These Ten-Year-Series (TYS) worked solutions with video explanations for 2002 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-3ff9e3@troppus.
Given that
${{u}_{n}}={{\text{e}}^{nx}}-{{\text{e}}^{\left( n+1 \right)x}}$,
find $\sum\limits_{n=1}^{N}{{{u}_{n}}}$ in terms of $N$ and $x$.
[2]
Hence determine the set of values of $x$ for which the infinite series
${{u}_{1}}+{{u}_{2}}+{{u}_{3}}+…$
is convergent and give the sum to infinity for cases where this exists.
[3]
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The $n$ th term of a series is ${{2}^{n-2}}+3n$. Find the sum of the first $N$ terms.
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A girl wishes to phone a friend but cannot remember the exact number. She knows that it is a five digit number, that it is even, and that it consists of the digits $2$, $3$, $4$, $5$, and $6$ in some order. Using this information, find the largest number of different wrong telephone numbers she could try.
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$O$ is the origin and $A$ is the point on the curve $y=\tan x$ where $x=\frac{1}{3}\pi $.
(i)
Calculate the area of the region $R$ enclosed by the arc $OA$, the $x$-axis and the line $x=\frac{1}{3}\pi $, giving your answer in an exact form.
[3]
(i) Calculate the area of the region $R$ enclosed by the arc $OA$, the $x$-axis and the line $x=\frac{1}{3}\pi $, giving your answer in an exact form.
[3]
(ii)
The region $S$ is enclosed by the arc $OA$, the $y$-axis and the line $y=\sqrt{3}$. Find the volume of the solid of revolution formed when $S$ is rotated through $360{}^\circ $ about the $x$-axis, giving your answer in an exact form.
[6]
(ii) The region $S$ is enclosed by the arc $OA$, the $y$-axis and the line $y=\sqrt{3}$. Find the volume of the solid of revolution formed when $S$ is rotated through $360{}^\circ $ about the $x$-axis, giving your answer in an exact form.
[6]
(iii)
Find $\int_{0}^{\sqrt{3}}{{{\tan }^{-1}}y\,\text{d}y}$.
[3]
(iii) Find $\int_{0}^{\sqrt{3}}{{{\tan }^{-1}}y\,\text{d}y}$.
[3]
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Given that
$\arg (a+b\text{i})=\theta $ ,
Where $a>0$, $b>0$, find, in terms of $\theta $ and $\pi $, the value of
(i)
$\arg (-a+\text{i}b)$,
[1]
(i) $\arg (-a+\text{i}b)$,
[1]
(ii)
$\arg (-a-\text{i}b)$,
[1]
(ii) $\arg (-a-\text{i}b)$,
[1]
(iii)
$\arg (b+\text{i}a)$.
[1]
(iii) $\arg (b+\text{i}a)$.
[1]
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A room contains $n$ randomly chosen people
(i)
Assume that a randomly chosen person is equally likely to have been born on any day of the week. The probability that the people in the room were all born on different days of the week is denoted by $P$.
(a) Find $P$ in the case $n=3$.
(b) Show that $P=\frac{120}{343}$ in the case $n=4$.
(i) Assume that a randomly chosen person is equally likely to have been born on any day of the week. The probability that the people in the room were all born on different days of the week is denoted by $P$.
(a) Find $P$ in the case $n=3$.
(b) Show that $P=\frac{120}{343}$ in the case $n=4$.
(ii)
Assume now that a randomly chosen person is equally likely to have been born in any month of the year. Find the smallest value of $n$ such that the probability that the people in the room were all born in different months of the year is less than $0.5$.
[5]
(ii) Assume now that a randomly chosen person is equally likely to have been born in any month of the year. Find the smallest value of $n$ such that the probability that the people in the room were all born in different months of the year is less than $0.5$.
[5]
(iii)
Assume now that a randomly chosen person is equally likely to be born on any of the $365$ days in the year. It is given that, for the case $n=21$, the probability that the people were all born on different days of the year is $0.55631$, correct to $5$ places of decimals. Without the use of a graphic calculator, find the smallest value of $n$ such that at least two of the people were born on the same day of the year exceeds $\frac{1}{2}$.
(iii) Assume now that a randomly chosen person is equally likely to be born on any of the $365$ days in the year. It is given that, for the case $n=21$, the probability that the people were all born on different days of the year is $0.55631$, correct to $5$ places of decimals. Without the use of a graphic calculator, find the smallest value of $n$ such that at least two of the people were born on the same day of the year exceeds $\frac{1}{2}$.
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