# 2008 A Level H2 Math

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Paper 1
Paper 2
##### 2008 A Level H2 Math Paper 1 Question 1

The diagram shows the curve with equation $y={{x}^{2}}$. The area of the region bounded by the curve, the lines $x=1$, $x=2$ and the $x$-axis is equal to the area of the region bounded by the curve, the lines $y=a$, $y=4$ and the $y$-axis, where $a<4$. Find the value of $a$.

[4]

##### 2008 A Level H2 Math Paper 1 Question 10 [Modified] (ii)

A student puts $\$10$on 1 January 2009 into a bank account which pays compound interest at a rate of$2\%$per month on the last day of each month. She puts a further$\$10$ into the account on the first day of each subsequent month.

(a)

How much in total is in the account at the end of $2$ years?

(a) How much in total is in the account at the end of $2$ years?

(b)

After how many complete months will the total in the account first exceed $\$2000$? (b) After how many complete months will the total in the account first exceed$\$2000$?

##### 2008 A Level H2 Math Paper 1 Question 11 [Modified]

The equations of three planes ${{p}_{1}}$, ${{p}_{2}}$, ${{p}_{3}}$ are

$2x-5y+3z=3$
$3x+2y-5z=-5$
$5x+\lambda y+17z=\mu$

respectively, where $\lambda$ and $\mu$ are constants. The planes ${{p}_{1}}$ and ${{p}_{2}}$ intersect in a line $\ell$.

(i)

Find a vector equation of $\ell$.

(i) Find a vector equation of $\ell$.

(ii)

Given that the line $\ell$ is contained in the plane ${{p}_{3}}$, find $\lambda$ and $\mu$.

(ii) Given that the line $\ell$ is contained in the plane ${{p}_{3}}$, find $\lambda$ and $\mu$.

(iii)

Given instead that the line $\ell$ does not intersect ${{p}_{3}}$, what can be said about the values of $\lambda$ and $\mu$?

(iii) Given instead that the line $\ell$ does not intersect ${{p}_{3}}$, what can be said about the values of $\lambda$ and $\mu$?

(iv)

If the equation of ${{p}_{3}}$ is $5x-22y+17z=5$, find the shortest distance between $\ell$ and plane ${{p}_{3}}$.

(iv) If the equation of ${{p}_{3}}$ is $5x-22y+17z=5$, find the shortest distance between $\ell$ and plane ${{p}_{3}}$.

(v)

Find the Cartesian equation of the plane which contains $\ell$ and the point $(1,\,-1,\,3)$.

(v) Find the Cartesian equation of the plane which contains $\ell$ and the point $(1,\,-1,\,3)$.

##### 2008 A Level H2 Math Paper 2 Question 1Â

Let $\text{f}\left( x \right)={{\text{e}}^{x}}\sin x$.

(i)

Sketch the graph of $y=\text{f}\left( x \right)$ for $-3\le x\le 3$.

(i) Sketch the graph of $y=\text{f}\left( x \right)$ for $-3\le x\le 3$.

(ii)

Using MF26, find the series expansion of $\text{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.

(ii) Using MF26, find the series expansion of $\text{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.

Denote the answer to part (ii) by $\text{g}\left( x \right)$.

(iii)

On the same diagram as in part (i), sketch the graph of $y=\text{g}\left( x \right)$. Label the two graphs clearly.

(iii) On the same diagram as in part (i), sketch the graph of $y=\text{g}\left( x \right)$. Label the two graphs clearly.

(iv)

Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\text{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\text{f}\left( x \right)$.

(iv) Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\text{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\text{f}\left( x \right)$.