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

2001 A Level H2 Math

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2001 A Level H2 Math Paper 1 Question 2

Find the equation of the tangent to the curve $y={{e}^{x}}$ at the point where $x=a$. Hence find the equation of the tangent to the curve $y={{e}^{x}}$ which passes through the origin. The straight line $y=mx$ intersects the curve $y={{e}^{x}}$ in two distinct points. Write down an inequality for $m$.

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2001 TYS 2001

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2001 A Level H2 Math Paper 1 Question 18 [Modified]

(i)

Find the exact value of $\int_{0}^{1}{\frac{1}{1+{{x}^{2}}}\,\text{d}x}$.

(i) Find the exact value of $\int_{0}^{1}{\frac{1}{1+{{x}^{2}}}\,\text{d}x}$.

(ii)

The graph of $y\,=\,\frac{1}{1+{{x}^{2}}}$, for $0\le x\le 1$, is shown in the diagram. Rectangles, each of width $\frac{1}{n}$, are drawn under the curve. Show that the total area $A$ of all $n$ rectangles is given by $A\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{2}}}+…+\frac{1}{2} \right\}$.

State the limit of $A$ as $n\to \infty $.

(ii) The graph of $y\,=\,\frac{1}{1+{{x}^{2}}}$, for $0\le x\le 1$, is shown in the diagram. Rectangles, each of width $\frac{1}{n}$, are drawn under the curve. Show that the total area $A$ of all $n$ rectangles is given by $A\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{2}}}+…+\frac{1}{2} \right\}$.

State the limit of $A$ as $n\to \infty $.

2001 TYS 2001

(iii)

It is given that $B\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{4}}}+…+\frac{1}{2} \right\}$. Find using the aid of a calculator the limit of $B$ as $n\to \infty $ correct to $3$ significant figures.

(iii) It is given that $B\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{4}}}+…+\frac{1}{2} \right\}$. Find using the aid of a calculator the limit of $B$ as $n\to \infty $ correct to $3$ significant figures.

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2001 TYS 20012001 TYS 2001
2001 TYS 20012001 TYS 2001
2001 TYS 2001
2001 TYS 20012001 TYS 2001
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2001 A Level H2 Math Paper 2 Question 6

A random variable $X$ has the probability distribution given in the following table:

2001 TYS 2001

(i)

Given that $\text{E}\left( X \right)=4$, find $p$ and $q$.

(i) Given that $\text{E}\left( X \right)=4$, find $p$ and $q$.

(ii)

Show that $\text{Var}\left( X \right)=1$.

(ii) Show that $\text{Var}\left( X \right)=1$.

(iii)

Find $\text{E}\left( \left| X-4 \right| \right)$.

(iii) Find $\text{E}\left( \left| X-4 \right| \right)$.

(iv)*

Ten independent observations of $X$ are taken. Find the probability that the value of $3$ is obtained at most $3$ times.

(iv)* Ten independent observations of $X$ are taken. Find the probability that the value of $3$ is obtained at most $3$ times.

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2001 TYS 2001


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2001 TYS 2001


2001 TYS 2001


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2001 A Level H2 Math Paper 2 Question 15 (b)

By expanding $\left( \mathbf{b}-\mathbf{c} \right)\,\centerdot \,\left( \mathbf{b}-\mathbf{c} \right)$, simplify

${{\left| \mathbf{b} \right|}^{2}}+{{\left| \mathbf{c} \right|}^{2}}-\left( \mathbf{b}-\mathbf{c} \right)\,\centerdot \,\left( \mathbf{b}-\mathbf{c} \right)$

By taking $\mathbf{b}=\overrightarrow{AC}$ and $\mathbf{c}=\overrightarrow{AB}$, deduce the cosine formula for triangle $ABC$.

[4]

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2001 A Level H2 Math Paper 2 Question 10
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