These Ten-Year-Series (TYS) worked solutions with video explanations for 2019 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
A function is defined as $\text{f}\left( x \right)=2{{x}^{3}}-6{{x}^{2}}+6x-12$.
(i)
Show that $\text{f}\left( x \right)$ can be written in the form $p\left\{ {{\left( x+q \right)}^{3}}+r \right\}$, where $p,q$ and $r$ are constants to be found.
[2]
(i) Show that $\text{f}\left( x \right)$ can be written in the form $p\left\{ {{\left( x+q \right)}^{3}}+r \right\}$, where $p,q$ and $r$ are constants to be found.
[2]
(ii)
Hence, or otherwise, describe a sequence of transformations that transform the graph of $y={{x}^{3}}$ onto the graph of $y=\text{f}\left( x \right)$.
[3]
(ii) Hence, or otherwise, describe a sequence of transformations that transform the graph of $y={{x}^{3}}$ onto the graph of $y=\text{f}\left( x \right)$.
[3]
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(i)
Sketch the graph of $y=\left| {{2}^{x}}-10 \right|$, giving the exact values of any points where the curve meets the axes.
[3]
(i) Sketch the graph of $y=\left| {{2}^{x}}-10 \right|$, giving the exact values of any points where the curve meets the axes.
[3]
(ii)
Without using a calculator, and showing all your working, find the exact interval, or intervals, for which $\left| {{2}^{x}}-10 \right|\le 6$. Give your answer in its simplest form.
[3]
(ii) Without using a calculator, and showing all your working, find the exact interval, or intervals, for which $\left| {{2}^{x}}-10 \right|\le 6$. Give your answer in its simplest form.
[3]
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(a)
An arithmetic series has first term $a$ and common difference $2a,$where $a\ne 0.$ A geometric series has first term $a$ and common ratio $2.$ The ${{k}^{\text{th}}}$ term of the geometric series is equal to the sum of the first $64$ terms of the arithmetic series. Find the value of $k.$
[3]
(a) An arithmetic series has first term $a$ and common difference $2a,$where $a\ne 0.$ A geometric series has first term $a$ and common ratio $2.$ The ${{k}^{\text{th}}}$ term of the geometric series is equal to the sum of the first $64$ terms of the arithmetic series. Find the value of $k.$
[3]
(b)
A geometric series has first term $f$ and common ratio $r,$ where $f$, $r\in \mathbb{R}$ and $f\ne 0$. The sum of the first four terms of the series is $0.$ Find the possible values of $f$ and $r.$ Find also, in terms of $f,$ the possible values of the sum of the first $n$ terms of the series.
[4]
(b) A geometric series has first term $f$ and common ratio $r,$ where $f$, $r\in \mathbb{R}$ and $f\ne 0$. The sum of the first four terms of the series is $0.$ Find the possible values of $f$ and $r.$ Find also, in terms of $f,$ the possible values of the sum of the first $n$ terms of the series.
[4]
(c)
The first term of an arithmetic series is negative. The sum of the first four terms of the series is $14$ and the product of the first four terms of the series is $0.$ Find the ${{11}^{\text{th}}}$ term of the series.
[4]
(c) The first term of an arithmetic series is negative. The sum of the first four terms of the series is $14$ and the product of the first four terms of the series is $0.$ Find the ${{11}^{\text{th}}}$ term of the series.
[4]
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(i)
The complex number $w$ can be expressed as $\cos \,\theta +\text{i sin}\,\theta .$
(i) The complex number $w$ can be expressed as $\cos \,\theta +\text{i sin}\,\theta .$
(a) Show that $w+\frac{1}{w}$ is a real number.
[2]
(b) Show that $\frac{w-1}{w+1}$ can be expressed as $k\tan \frac{1}{2}\theta ,$ where $k$ is a complex number to be found.
[4]
(ii)
[5]
[5]
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A curve $C$ has parametric equations
$x=a( 2\cos \theta -\cos 2\theta)$,
$y=a( 2\sin \theta -\sin 2\theta)$,
for $0\le \theta \le 2\pi .$
(i)
[2]
[2]
(ii)
[2]
[2]
(iii)
Show that the area enclosed by the $x$-axis, and the part of $C$ above the $x$-axis, is given by
$\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{a}^{2}}\left( 4{{\sin }^{2}}\theta -6\sin \theta \sin 2\theta +2{{\sin }^{2}}2\theta \right)}\,\text{d}\theta ,$
where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.
[3]
(iii) Show that the area enclosed by the $x$-axis, and the part of $C$ above the $x$-axis, is given by
$\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{a}^{2}}\left( 4{{\sin }^{2}}\theta -6\sin \theta \sin 2\theta +2{{\sin }^{2}}2\theta \right)}\,\text{d}\theta ,$
where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.
[3]
(iv)
[5]
[5]
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Scientists are investigating how the temperature of water changes in various environments.
(i)
The scientists begin by investigating how hot water cools.
The water is heated in a container and then placed in a room which is kept at a constant temperature of $16\,{}^\circ \text{C}$. The temperature of the water $t$ minutes after it is placed in the room is $\theta \,{}^\circ \text{C}\text{.}$ This temperature decreases at a rate proportional to the difference between the temperature of the water and the temperature of the room. The temperature of the water falls from a value of $80\,{}^\circ \text{C}$ to $32\,{}^\circ \text{C}$ in the first $30$ minutes.
(i) The scientists begin by investigating how hot water cools.
The water is heated in a container and then placed in a room which is kept at a constant temperature of $16\,{}^\circ \text{C}$. The temperature of the water $t$ minutes after it is placed in the room is $\theta \,{}^\circ \text{C}\text{.}$ This temperature decreases at a rate proportional to the difference between the temperature of the water and the temperature of the room. The temperature of the water falls from a value of $80\,{}^\circ \text{C}$ to $32\,{}^\circ \text{C}$ in the first $30$ minutes.
(a) Write down the differential equation for this situation. Solve this differential equation to get $\theta $ as an exact function of $t$.
[6]
(b) Find the temperature of the water $45$ minutes after it is placed in the room.
[1]
(ii)
The scientists then model the thickness of ice on a pond.
In winter the surface of the water in the pond freezes. Once the thickness of the ice reaches $3$ cm, it is safe to skate on the ice. The freezing of the water is modelled by a differential equation in which the rate of change of the thickness of ice is inversely proportional to its thickness. It is given that $T=0$ when $t=0.$ After $60$ minutes, the ice is $1$ cm thick.
Find the time from when freezing commences until the ice is first safe to skate on.
[6]
(ii) The scientists then model the thickness of ice on a pond.
In winter the surface of the water in the pond freezes. Once the thickness of the ice reaches $3$ cm, it is safe to skate on the ice. The freezing of the water is modelled by a differential equation in which the rate of change of the thickness of ice is inversely proportional to its thickness. It is given that $T=0$ when $t=0.$ After $60$ minutes, the ice is $1$ cm thick.
Find the time from when freezing commences until the ice is first safe to skate on.
[6]
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A ray of light passes from air into a material made into a rectangular prism. The ray of light is sent in direction $\left( \begin{matrix}-2 \\-3 \\ -6 \\\end{matrix} \right)$ from a light source at the point $P$ with coordinates $\left( 2,\,2,\,4 \right).$ The prism is placed so that the ray of light passes through the prism, entering at the point $Q$ and emerging at the point $R$ and is picked up by a sensor at point $S$ with coordinates $\left( -5,\,-6,\,-7 \right).$ The acute angle between $PQ$ and the normal to the top of the prism at $Q$ is $\theta $and the acute angle between $QR$ and the same normal is $\beta$ (see diagram).
It is given that the top of the prism is a part of the plane $x+y+z=1,$ and that the base of the prism is a part of the plane $x+y+z=-9.$ It is also given that the ray of light along $PQ$ is parallel to the ray of light along $RS$ so that $P,\,Q,\,R$ and $S$ lie in the same plane.
(i)
[5]
[5]
(ii)
[3]
[3]
(iii)
[3]
[3]
Snell’s law states that $\sin \theta =k\sin \beta$, where $k$ is a constant called the refractive index.
(iv)
[1]
[1]
(v)
[1]
[1]
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You are given that $I=\int{x{{\left( 1-x \right)}^{\frac{1}{2}}}}\text{d}x$.
(i)
Use integration by parts to find an expression for $I$.
[2]
(i) Use integration by parts to find an expression for $I$.
[2]
(ii)
Use the substitution ${{u}^{2}}=1-x$ to find another expression for $I$.
[2]
(ii) Use the substitution ${{u}^{2}}=1-x$ to find another expression for $I$.
[2]
(iii)
Show algebraically that your answers to parts (i) and (ii) differ by a constant.
[2]
(iii) Show algebraically that your answers to parts (i) and (ii) differ by a constant.
[2]
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(i)
Sketch the graph of $y=\frac{2-x}{3{{x}^{2}}+5x-8}$. Give the equations of the asymptotes and the coordinates of the point(s) where the curve crosses either axis.
[4]
(i) Sketch the graph of $y=\frac{2-x}{3{{x}^{2}}+5x-8}$. Give the equations of the asymptotes and the coordinates of the point(s) where the curve crosses either axis.
[4]
(ii)
Solve the inequality $\frac{2-x}{3{{x}^{2}}+5x-8}>0$.
[1]
(ii) Solve the inequality $\frac{2-x}{3{{x}^{2}}+5x-8}>0$.
[1]
(iii)
Hence solve the inequality $\frac{x-2}{3{{x}^{2}}+5x-8}>0$.
[1]
(iii) Hence solve the inequality $\frac{x-2}{3{{x}^{2}}+5x-8}>0$.
[1]
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A solid cylinder has radius $r$ cm, height $h$ cm and total surface area $900$ cm$^{2}$. Find the exact value of the maximum possible volume of the cylinder. Find also the ratio $r:h$ that gives this maximum volume.
[7]
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(i)
Given that $\text{f}\left( x \right)=\sec 2x$, find $\text{f}’\left( x \right)$ and $\text{f}”\left( x \right)$. Hence, or otherwise, find the Maclaurin series for $\text{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.
[5]
(i) Given that $\text{f}\left( x \right)=\sec 2x$, find $\text{f}’\left( x \right)$ and $\text{f}”\left( x \right)$. Hence, or otherwise, find the Maclaurin series for $\text{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.
[5]
(ii)
Use your series from part (i) to estimate $\int_{0}^{0.02}{\sec 2x\text{d}x}$, correct to $5$ decimal places.
[2]
(ii) Use your series from part (i) to estimate $\int_{0}^{0.02}{\sec 2x\text{d}x}$, correct to $5$ decimal places.
[2]
(iii)
Use your calculator to find $\int_{0}^{0.02}{\sec 2x\text{d}x}$, correct to $5$ decimal places.
[1]
(iii) Use your calculator to find $\int_{0}^{0.02}{\sec 2x\text{d}x}$, correct to $5$ decimal places.
[1]
(iv)
Comparing your answers to parts (ii) and (iii), and with reference to the value of $x$, comment on the accuracy of your approximations.
[2]
(iv) Comparing your answers to parts (ii) and (iii), and with reference to the value of $x$, comment on the accuracy of your approximations.
[2]
(v)
Explain why a Maclaurin series for $\text{g}\left( x \right)=\operatorname{cosec}2x$ cannot be found.
[1]
(v) Explain why a Maclaurin series for $\text{g}\left( x \right)=\operatorname{cosec}2x$ cannot be found.
[1]
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With reference to the origin $O,$ the points $A,\,B,\,C$ and $D$ are such that $\overrightarrow{OA}=\mathbf{a}\text{,}$$\overrightarrow{OB}=\mathbf{b}\text{,}$$\overrightarrow{OC}=2\mathbf{a}+4\mathbf{b}$and $\overrightarrow{OD}=\mathbf{b}+5\mathbf{a}\text{.}$ The lines $BD$ and $AC$ cross at $X$(see diagram).
(i)
[4]
[4]
The point $Y$ lies on $CD$ and is such that the points $O,\,X$ and $Y$ are collinear.
(ii)
[6]
[6]
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A company produces drinking mugs. It is known that, on average, $8\%$ of the mugs are faulty. Each day the quality manager collects $50$ of the mugs at random and checks them; the number of faulty mugs found is the random variable $F$.
(i)
State, in the context of the question, two assumptions needed to model $F$ by a binomial distribution.
[2]
(i) State, in the context of the question, two assumptions needed to model $F$ by a binomial distribution.
[2]
You are now given that $F$ can be modelled by a binomial distribution.
(ii)
Find the probability that, on a randomly chosen day, at least $7$ faulty mugs are found.
[2]
(ii) Find the probability that, on a randomly chosen day, at least $7$ faulty mugs are found.
[2]
(iii)
The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of $5$ days, at least $7$ faulty mugs are found on no more than $2$ days.
[2]
(iii) The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of $5$ days, at least $7$ faulty mugs are found on no more than $2$ days.
[2]
The company also makes saucers. The number of faulty saucers also follows a binomial distribution. The probability that a saucer is faulty is $p$. Faults on saucers are independent of faults on mugs.
(iv)
Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
(iv) Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
The mugs and saucers are sold in sets of $2$ randomly chosen mugs and $2$ randomly chosen saucers. The probability that a set contains at most $1$ faulty item is $0.97$.
(v)
Write down an equation satisfied by $p$. Hence find the value of $p$.
[4]
(v) Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
Assumptions:
Assumptions:
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