Ten-Year-Series (TYS) Solutions | Past Year Exam Questions

1995 A Level H2 Math

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1995 A Level H2 Math Paper 1 Question 16 [Modified]

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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1995 A Level H2 Math Paper 1 Question 17

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 $R$.

[3]

(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.

[3]

(ii) Find the exact value of the volume generated when $R$ is rotated through two right angles about the $x$- axis.

[3]

(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]

(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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1995 A Level H2 Math Paper 2 Question 6

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 $\frac{1}{20}$.

[1]

(a) Show that the probability that the first $3$ balls drawn are red is $\frac{1}{20}$.

[1]

(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 $\frac{3}{20}$.

[2]

(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 $\frac{3}{20}$.

[2]

(c)

Find the probability that the third red ball appears on the fifth draw.

[2]

(c) Find the probability that the third red ball appears on the fifth draw.

[2]

(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]

(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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1995 A Level H2 Math Paper 2 Question 12 [Modified]

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.

[2]

(i) Find the coordinates of the axial intercepts algebraically.

[2]

(ii)

Show that the Cartesian equation of the curve is $\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=\frac{1}{4}$.

[2]

(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 $(-2,0)$.

[3]

(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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