+x^3/3+\cdots\)
\end_inset 

, find the series expansions of the following functions: 
\layout Standard


\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \tan x\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
(up to the 
\begin_inset Formula \( x^5\)
\end_inset 

 term)
\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \sin x \cos x\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
(up to the 
\begin_inset Formula \( x^5\)
\end_inset 

 term)
\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{e^x +e^{-x}}{2} \)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
(up to the 
\begin_inset Formula \( x^5\)
\end_inset 

 term)
\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ln(e^x)\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
(up to the 
\begin_inset Formula \( x^3\)
\end_inset 

 term)
\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(v)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac{1-\cos^2x}{x^2}\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
(up to the 
\begin_inset Formula \( x^2\)
\end_inset 

 term). 
\latex latex 

\backslash et
\newline 

\latex default 

\latex latex 

\backslash notes 
\latex default 
Do part (ii) without using a trig.
 formula, and compare your answer with the expansion for 
\begin_inset Formula \( \sin(2x)\)
\end_inset 

.
 The function in part (iii) is 
\begin_inset Formula \( \cosh x\)
\end_inset 

, of which more later.
 The interesting thing about the function in part (v) is that the series
 shows it is `well-behaved' at 
\begin_inset Formula \( x=0\)
\end_inset 

, despite appearances. 
\layout Standard


\latex latex 

\backslash nq{Proof by induction}
\latex default 
Prove by induction that the following results are valid. 
\layout Standard


\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds a+ar+ar^2 + \cdots + ar^{n-1} = a
\left(\frac{1-r^n}{1-r^{\phantom{n}}}\right).\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2.\)
\end_inset 


\latex latex 
 
\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash note  
\latex default 
Methods of proof will be discussed in some detail in our first term course
 
\shape slanted 
Numbers and Sets
\shape default 
.
 Most students will have met mathematical induction at school
\latex latex 

\backslash ,
\latex default 
; if you haven't, you will probably want to try out this straightforward
 but important method of proof. 
\layout Standard


\latex latex 

\backslash nq {Arithmetic and geometric progressions}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Find the sum of all the odd integers from 11 to 99. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Evaluate 
\begin_inset Formula \( 6+3+ \frac{3}{2} +\frac{3}{4} +  \cdots\)
\end_inset 

 . 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Find 
\begin_inset Formula \( \sin\theta +2\sin^3\theta + 4\sin^5\theta +\cdots\)
\end_inset 


\latex latex 

\backslash  
\latex default 
. 
\latex latex 

\backslash qquad
\newline 

\latex default 
(What ranges of values of 
\begin_inset Formula \( \theta\)
\end_inset 

 are allowed?). 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 Estimate roughly the approximate number of times a piece of paper has to
 be torn in half, placing the results of each tearing in a stack and then
 doing the next tearing, for the stack of paper to reach the moon. 
\layout Standard


\latex latex 

\backslash note 
\latex default 
The dots in (ii) and (iv) indicate that the series has an infinite number
 of terms.
 For part (iii), recall that the expansion 
\begin_inset Formula \( a(1+r+r^2+\cdots)\)
\end_inset 

 only converges if 
\begin_inset Formula \( -1< r <1\)
\end_inset 

.
 You may find the approximation 
\begin_inset Formula \( 10^3=2^{10}\)
\end_inset 

 useful for part (iv).
 The distance from the Earth to the Moon is about 
\begin_inset Formula \( 4\times10^5\)
\end_inset 

 km. 
\layout Section*

Trigonometry
\layout Standard

It is not necessary to learn all the various trigonometrical formulae; but
 you should certainly know what is available.
 The double-angle formulae (such as 
\begin_inset Formula \( \tan 2x = 2\tan x
/(1-\tan^2x)\)
\end_inset 

) are worth learning, as are the basic formulae for 
\begin_inset Formula \( \sin(A\pm B)\)
\end_inset 

, 
\begin_inset Formula \( \cos(A\pm B)\)
\end_inset 

 and 
\begin_inset Formula \( \tan (A\pm B)\)
\end_inset 

, the Pythagoras-type identities (such as 
\begin_inset Formula \( \sec^2 x = 1+ \tan^2 x \)
\end_inset 

) and a few special values (such as 
\begin_inset Formula \( \sin \pi/4 = 1/\surd2\)
\end_inset 

) that can be deduced from right-angled triangles with sides 
\begin_inset Formula \( (1,1,\surd2)\)
\end_inset 

 or 
\begin_inset Formula \( (1,\surd3,2)\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Basic identities}
\latex default 
Starting from the identity 
\begin_inset Formula \( \sin(A+B)=\sin A \cos B + \cos A \sin
B\)
\end_inset 

, use the basic properties of the trig.
 functions (such as 
\begin_inset Formula \( \sin(-A)=-\sin A\)
\end_inset 

) to prove the following: 
\layout Standard


\latex latex 

\backslash subq{i}
\latex default 
 
\begin_inset Formula \( \sin(A-B)=\sin A \cos B - \cos A \sin
B\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 
\begin_inset Formula \( \cos(A+B)=\cos A \cos B -\sin A \sin
B\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 
\begin_inset Formula \( \ds \tan (A+B) =\frac{\tan A +\tan B}{1-\tan A \tan B}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 
\begin_inset Formula \( \sin C + \sin D = 2 \sin \half(C+D) \cos\half(C-D)\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{v}
\latex default 
 
\begin_inset Formula \( \ds \tan^{-1}a  +\tan^{-1} b  = \tan^{-1}\left(
 \frac{a+b}{1-ab} \right)\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash notes 
\latex default 
For part (ii), recall that 
\begin_inset Formula \( \cos A =\sin (\pi/2 -A)\)
\end_inset 

.
 You can use part (iii) to help with part (v). 
\layout Standard


\latex latex 

\backslash nq {Trig. equations}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
Solve the following equations: 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
 (a)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \sin (x+ {\textstyle\frac{\pi}{6}}) 
+ \sin (x- {\textstyle \frac{\pi}{6}}) =\frac {\surd3} 2\)
\end_inset 

 ; 
\latex latex 

\backslash >
\latex default 
(b)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \cos x +\cos 2x + \cos 3x =0 \)
\end_inset 

 .
\newline 

\latex latex 
 
\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Write down the value of 
\begin_inset Formula \( \cot (\pi/6)\)
\end_inset 

 and use a double angle formula to show that 
\begin_inset Formula \( \cot(\pi/12)\)
\end_inset 

 satisfies the equation 
\begin_inset Formula \( c^2-2\surd3 c-1=0\)
\end_inset 

.
 Deduce that 
\begin_inset Formula \( \cot (\pi/12)=2+\surd3\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq {Trig. identities using Pythagoras}
\latex default 
Prove the following identities: 
\layout Standard


\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
 (i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\tan\theta + \cot\theta = \sec\theta {\textrm{\, cosec\,}}\theta\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (ii)
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( \ds(\cot \theta +{\textrm{cosec\,}}\theta)^2
=\frac{1+\cos\theta}{1-\cos\theta}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\cos\theta = \frac{1-t^2}{1+t^2}\,; \qquad
\sin\theta = \frac{2t}{1+t^2}\,; \qquad
\tan\theta = \frac {2t}{1-t^2},\qquad \)
\end_inset 

 where 
\begin_inset Formula \( t=\tan \half\theta\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Section*

Complex numbers
\layout Standard

No prior knowledge of complex numbers is assumed in our first year course
 because some A-level syllabuses do not cover complex numbers.
 Nevertheless, the material is lectured quite fast, so if you have not met
 complex numbers before, working through the following examples with the
 help of a textbook would make the first few weeks of the course more comfortabl
e.
 A complex number 
\begin_inset Formula \( z\)
\end_inset 

 can be written as 
\begin_inset Formula \( x+iy\)
\end_inset 

, where 
\begin_inset Formula \( x\)
\end_inset 

 is the real part, 
\begin_inset Formula \( y\)
\end_inset 

 is the imaginary part and 
\begin_inset Formula \( i^2=-1\)
\end_inset 

.
 The modulus of 
\begin_inset Formula \( z\)
\end_inset 

 (written 
\begin_inset Formula \( |z|\)
\end_inset 

 or 
\begin_inset Formula \( r\)
\end_inset 

) is 
\begin_inset Formula \( \surd(x^2+y^2)\)
\end_inset 

 and the argument (written 
\begin_inset Formula \( \arg z\)
\end_inset 

 or 
\begin_inset Formula \( \theta\)
\end_inset 

) is defined by 
\begin_inset Formula \( x=r\cos\theta\)
\end_inset 

, 
\begin_inset Formula \( y=r\sin\theta\)
\end_inset 

 and 
\begin_inset Formula \( -\pi < \theta \leq \pi\)
\end_inset 

.
 The complex conjugate of 
\begin_inset Formula \( z\)
\end_inset 

 (written 
\begin_inset Formula \( z^*\)
\end_inset 

) is 
\begin_inset Formula \( x-iy\)
\end_inset 

.
 The inverse, 
\begin_inset Formula \( z^{-1}\)
\end_inset 

, of 
\begin_inset Formula \( z\)
\end_inset 

 is the complex number that satisfies 
\begin_inset Formula \( z^{-1}z=1\)
\end_inset 

 (for 
\begin_inset Formula \( z\ne0\)
\end_inset 

). 
\layout Standard


\latex latex 

\backslash nq{Algebra of complex numbers}
\latex default 
Use the definitions above with 
\begin_inset Formula \( z_1=x_1+iy_1\)
\end_inset 

 and 
\begin_inset Formula \( z_2=x_2+iy_2\)
\end_inset 

 to show: 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( z_1z_2 = (x_1x_2-y_1y_2) +i(x_1y_2+y_1x_2)\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds |z|^2 = zz^*\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds z^{-1} = \frac{z^*}{{|z|}^2}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( |z_1z_2| = |z_1||z_2|\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\newline 

\latex latex 
 
\backslash >
\latex default 
(v)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \arg (z_1z_2) = \arg z_1+ \arg z_2\)
\end_inset 


\latex latex 
 
\backslash qquad 
\latex default 
(assume that 
\begin_inset Formula \( 0 < \arg z_1\leq \pi/4\)
\end_inset 

 and 
\begin_inset Formula \( 0 < \arg z_2\leq \pi/4\)
\end_inset 

). 
\latex latex 

\backslash et
\newline 

\latex default 

\noindent 
Give a sketch of the 
\begin_inset Formula \( x\)
\end_inset 

--
\begin_inset Formula \( y\)
\end_inset 

 plane (called also the complex plane or the Argand diagram) showing the
 points representing the complex numbers 
\begin_inset Formula \( z_1 =1-i\)
\end_inset 

, 
\begin_inset Formula \( z_2 = -\surd 3 +i\)
\end_inset 

.
 Verify results (iii), (iv) and (v) for these numbers. 
\layout Standard


\latex latex 

\backslash nq {De Moivre's theorem}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Show by means of series expansions that 
\begin_inset Formula \[ 
\cos \theta +i \sin \theta = e^{i\theta}
\]
\end_inset 

 and deduce that 
\begin_inset Formula \( \cos\theta - i\sin\theta =e^{-i\theta}\)
\end_inset 

 and that 
\begin_inset Formula \( z=re^{i\theta}\)
\end_inset 

. 
\layout Standard


\noindent 
Deduce also that 
\begin_inset Formula \( \cos\theta = \half (e^{i\theta} + e^{-i\theta})\)
\end_inset 

 and 
\begin_inset Formula \( \sin \theta = \frac{1}{2i} (e^{i\theta} - e^{-i\theta})\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Use the above result to show that 
\begin_inset Formula \( r e^{i\theta}=1\)
\end_inset 

 (for real 
\begin_inset Formula \( \theta\)
\end_inset 

 and 
\begin_inset Formula \( r>0\)
\end_inset 

) if and only if 
\begin_inset Formula \( r=1\)
\end_inset 

 and 
\begin_inset Formula \( \theta =2n\pi \)
\end_inset 

 for some integer 
\begin_inset Formula \( n\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Use part (ii) to find the three distinct roots of the equation 
\begin_inset Formula \( z^3=1\)
\end_inset 

.
 Draw them on the complex plane and convert them from modulus-argument form
 to real--imaginary form. 
\layout Standard


\latex latex 

\backslash nq{Geometry of the complex plane}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
Show by transforming to cartesian coordinates that the equation 
\begin_inset Formula \( |z-c| = r\)
\end_inset 

, where 
\begin_inset Formula \( c\)
\end_inset 

 is a complex number, describes a circle. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
Show that the equation 
\begin_inset Formula \( \arg z=\alpha\)
\end_inset 

, where 
\begin_inset Formula \( \alpha\)
\end_inset 

 is a constant, describes a line. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Show by means of a diagram that 
\begin_inset Formula \( |z_1+z_2| \leq |z_1| +|z_2|\)
\end_inset 

 for any two complex numbers.
 Deduce that 
\begin_inset Formula \( |z_1-z_3| \leq |z_1| +|z_3|\)
\end_inset 

 and 
\begin_inset Formula \( |z_4-z_2| \geq |z_4| -|z_2|\)
\end_inset 

 for any complex numbers 
\begin_inset Formula \( z_1\)
\end_inset 

, 
\begin_inset Formula \( z_2\)
\end_inset 

, 
\begin_inset Formula \( z_3\)
\end_inset 

 and 
\begin_inset Formula \( z_4\)
\end_inset 

.
 Under what circumstances is 
\begin_inset Formula \( |z_1+z_2| = |z_1|
+|z_2|\)
\end_inset 

? 
\layout Section*

Hyperbolic functions
\layout Standard

Prior knowledge of hyperbolic functions is not assumed for our mathematics
 course, but it is worth getting to know the definitions and basic properties,
 which are given below.
 Hyperbolic functions are very similar to trigonometric functions, and many
 of their properties are direct analogues of the properties of trigonometric
 functions.
 The definitions are 
\begin_inset Formula \[ 
\sinh x = \frac{e^x-e^{-x}}{2}, 
\quad
\cosh x = \frac{e^x+e^{-x}}{2}, 
\quad
\tanh x = \frac{e^x-e^{-x}} {e^x+e^{-x}}
\]
\end_inset 

 and 
\begin_inset Formula \( \cosech x = (\sinh x)^{-1}\)
\end_inset 

, 
\begin_inset Formula \( \sech x = (\cosh x)^{-1}\)
\end_inset 

, 
\begin_inset Formula \( \coth x = (\tanh x)^{-1}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Basic properties}
\latex default 
Give a rough sketch of the graphs of the six hyperbolic functions.
 Show from the above definitions that 
\layout Standard


\latex latex 

\backslash subq{i}
\latex default 
 
\begin_inset Formula \(  \cosh^2 x - \sinh^2 x = 1\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 
\begin_inset Formula \( \ds \frac{\d{(}\sinh x)}{\d {x}}  = \cosh x\,;
\qquad  \frac{\d{(}\cosh x) }{\d {x}}  = \sinh x\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 
\begin_inset Formula \( \cosh  (x+y) = \cosh x \cosh y + \sinh x \sinh y\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 
\begin_inset Formula \( \cosh (ix) = \cos x, \qquad \sinh (ix) = i \sin x,\qquad
\cos(ix)=\cosh(x), \qquad \sin (ix)=i\sinh x\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash notes 
\latex default 
You can use (iv) to prove (i) and (iii) using the corresponding trig.
 identities.
 In fact, (iv) is behind the rule which says that any formula involving
 trig.
 functions becomes the corresponding formula involving hyperbolic functions
 if you change the sign of every product of two odd functions (such as 
\begin_inset Formula \( \sin x\)
\end_inset 

 or 
\begin_inset Formula \( \tan x\)
\end_inset 

). 
\layout Standard


\latex latex 

\backslash nq{Further properties}
\latex default 
Use the definitions, and the results of the previous question, to show that
 
\layout Standard


\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( \sech^2 x = 1- \tanh^2 x\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (ii) 
\begin_inset Formula \( \ds \frac{\d{(}\tanh x) }{\d {x}}  = \sech^2x\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
(iii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( \ds \frac{\d{^}2(\sinh x)}{\d {x}^2}  = \sinh  x\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \sinh^{-1}x = \ln\left(x +\surd(x^2+1)\right)\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\latex default 

\latex latex 

\backslash note 
\latex default 
For part (iv), note that 
\begin_inset Formula \( \sinh^{-1}x\)
\end_inset 

 is the inverse function, not the reciprocal. 
\layout Section*

Conic sections
\layout Standard

You should be familiar with basic coordinate geometry (lines, circles, tangents
 and normals to curves, etc).
 Conic sections (parabola, ellipse and hyperbola) are introduced in the
 first term course Algebra and Geometry, but you would find it very useful
 to do a little preliminary work on coordinate and parametric equations
 for conic sections if you have not seen them before.
 They are not only important in pure mathematics: they also arise, for example,
 as celestial orbits in the second term Dynamics course. 
\layout Standard


\latex latex 

\backslash nq{Basic definitions}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Parabola.
 The point 
\begin_inset Formula \( (x,y)\)
\end_inset 

 has the property that its distance from the point 
\begin_inset Formula \( (a,0)\)
\end_inset 

 is equal to its distance from the line 
\begin_inset Formula \( x=-a\)
\end_inset 

.
 Sketch the locus of the point using this information.
 Show that 
\begin_inset Formula \( y^2=4ax\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Ellipse and hyperbola.
 The point 
\begin_inset Formula \( (x,y)\)
\end_inset 

 has the property that its distance from the point 
\begin_inset Formula \( (ae,0)\)
\end_inset 

 is 
\begin_inset Formula \( e\)
\end_inset 

 times its distance from the line 
\begin_inset Formula \( x=ae^{-1}\)
\end_inset 

.
 Sketch the locus of the point for 
\begin_inset Formula \( 0<e <1\)
\end_inset 

 and for 
\begin_inset Formula \( e>1\)
\end_inset 

.
 Show that 
\begin_inset Formula \( x^2/a^2 + y^2/b^2 =1\)
\end_inset 

 if 
\begin_inset Formula \( 0<e <1\)
\end_inset 

 and that 
\begin_inset Formula \( x^2/a^2 - y^2/b^2 =1\)
\end_inset 

 if 
\begin_inset Formula \( e>1\)
\end_inset 

, where 
\begin_inset Formula \( b=a\sqrt{|1-e^2|}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Rectangular hyperbola.
 Show that the hyperbola 
\begin_inset Formula \( x^2/a^2-y^2/b^2=1\)
\end_inset 

 becomes the rectangular hyperbola 
\begin_inset Formula \( XY=1\)
\end_inset 

 in the new coordinates given by 
\begin_inset Formula \( x=a(X+Y)/2\)
\end_inset 

, 
\begin_inset Formula \( y=b(X-Y)/2\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Parametric equations}
\latex default 
Show that the curves with parametric form 
\begin_inset Formula \( (x,y)=
(at^2,2at)\)
\end_inset 

, 
\begin_inset Formula \( (a\cos \theta, b\sin\theta)\)
\end_inset 

 and 
\begin_inset Formula \( (a\cosh \theta, b\sinh\theta)\)
\end_inset 

 are conic sections. 
\layout Standard


\latex latex 

\backslash nq{Polar coordinates}
\latex default 
By converting to cartesian coordinates, show that: 
\layout Standard


\latex latex 

\backslash subq{i}
\latex default 
 
\begin_inset Formula \( r\cos(\theta-\alpha)=c\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 
 describes a straight line
\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 
\begin_inset Formula \( r=\ell\cos\theta\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 
describes a circle
\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 
\begin_inset Formula \( r^2 \cos(\theta +\alpha)\cos(\theta -\alpha)  =
c^2\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 
 describes a hyperbola
\latex latex 

\backslash ,
\latex default 
; 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 
\begin_inset Formula \( r^{-1} = k \cos\theta + m\)
\end_inset 

 can describe any conic section, the type depending on the values of 
\begin_inset Formula \( k\)
\end_inset 

 and 
\begin_inset Formula \( m\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash notes 
\latex default 
Here, 
\begin_inset Formula \( c\)
\end_inset 

, 
\begin_inset Formula \( k\)
\end_inset 

, 
\begin_inset Formula \( \ell\)
\end_inset 

, 
\begin_inset Formula \( m\)
\end_inset 

 and 
\begin_inset Formula \( \alpha\)
\end_inset 

 are constants.
 Special cases, such as 
\begin_inset Formula \( \alpha=0\)
\end_inset 

 in part (iii), can be ignored.
 In each case, you should give a careful description (
\begin_inset Formula \( x\)
\end_inset 

 and 
\begin_inset Formula \( y\)
\end_inset 

 intercepts, 
\begin_inset Formula \( e\)
\end_inset 

, etc) of the curves. 
\layout Standard


\latex latex 

\backslash newpage
\newline 

\latex default 

\layout Section*

Matrices and vectors
\layout Standard

Matrices and vectors arise in all branches of mathematics and form an indispensa
ble
 part of a mathematician's toolkit.
 You don't need to know much about them at the moment: they are dealt with
 in detail in the first term course Algebra and Geometry.
 However, it is as well to get to know the rules of matrix multiplication
 and the definition of a determinant.
 We consider here only 
\begin_inset Formula \( 2\times2\)
\end_inset 

 matrices.
 It is also worth understanding the basic geometric uses of vectors.
 The rule for multiplying the matrix 
\begin_inset Formula \( \ds \pmatrix {a&b\cr c&d\cr}\)
\end_inset 

 by the matrix 
\begin_inset Formula \( \ds \pmatrix {p&q\cr r&s\cr} \)
\end_inset 

 is 
\begin_inset Formula \[ 
 \ds \pmatrix {a&b\cr c&d\cr}
 \ds \pmatrix {p&q\cr r&s\cr}
 = \ds \pmatrix {ap+br&aq+bs\cr cp+dr&cq+ds\cr}.
\]
\end_inset 


\layout Standard


\latex latex 

\backslash nq{Matrix multiplication}
\latex default 
Let 
\begin_inset Formula \( \ds \bA =\pmatrix{1&2\cr 1&3\cr}\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 
 
\begin_inset Formula \( \ds \bB =\pmatrix{1&0\cr -1&1\cr}\)
\end_inset 

, 
\latex latex 

\backslash 
\latex default 
 
\begin_inset Formula \( \ds \bC = \pmatrix{2&1\cr 1&2\cr}\)
\end_inset 

 and 
\begin_inset Formula \( \ds \bI=\pmatrix{1&0\cr 0&1\cr}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{i}
\latex default 
 Show that 
\begin_inset Formula \( \bA\bB \ne \bB\bA\)
\end_inset 

.
 (This shows that matrix multiplication is not commutative.) 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Show that 
\begin_inset Formula \( (\bA\bB)\bC = \bA(\bB\bC)\)
\end_inset 

.
 (This illustrates that matrix multiplication is associative). 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Without using any formulae, find a matrix 
\begin_inset Formula \( 
\ds \bD=\pmatrix{a&b\cr c&d\cr}\)
\end_inset 

 such that 
\begin_inset Formula \( \bA\bD=\bI\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 Show that 
\begin_inset Formula \( (\bA\bB)^\T = \bB^\T\bA^\T\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{v}
\latex default 
 Show that 
\begin_inset Formula \( \det{\bA\bB}=\det\bA \det \bB\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash notes 
\latex default 
The symbol 
\begin_inset Formula \( {}^\T\)
\end_inset 

 denotes the transpose of the matrix, i.e.
 the matrix obtained by exchanging rows and columns: 
\begin_inset Formula \( \ds \pmatrix {a&b\cr c&d\cr}^{\!\T}
=\pmatrix {a&c\cr b&d\cr}\)
\end_inset 

.
 The determinant of 
\begin_inset Formula \( \ds \pmatrix {a&b\cr c&d\cr}\)
\end_inset 

 is 
\begin_inset Formula \( ad-bc\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq {Position vectors}
\latex default 
 Show that the points with position vectors 
\begin_inset Formula \[  \pmatrix{1\cr0\cr1\cr}\,\,, \,\, \,\,\,\,
\pmatrix{
2\cr
1\cr
0\cr
} \,\,, \,\,\,\,\
\pmatrix{
4\cr
3\cr
-2\cr
} \,\,,\]
\end_inset 

 lie on a straight line and give the equation of the line in the two forms
 
\begin_inset Formula \[ 
\mbox{(i)} \,\, \,\,\,\,{\textbf{r}} = {\textbf{a}} + \lambda {\textbf{b}}
\hspace{1cm} 
\mbox{(ii)} \,\,\,\,\,\,{ x-x_0 \over c} = {y-y_0 \over d} = {z-z_0 \over e}.
\]
\end_inset 


\layout Standard


\latex latex 

\backslash nq{Scalar products}
\latex default 
 The three vectors 
\begin_inset Formula \( \bA\)
\end_inset 

, 
\begin_inset Formula \( \bB\)
\end_inset 

 and 
\begin_inset Formula \( \bC\)
\end_inset 

 are defined by 
\begin_inset Formula \[ 
 {\textbf{A}} = (1,3,4), \qquad
{\textbf{B}} = (2,1,3),\qquad 
{\textbf{C}} = (3,3,2).
\]
\end_inset 


\layout Standard


\latex latex 

\backslash subq{i}
\latex default 
 Order the vectors by magnitude. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Use the scalar product to find the angles between the pairs of vectors
 (a) 
\begin_inset Formula \( \hbox{\textbf{A}}\)
\end_inset 

 and 
\begin_inset Formula \( \hbox{\textbf{B}}\)
\end_inset 

, (b) 
\begin_inset Formula \( \hbox{\textbf{B}}\)
\end_inset 

 and 
\begin_inset Formula \( \hbox{\textbf{C}}\)
\end_inset 

, leaving your answer in the form of an inverse cosine. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Find the lengths of the projections of the vectors (a) 
\begin_inset Formula \( \hbox{\textbf{A}}\)
\end_inset 

 onto 
\begin_inset Formula \( \hbox{\textbf{B}}\)
\end_inset 

, (b) 
\begin_inset Formula \( \hbox{\textbf{B}}\)
\end_inset 

 onto 
\begin_inset Formula \( \hbox{\textbf{A}}\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 Find an equation of the plane, in the form 
\begin_inset Formula \( \br = \ba +\lambda \bb+ \mu \bc\)
\end_inset 

, through the points with position vectors 
\begin_inset Formula \( \bA\)
\end_inset 

, 
\begin_inset Formula \( \bB\)
\end_inset 

 and 
\begin_inset Formula \( \bC\)
\end_inset 

.
 Show that the normal to this plane is 
\begin_inset Formula \( (1,0,1)\)
\end_inset 

 and find an equation of the plane in the form 
\begin_inset Formula \( \br.\bn = p\)
\end_inset 

, where 
\begin_inset Formula \( \bn \)
\end_inset 

 is a unit vector. 
\layout Standard


\latex latex 

\backslash note{The projection of  
\backslash (
\backslash hbox{
\backslash textbf{A}}
\backslash ) onto 
\backslash (
\backslash hbox{
\backslash textbf{B}}
\backslash ) is the 
\newline 
component of 
\backslash (
\backslash textbf{A}
\backslash ) in the direction of the vector 
\backslash (
\backslash textbf{B}
\backslash ).}
\latex default 

\layout Section*

Differentiation
\layout Standard

Differentiation of standard functions, products, quotients, implicit function
 expressions, and functions of a function (using the chain rule) should
 be routine. 
\latex latex 

\backslash nq{Direct differentiation}
\latex default 
 Differentiate 
\begin_inset Formula \( y(x)\)
\end_inset 

 with respect to 
\begin_inset Formula \( x\)
\end_inset 

 in the following cases: 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y=\ln \left( x+\surd(1+x^2)\right)\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y= a^x\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y=x^x\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds y=\sin^{-1} \frac x {\sqrt{1+x^2}}\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\latex default 

\latex latex 

\backslash notes 
\latex default 
Simplify your answer to part (i).
 For (ii), see the definition in question 5(iv).
 Can you see why the answer to (iv) is surprisingly simple? 
\layout Standard


\latex latex 

\backslash nq{Parametric differentiation}
\latex default 
Show that if 
\begin_inset Formula \( x=a\cos\theta\)
\end_inset 

, 
\begin_inset Formula \( y=b\sin\theta\)
\end_inset 

 then 
\begin_inset Formula \( \ds \frac{\d{^}2y}{\d {x}^2}<0\)
\end_inset 

 for 
\begin_inset Formula \( y>0\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Stationary points}
\latex default 
Find the stationary points of the function 
\begin_inset Formula \[ 
\f(x) = \frac{x}{x^2+a^2},
\]
\end_inset 

 where 
\begin_inset Formula \( a>0\)
\end_inset 

, classifying them as either maximum or minimum.
 Sketch the curve (without using a calculator). 
\layout Standard


\latex latex 

\backslash newpage
\newline 

\latex default 

\layout Section*

Integration
\layout Standard

You need to be able to recognise standard integrals (without having to leaf
 through a formula book) and evaluate them (referring to your formula book,
 if necessary).
 You need to be familiar with the techniques of integration by parts and
 by substitution. 
\layout Standard


\latex latex 

\backslash nq {Indefinite integrals}
\latex default 
 Calculate the following integrals: 
\layout Standard


\latex latex 

\backslash bt 
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int a^x \, \d {x}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \frac{1}{x^2 -2x+6}\,\d {x}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int e^{ax} \cos (bx)\, \d {x}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int e^{ax}e^{ibx}\, \d {x} \)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(v)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \frac{1}{1-x^3}\,\d {x}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (vi)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \cosec x\, \d {x} \)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(vii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \sec x\, \d {x} \)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
 (viii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \frac{1}{ \surd(c^2 + m^2y^2)}\,\d {y}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(ix) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \int \tan^{-1} x\, \d {x}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(x) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds  \int x^3 e^{x^2} \d {x} \)
\end_inset 


\latex latex 

\backslash ,
\latex default 
. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash notes 
\latex default 
For part (i), see question 5(iv).
 For part (ii), see question 4(i).
 Note that you can obtain (iii) from (iv) by taking the real part.
 Use partial fractions for part (v).
 For (vi) and (vii), use the substitution 
\begin_inset Formula \( t= \tan (x/2)\)
\end_inset 

 rather than the trick of multiplying top and bottom by e.g. 
\begin_inset Formula \( \sec x + \tan x\)
\end_inset 

.
 Try also deriving (vii) from (vi) by means of the substitution 
\begin_inset Formula \( y=\pi/2 -x\)
\end_inset 

.
 For (viii), either substitute 
\begin_inset Formula \( my = c\sinh \theta\)
\end_inset 

 or, if you are not keen on hyperbolic functions, see question 24(i).
 Use integration by parts for (ix) and (x). 
\layout Section*

Differential equations
\layout Standard

There is a course on differential equations in the first term for which
 very little knowledge is assumed. 
\layout Standard


\latex latex 

\backslash nq{First order equations}
\latex default 
 Find the general solution (i.e.
 with a constant of integration) of the following equations. 
\latex latex 

\backslash bt
\newline 

\latex default 

\latex latex 

\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds y\frac{\d {y}}{\d {x}} = x\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{\d {y}}{\d {x}} = my\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
;
\newline 

\latex latex 
 
\backslash >
\latex default 
(iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds x \frac{\d {y}}{\d {x}} = y +1\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{\d {y}}{\d {x}} -y\tan x  = 1\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\newline 

\latex latex 
 
\backslash >
\latex default 
(v)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{\d {y}}{\d {x}} = \sqrt{c^2-k^2y^2}\)
\end_inset 


\latex latex 

\backslash ,
\latex default 
; 
\latex latex 

\backslash >
\latex default 
(vi)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds z\frac{\d {z}}{\d {y}} + k^2 y=0\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\latex default 

\latex latex 

\backslash note 
\latex default 
For part (iv), find a function (`integrating factor') 
\begin_inset Formula \( w(x)\)
\end_inset 

 such that the equation can be written 
\begin_inset Formula \( \ds \frac{\d {(}wy)}{\d {x}} = w\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash nq{Second order equations}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Show, by substituting 
\begin_inset Formula \( y=e^{mx}\)
\end_inset 

 into the equation, that there are two values of 
\begin_inset Formula \( m\)
\end_inset 

 for which 
\begin_inset Formula \( e^{mx}\)
\end_inset 

 satisfies 
\begin_inset Formula \[ 
a \frac{\d{^}2 y}{\d {x}^2} +b \frac{\d {y}}{\d {x}} +c y =0,
\eqno(*)\]
\end_inset 

 where 
\begin_inset Formula \( a\)
\end_inset 

, 
\begin_inset Formula \( b\)
\end_inset 

 and 
\begin_inset Formula \( c\)
\end_inset 

 are constants. 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Show (by substitution) that if both 
\begin_inset Formula \( y_1\)
\end_inset 

 and 
\begin_inset Formula \( y_2\)
\end_inset 

 satisfy the equation 
\begin_inset Formula \( (*)\)
\end_inset 

, then so also does the function 
\begin_inset Formula \( y\)
\end_inset 

 defined by 
\begin_inset Formula \( y=Ay_1+By_2\)
\end_inset 

, where 
\begin_inset Formula \( A\)
\end_inset 

 and 
\begin_inset Formula \( B\)
\end_inset 

 are constants. 
\layout Standard


\latex latex 

\backslash subq{iii}
\latex default 
 Find the two values of 
\begin_inset Formula \( m\)
\end_inset 

 for which 
\begin_inset Formula \( e^{mx}\)
\end_inset 

 satisfies the equation 
\begin_inset Formula \[ 
\frac{\d{^}2 y}{\d {x}^2} +3 \frac{\d {y}}{\d {x}} +2 y =0
\]
\end_inset 

 and write down a solution that contains two arbitrary constants. 
\layout Standard

You have just solved the above equation (
\begin_inset Formula \( y''+3y'+2y=0\)
\end_inset 

) by guessing solutions.
 Instead, let 
\begin_inset Formula \( z=y'+2y\)
\end_inset 

.
 Then show that 
\begin_inset Formula \( z'+z=0\)
\end_inset 

, solve for 
\begin_inset Formula \( z\)
\end_inset 

 and then solve the resulting first-order differential equation of 
\begin_inset Formula \( y\)
\end_inset 

, thereby proving that the solution you found above is in fact the most
 general solution. 
\layout Standard


\latex latex 

\backslash subq{iv}
\latex default 
 Show by substitution that 
\begin_inset Formula \( e^{px}\sin qx\)
\end_inset 

 satisfies the equation 
\begin_inset Formula \[ 
\frac{\d{^}2y}{\d {x}^2} -2 p \frac {\d {y}}{\d {x}} +(p^2+q^2)y=0.
\]
\end_inset 

 Find the general solution by the method outlined in the second paragraph
 of (iii) above. 
\layout Standard


\latex latex 

\backslash nq{Simple harmonic motion}
\latex default 

\latex latex 

\backslash subq{i}
\latex default 
 Show that the equation 
\begin_inset Formula \[ 
\frac{\d {^}2y}{\d {x}^2}+k^2 y=0
\]
\end_inset 

 can be written as 
\begin_inset Formula \( \ds z\frac{\d {z}}{\d {y}} + k^2 y=0\)
\end_inset 

, where 
\begin_inset Formula \( \ds z=\frac{\d {y}}{\d {x}}\)
\end_inset 

.
 Show that 
\begin_inset Formula \( z=\surd (c^2 - k^2y^2)\)
\end_inset 

, where 
\begin_inset Formula \( c\)
\end_inset 

 is a constant of integration, (compare questions 28(v) and 28(vi)) and
 hence show that 
\begin_inset Formula \[ 
y=R \sin k(x-x_0),
\]
\end_inset 

 where 
\begin_inset Formula \( R\)
\end_inset 

 and 
\begin_inset Formula \( x_0\)
\end_inset 

 are constants.
 (
\begin_inset Formula \( R= c/k\)
\end_inset 

 and 
\begin_inset Formula \( x_0 \)
\end_inset 

 is a new constant of integration.) 
\layout Standard


\latex latex 

\backslash subq{ii}
\latex default 
 Repeat the steps of part (i) to show that the general solution of 
\begin_inset Formula \[ 
  \frac{\d {^}2 y}{\d {x}^2} -k^2 y=0
\]
\end_inset 

 is 
\begin_inset Formula \[ 
y = R\sinh k(x-x_0).
\]
\end_inset 

 Show also that this can be written in the form 
\layout Standard


\begin_inset Formula \[ 
y = Ae^{kx} + Be^{-kx}
.
\]
\end_inset 


\layout Standard


\latex latex 

\backslash newpage
\newline 

\latex default 

\layout Section*

Answers
\layout Standard


\latex latex 

\backslash setcounter{qnumber}{0}
\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( (x-2)(x-1)\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( 3x(x-2)(x+1)\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
(iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( ( x- \half(1-\surd5))(x-\half(1+\surd5))\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( (x-1)(x^2+x+1)\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
(v)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( (x-3)(x-1)^2(x+2)\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash na
\newline 

\latex default 

\begin_inset Formula \( x<-1\)
\end_inset 

 or 
\begin_inset Formula \( 0<x<3\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac1{x-1}-\frac1{x+1}\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( \ds\frac13\left( \frac1{x+1}-\frac{x-2}{x^2-x+1}\right)\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac1{(x+1)^2}-\frac1{x+1} +\frac1{x-2}\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(iv)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds 1-\frac1{x-2}+\frac2{x+1}\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash na 
\latex default 
(i) 5; 
\latex latex 

\backslash  
\latex default 
 (ii) 2; 
\latex latex 

\backslash  
\latex default 
(iii) 
\begin_inset Formula \( -3\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 
(smallest when 
\begin_inset Formula \( \sin x=-1\)
\end_inset 

). 
\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac{3^k2^{10-k}10!}{k!(10-k)!}\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( 1+6x+21x^2 +50x^3 +90x^4\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 Same as (ii) 
\latex latex 

\backslash >
\latex default 
(iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  \surd2 \left(1+\frac{1}{4} x-\frac1{32}x^2+\frac1{128}x^3\right)\)
\end_inset 

.
\newline 

\latex latex 
 
\backslash et
\newline 

\latex default 

\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x +\frac13 x^3 +\frac2{15}x^5\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( x-\frac23x^3 + \frac{2}{15}x^5 \)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( 1+\frac1{2}x^2 +\frac{1}{24}x^4\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
(v) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( 1-\frac13x^2\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( 2475\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ii) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( 12\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds\frac{\sin\theta}{\cos2\theta}\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
 (
\begin_inset Formula \( -\pi/4+n\pi<\theta<\pi/4+n\pi\)
\end_inset 

) 
\latex latex 

\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 about 42 times. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (ia)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( n\pi + (-1)^n\pi/6\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(ib) 
\latex latex 

\backslash >
\latex default 

\begin_inset Formula \( n\pi/2 +\pi/4, \,\,2n\pi\pm 2\pi/3\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash na 
\latex default 

\begin_inset Formula \( |z_1| = \surd2\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( \arg z_1 = -\pi/4\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 
 
\begin_inset Formula \( |z_2| = 2\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( \arg z_2=5\pi/6\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash na 
\latex default 
(iii) 1, 
\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( e^{2i\pi/3}\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( e^{-2i\pi/3}\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 
or 1, 
\begin_inset Formula \( (-1\pm
i\surd3)/2\)
\end_inset 

.
\latex latex 

\backslash addtocounter{qnumber}{6}
\latex default 

\latex latex 
 
\backslash na 
\latex default 
(iii) 
\begin_inset Formula \( \pmatrix{3&-2\cr -1 & 1\cr}\)
\end_inset 

. 
\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
(i)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \br = \pmatrix{1\cr 0\cr 1\cr} +\lambda \pmatrix{1\cr 1\cr
-1\cr}\)
\end_inset 


\latex latex 

\backslash >
\latex default 
 (ii)
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x-1=y=1-z\)
\end_inset 


\latex latex 
 
\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash na 
\latex default 
(i) 
\latex latex 

\backslash  
\latex default 
 
\begin_inset Formula \( |\bA|>|\bC|>|\bB|\)
\end_inset 

; 
\latex latex 

\backslash  
\latex default 

\latex latex 

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\latex default 

\latex latex 

\backslash  
\latex default 
 (ii) 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( \cos^{-1} 17/(2\sqrt{91})\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash 
\latex default 
 
\begin_inset Formula \( \cos^{-1} 15/(2\sqrt{77})\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\latex latex 

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\latex default 

\latex latex 

\backslash 
\latex default 
 (iii) 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( 17/\sqrt{14}\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( 17/\sqrt{26}\)
\end_inset 


\newline 

\latex latex 
 
\backslash hbox 
\latex default 
to 
\latex latex 

\backslash parindent{}
\latex default 

\hfill 
 (iv) 
\begin_inset Formula \( \br=\pmatrix{1\cr3\cr4} +\lambda \pmatrix{1\cr -2\cr-1\cr}
+\mu\pmatrix{2\cr 0\cr -2}\)
\end_inset 

, 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash 
\latex default 
 
\begin_inset Formula \( \br.\pmatrix{1/\surd2\cr0\cr1/\surd2\cr}=5/\surd2\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  \ds \frac{1}{\sqrt{x^2+1}}\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  a^x \ln a \)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
(iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x^x(1+\ln x)\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
(iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{1}{1+x^2}\)
\end_inset 


\latex latex 

\backslash qquad 
\latex default 
(
\begin_inset Formula \( \ds \sin^{-1}\frac x {\sqrt{x^2+1}}=\tan^{-1}x\)
\end_inset 

). 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash na 
\newline 

\latex default 
Maximum at 
\begin_inset Formula \( \ds \big(a, \frac1 {2a}\big)\)
\end_inset 

, 
\latex latex 

\backslash 
\latex default 
 minimum at 
\begin_inset Formula \( \ds \big(-a, -\frac1 {2a}\big)\)
\end_inset 

. 
\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac1{\ln a} a^x+C\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac 1{\surd5} \tan^{-1} \frac{x-1}{\surd5}+C\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{1}{a^2+b^2} (a\cos bx + b \sin bx)e^{ax} +C\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac{1}{a+ib} e^{(a+ib)x} +C\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (v) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ds \frac16 \ln \frac{x^2+x+1}{(x-1)^2}
+\frac1{\surd3}\tan^{-1}\left(\frac{2x+1}{\surd3}\right) +C\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (vi) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \ln \tan (x/2)+C\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (vii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  \ds \ln\left(\frac{1+\tan (x/2)}{1-\tan(x/2)}\right)+C\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (viii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  m^{-1} \left( \ln(my) + \sqrt{m^2y^2 +c^2}\right)+C\)
\end_inset 


\latex latex 
 
\backslash  
\latex default 

\latex latex 

\backslash 
\latex default 
 or 
\latex latex 

\backslash  
\latex default 

\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( m^{-1} \sinh^{-1}(my/c)+C\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (ix) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( x\tan^{-1} x - \half \ln (1+x^2)+C\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (x) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( \half(x^2-1)e^{x^2}+C\)
\end_inset 


\latex latex 
 
\backslash et 
\newline 

\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
 (i) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y^2 =x^2 +C\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (ii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y=Ce^{mx}\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y=Cx-1\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( y=C\sec x +\tan x\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (v) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \(  ky = c \sin k(x-x_0)\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (vi) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( z^2+k^2y^2=C\)
\end_inset 


\latex latex 
 
\backslash et
\newline 

\newline 

\newline 

\latex default 

\layout Standard


\latex latex 

\backslash stepcounter{qnumber}
\latex default 

\latex latex 
 
\backslash bt{
\backslash textbf{
\backslash theqnumber.}}
\latex default 

\latex latex 
 
\backslash >
\latex default 
(i) 
\latex latex 

\backslash >
\latex default 
 Solution of 
\begin_inset Formula \( am^2+bm+c=0\)
\end_inset 


\latex latex 
 
\backslash >
\latex default 
 (iii) 
\latex latex 

\backslash >
\latex default 
 
\begin_inset Formula \( m=-1\)
\end_inset 

 and 
\begin_inset Formula \( m=-2\)
\end_inset 

; 
\latex latex 

\backslash  
\latex default 

\begin_inset Formula \( y=Ae^{-x} +B e^{-2x}\)
\end_inset 


\newline 

\latex latex 
 
\backslash >
\latex default 
 (iv) 
\latex latex 

\backslash >
\latex default 
 Set 
\begin_inset Formula \( z= y' -(p+iq)y\)
\end_inset 

. 
\latex latex 

\backslash et
\newline 

\newline 

\latex default 

\the_end