These Ten-Year-Series (TYS) worked solutions with video explanations for 2007 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
Express
$\frac{2n+3}{n\left( n+1 \right)}$
in partial fractions and hence use the method of differences to find
${{\sum\limits_{n=1}^{N}{\frac{2n+3}{n\left( n+1 \right)}\left( \frac{1}{3} \right)}}^{n+1}}$
in terms of $N$.
[4]
Deduce the value of
$\sum\limits_{n=1}^{\infty }{\frac{2n+3}{n\left( n+1 \right)}{{\left( \frac{1}{3} \right)}^{n+1}}}$
[1]
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The current $I$ in an electric circuit at time $t$ satisfies the differential equation
$4\frac{\text{d}I}{\text{d}t}=2-3I$.
Find $I$ in terms of $t$, given that $I=2$ when $t=0$.
[6]
State what happens to the current in this circuit for large values of $t$.
[1]
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The line $l$ passes through the points $A$ and $B$ with coordinates of $(1,2,4)$ and $(-2,3,1)$ respectively.
The plane $p$ has equation $3x-y+2z=17$. Find
(i)
the coordinates of the point of intersection of $l$ and $p$,
[5]
(i) the coordinates of the point of intersection of $l$ and $p$,          Â
   [5]
(ii)
the acute angles between $l$ and $p$,
[3]
(ii)Â the acute angles between $l$ and $p$,
[3]
(iii)
the perpendicular distance from $A$ to $p$ .
(iii) the perpendicular distance from $A$ to $p$.
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A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.
(i)
Sketch the curve.
[2]
(i) Sketch the curve.
[2]
(ii)
The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.
[6]
(ii) The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.
[6]
(iii)
Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\text{t}\,\,\text{d}t}$, and use the substituition $\sin t=u$ to find this area.
[5]
(iii)
Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\text{t}\,\,\text{d}t}$, and use the substituition $\sin t=u$ to find this area.
[5]
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A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{1}}=1$ and
${{u}_{n+1}}={{u}_{n}}-\frac{2n+1}{{{n}^{2}}{{\left( n+1 \right)}^{2}}}$, for all $n\ge 1$.
(i)
Given that ${{u}_{n}}=\frac{1}{{{n}^{2}}}$, find $\sum\limits_{n=1}^{N}{\frac{2n+1}{{{n}^{2}}{{\left( n+1 \right)}^{2}}}}$.
[2]
(i) Given that ${{u}_{n}}=\frac{1}{{{n}^{2}}}$, find $\sum\limits_{n=1}^{N}{\frac{2n+1}{{{n}^{2}}{{\left( n+1 \right)}^{2}}}}$.
[2]
(ii)
Give a reason why the series in part (i) is convergent and state the sum to infinity.
[2]
(ii) Give a reason why the series in part (i) is convergent and state the sum to infinity.
[2]
(iii)
Use your answer to part (i) to find $\sum\limits_{n=2}^{N}{\frac{2n-1}{{{n}^{2}}{{\left( n-1 \right)}^{2}}}}$.
[2]
(iii) Use your answer to part (i) to find $\sum\limits_{n=2}^{N}{\frac{2n-1}{{{n}^{2}}{{\left( n-1 \right)}^{2}}}}$.
[2]
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Chickens and turkeys are sold by weight. The masses, in kg, of chickens and turkeys are modelled as having independent normal distributions with means and standard deviations as shown in the table.
Chickens are sold at $\$3$ per kg and turkeys at $\$5$ per kg.
(i)
Find the probability that a randomly chosen chicken has a selling price exceeding $\$7$.
(i) Find the probability that a randomly chosen chicken has a selling price exceeding $\$7$.
(ii)
Find the probability of the event that both a randomly chosen chicken has a selling price exceeding $\$7$ and a randomly chosen turkey has a selling price exceeding $\$55$.
(ii) Find the probability of the event that both a randomly chosen chicken has a selling price exceeding $\$7$ and a randomly chosen turkey has a selling price exceeding $\$55$.
(iii)
Find the probability that the total selling price of a randomly chosen chicken and a randomly chosen turkey is more than $\$62$.
(iii) Find the probability that the total selling price of a randomly chosen chicken and a randomly chosen turkey is more than $\$62$.
(iv)
Explain why the answer to part (iii) is greater that the answer to part (ii).
(iv) Explain why the answer to part (iii) is greater that the answer to part (ii).
(v)
Find the probability that the cost of sixteen chickens is more than the cost of $2$ turkeys.
(v) Find the probability that the cost of sixteen chickens is more than the cost of $2$ turkeys.Â
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