These Ten-Year-Series (TYS) worked solutions with video explanations for 2001 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
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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(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 $.
(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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A random variable $X$ has the probability distribution given in the following table:
(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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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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