1995 A Level H2 Math

2021

2022

2023

2020

2019

2018

2017

2016

2015

2014

2013

2012

2011

2010

2009

2008

2007

2006

2005

2004

2003

2002

2001

2000

1995

1994

1993

1991

1990

1986

1984

1983

1982

Â  Â 1971-1980

1978

1977

1973

Paper 1
Paper 2
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.

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]

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]

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]