These Ten-Year-Series (TYS) worked solutions with video explanations for 1995 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
Use the substitution $u={{e}^{x}}$ to show that $\int_{0}^{1}{\frac{{{\text{e}}^{x}}-1}{{{\text{e}}^{x}}+1}}\text{ d}x=\int_{1}^{\text{e}}{\frac{u-1}{u\left( u+1 \right)}}\text{ d}u$. Hence, by using partial fractions, find the exact value of $\int_{0}^{1}{\frac{{{\text{e}}^{x}}-1}{{{\text{e}}^{x}}+1}}\text{ d}x$. Suggest an alternative method.
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A curve $C$ has equation $y=\frac{1}{\sqrt{4-{{x}^{2}}}}$ for $-1\le x\le 1$. The region $R$ is enclosed by $C$, the $x$- axis and the lines $x=-1$ and $x=1$.
(i)
Find the exact value of the area of $R$.
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(i) Find the exact value of the area of $R$.
[3]
(ii)
Find the exact value of the volume generated when $R$ is rotated through two right angles about the $x$- axis.
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(ii) Find the exact value of the volume generated when $R$ is rotated through two right angles about the $x$- axis.
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(iii)
Show that the volume generated when $R$ is rotated through two right angles about the $y$- axis is $\pi \,(4-2\sqrt{3})$.
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(iii) Show that the volume generated when $R$ is rotated through two right angles about the $y$- axis is $\pi \,(4-2\sqrt{3})$.
[4]
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A bag contains $3$ red balls and $3$ green balls. Balls are drawn from the bag at random, one by one and without replacement.
(a)
Show that the probability that the first $3$ balls drawn are red is $\frac{1}{20}$.
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(a) Show that the probability that the first $3$ balls drawn are red is $\frac{1}{20}$.
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(b)
Find the probability that the first $3$ balls drawn consist of $2$ red balls and $1$ green ball (in an order). Hence, or otherwise show that the probability that the third red appear on the fourth draw is $\frac{3}{20}$.
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(b) Find the probability that the first $3$ balls drawn consist of $2$ red balls and $1$ green ball (in an order). Hence, or otherwise show that the probability that the third red appear on the fourth draw is $\frac{3}{20}$.
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(c)
Find the probability that the third red ball appears on the fifth draw.
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(c) Find the probability that the third red ball appears on the fifth draw.
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(d)
The random variable $X$ is the number of draws required up to and including the one on which the third red ball appears. Tabulate the probability distribution of $X$ and find $\text{E}\left( X \right)$.
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(d) The random variable $X$ is the number of draws required up to and including the one on which the third red ball appears. Tabulate the probability distribution of $X$ and find $\text{E}\left( X \right)$.
[4]
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A curve is defined parametrically by $x=\frac{5}{2}\cos \theta ,\,\,y=\frac{3}{2}\sin \theta \,\,,$for $0\le \theta \le 2\pi $.
(i)
Find the coordinates of the axial intercepts algebraically.
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(i) Find the coordinates of the axial intercepts algebraically.
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(ii)
Show that the Cartesian equation of the curve is $\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=\frac{1}{4}$.
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(ii) Show that the Cartesian equation of the curve is $\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=\frac{1}{4}$.
[2]
(iii)
Show that the distance of the point $P$ on the curve with parameter $p$ to the point $(2,0)$ is $\frac{5}{2}-2\cos \,p$. Find a similar expression for the distance between $P$ and $(-2,0)$.
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(iii) Show that the distance of the point $P$ on the curve with parameter $p$ to the point $(2,0)$ is $\frac{5}{2}-2\cos \,p$. Find a similar expression for the distance between $P$ and $(-2,0)$.
[3]
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