CIE Application of Differentiation (Additional Mathematics-2018)

$\newcommand{\D}{\displaystyle} \def\dydx{\displaystyle\frac{dy}{dx}}\def\frac{\dfrac}$
1 (CIE 2012, s, paper 11, question 5)
(i) Find the equation of the tangent to the curve $\D y = x^3 + 2x^2 - 3x + 4$ at the point where the curve crosses the y-axis. [4]
(ii) Find the coordinates of the point where this tangent meets the curve again. [3]

2 (CIE 2012, s, paper 11, question 10)
Variables $\D x$ and $\D y$ are such that $\D y = e^{2x} + e^{-2x} .$
(i) Find $\D \frac{dy}{dx}.$ [2]
(ii) By using the substitution $\D u = e^{2x} ,$ find the value of $\D y$ when $\dydx = 3.$ [4]
(iii) Given that $\D x$ is decreasing at the rate of 0.5 units $\D s^{-1},$ find the corresponding rate of changeof $\D y$ when $\D x = 1.$ [3]

3 (CIE 2012, s, paper 22, question 2)
(i) Given that $\D y =\sqrt{ (4x + 1)^3},$ find $\D \dydx.$ [2]
(ii) Hence find the approximate increase in $\D y$ as $\D x$ increases from 6 to $\D 6 + p,$ where $\D p$ is small. [2]

4 (CIE 2012, s, paper 22, question 8)
An open rectangular cardboard box with a square base is to have a volume of 256 cm$\D ^3.$ Find the dimensions of the box if the area of cardboard used is as small as possible. [7]

5 (CIE 2012, w, paper 12, question 11or)
A curve is such that $\D y = \frac{Ax^2+B}{x^2-2},$ where $\D A$ and $\D B$ are constants.
(i) Show that $\D\dydx=-\frac{2x(2A+B)}{(x^2-2)^2}.$  [4]

It is given that $\D y = -3$ and $\D \dydx = -10$ when $\D x = 1.$
(ii) Find the value of $\D A$ and of $\D B.$ [3]
(iii) Using your values of $\D A$ and $\D B,$ find the coordinates of the stationary point on the curve, and determine the nature of this stationary point. [4]


6 (CIE 2012, w, paper 13, question 2)
The rate of change of a variable $\D x$ with respect to time $\D t$ is $\D 4\cos^2t.$
(i) Find the rate of change of $\D x$ with respect to $\D t$ when $\D t = \frac{\pi}{6} .$ [1]

The rate of change of a variable $\D y$ with respect to time $\D t$ is $\D 3\sin t.$
(ii) Using your result from part (i), find the rate of change of $\D y$ with respect to $\D x$ when $\D t = \frac{\pi}{6}.$ [3]


7 (CIE 2012, w, paper 21, question 6)
The normal to the curve $\D y = x^3 + 6x^2 - 34x + 44$ at the point $\D P (2, 8)$ cuts the x-axis at $\D A$ and the y-axis at $\D B.$ Show that the mid-point of the line AB lies on the line $\D 4y = x + 9.$ [8]

8 (CIE 2012, w, paper 22, question 2)
The total surface area, $\D A$ cm$\D^2,$ of a solid cylinder with radius $\D r$ cm and height 5 cm is given by $\D A = 2\pi r^2 + 10\pi r.$ Given that $\D r$ is increasing at a rate of $\D \frac{0.2}{\pi}$ cm s$\D ^{-1},$ find the rate of increase of $\D A$ when $\D r$ is 6. [4]

9 (CIE 2012, w, paper 23, question 10) 

A track runs due east from $\D A$ to $\D B,$ a distance of 200 m. The point $\D C$ is 80 m due north of $\D B.$ A cyclist travels on the track from A to D, where D is $\D x$ m due west of B. The cyclist then travels in a straight line across rough ground from D to C. The cyclist travels at 10 m s$\D ^{-1}$ on the track and at 6 m $\D s^{-1}$ across rough ground.
(i) Show that the time taken, Ts, for the cyclist to travel from A to C is given by
$$T=\frac{200-x}{10}+\frac{\sqrt{(x^2+6400)}}{6}$$ [2]
(ii) Given that $\D x$ can vary, find the value of $\D x$ for which $\D T$ has a stationary value and the corresponding value of $\D T.$ [6]

10 (CIE 2013, s, paper 11, question 10)
The point $\D A,$ whose x-coordinate is 2, lies on the curve with equation $\D y= x^3- 4x^2+ x+ 1.$
(i) Find the equation of the tangent to the curve at A. [4]
This tangent meets the curve again at the point B.
(ii) Find the coordinates of B. [4]
(iii) Find the equation of the perpendicular bisector of the line AB. [4]

11 (CIE 2013, s, paper 12, question 6)
The normal to the curve $\D y + 2 = 3 \tan x,$ at the point on the curve where $\D x = \frac{3\pi}{4}$, cuts the y-axis at the point P. Find the coordinates of P. [6]


12 (CIE 2013, s, paper 21, question 3)
Variables $\D x$ and $\D y$ are related by the equation $\D y= 10- 4 \sin^2x$, where $\D 0\le x\le\frac{\pi}{2}.$
 Given that $\D x$ is increasing at a rate of 0.2 radians per second, find the corresponding rate of change of $\D y$ when $\D y = 8.$ [6]

13 (CIE 2013, s, paper 21, question 5) 

A piece of wire of length $96 \mathrm{~cm}$ is formed into the rectangular shape $P Q R S T U$ shown in the diagram. It is given that $P Q=T U=S R=x \mathrm{~cm} .$ It may be assumed that $P Q$ and $T U$ coincide and that $T S$ and $Q R$ have the same length.

(i) Show that the area, $A \mathrm{~cm}^{2}$, enclosed by the wire is given by $A=\frac{96 x-3 x^{2}}{2}$.[2]

(ii) Given that $x$ can vary, find the stationary value of $A$ and determine the nature of this stationary value. $[4]$

14 (CIE 2013, s, paper 21, question 6)

Find the equation of the normal to the curve $y=\frac{x^{2}+8}{x-2}$ at the point on the curve where $x=4$.

$[6]$

15 (CIE 2013, w, paper 13 , question 4) 

A curve has equation $y=\frac{\mathrm{e}^{2 x}}{(x+3)^{2}}$.

(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{A \mathrm{e}^{2 x}(x+2)}{(x+3)^{3}}$, where $A$ is a constant to be found.

(ii) Find the exact coordinates of the point on the curve where $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$.

16 (CIE $2013, \mathrm{w}$, paper 21, question 3$)$

(i) Given that $y=\left(\frac{1}{4} x-5\right)^{8}$, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.$[2]$

(ii) Hence find the approximate change in $y$ as $x$ increases from 12 to $12+p$, where $p$ is small.[2]

17 (CIE 2013, w, paper 21, question 7) 

The diagram shows a box in the shape of a cuboid with a square cross-section of side $x \mathrm{~cm}$. The volume of the box is $3500 \mathrm{~cm}^{3}$. Four pieces of tape are fastened round the box as shown. The pieces of tape are parallel to the edges of the box.

(i) Given that the total length of the four pieces of tape is $L \mathrm{~cm}$, show that $L=14 x+\frac{7000}{x^{2}} .[3]$

(ii) Given that $x$ can vary, find the stationary value of $L$ and determine the nature of this stationary value. $[5]$

18 ( CIE 2013, w, paper 23 , question 1)

Find the coordinates of the stationary points on the curve $y=x^{3}-6 x^{2}-36 x+16$.

19 (CIE 2013, w, paper 23, question 10) 

The diagram shows a container in the shape of a cone of height $120 \mathrm{~cm}$ and radius $30 \mathrm{~cm}$. Water is poured into the container at a rate of $20 \pi \mathrm{cm}^{3} \mathrm{~s}^{-1}$.

(i) At the instant when the depth of water in the cone is $h \mathrm{~cm}$ the volume of water in the cone is $V \mathrm{~cm}^{3}$. Show that $V=\frac{\pi h^{3}}{48}$.

(ii) Find the rate at which $h$ is increasing when $h=50$.

(iii) Find the rate at which the circular area of the water's surface is increasing when $h=50$. [4]

20 (CIE 2014, s, paper 12, question 9)

A solid circular cylinder has a base radius of $r \mathrm{~cm}$ and a volume of $4000 \mathrm{~cm}^{3}$.

(i) Show that the total surface area, $A \mathrm{~cm}^{2}$, of the cylinder is given by $A=\frac{8000}{r}+2 \pi r^{2}$.

(ii) Given that $r$ can vary, find the minimum total surface area of the cylinder, justifying that this area is a minimum.

21 (CIE 2014, s, paper 13, question 4) 

The diagram shows a thin square sheet of metal measuring $24 \mathrm{~cm}$ by $24 \mathrm{~cm}$. A square of side $x \mathrm{~cm}$ is cut off from each corner. The remainder is then folded to form an open box, $x \mathrm{~cm}$ deep, whose square base is shown shaded in the diagram.

(i) Show that the volume, $V \mathrm{~cm}^{3}$, of the box is given by $\quad V=4 x^{3}-96 x^{2}+576 x$.

(ii) Given that $x$ can vary, find the maximum volume of the box.

22 (CIE 2014, s, paper 13, question 6)

Find the equation of the normal to the curve $y=x\left(x^{2}-12\right)^{\frac{1}{3}} \quad$ at the point on the curve where $x=2$.

23 (CIE 2014, s, paper 21, question 8)

A sector of a circle of radius $r \mathrm{~cm}$ has an angle of $\theta$ radians, where $\theta<\pi .$ The perimeter of the sector is $30 \mathrm{~cm}$.

(i) Show that the area, $A \mathrm{~cm}^{2}$, of the sector is given by $A=15 r-r^{2}$.$[3]$

(ii) Given that $r$ can vary, find the maximum area of the sector.$[3]$

24 (CIE 2014, s, paper 22, question 12)

A curve has equation $y=x^{3}-9 x^{2}+24 x$.

(i) Find the set of values of $x$ for which $\frac{\mathrm{d} y}{\mathrm{~d} x} \geqslant 0 .$[4]

The normal to the curve at the point on the curve where $x=3$ cuts the $y$-axis at the point $P$.

(ii) Find the equation of the normal and the coordinates of $P$.

25 (CIE 2014, s, paper 23, question 8)

A curve is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=6 x^{2}-8 x+3$.

(i) Show that the curve has no stationary points.$[2]$

Given that the curve passes through the point $P(2,10)$,

(ii) find the equation of the tangent to the curve at the point $P$,

(iii) find the equation of the curve.$[4$

******

26 (CIE $2014, \mathrm{w}$, paper 11, question 1$)$

Find the coordinates of the stationary point on the curve $y=x^{2}+\frac{16}{x}$.

27 (CIE 2014, w, paper 23, question 9)

(i) Determine the coordinates and nature of each of the two turning points on the curve $\quad y=4 x+\frac{1}{x-2}$.

(ii) Find the equation of the normal to the curve at the point $(3,13)$ and find the $x$-coordinate of the point where this normal cuts the curve again.

28 (CIE 2015, s, paper 11, question 7) The point $A$, where $x=0$, lies on the curve $y=\frac{\ln \left(4 x^{2}+3\right)}{x-1}$. The normal to the curve at $A$ meets the $x$-axis at the point $B$.

(i) Find the equation of this normal.$[7]$

(ii) Find the area of the triangle $A O B$, where $O$ is the origin.[2]

29 (CIE 2015, s, paper 12, question 9) 

A curve has equation $y=4 x+3 \cos 2 x$. The normal to the curve at the point where $x=\frac{\pi}{4}$ meets the $x$-and $y$-axes at the points $A$ and $B$ respectively. Find the exact area of the triangle $A O B$, where $O$ is the origin. $\quad[8]$

30 (CIE 2015, s, paper 22, question 11) 

The diagram shows a cuboid of height $h$ units inside a right pyramid $O P Q R S$ of height 8 units and with square base of side 4 units. The base of the cuboid sits on the square base $P Q R S$ of the pyramid. Tho points $A, B, C$ and $D$ are corners of the cuboid and lie on the edges $O P, O Q, O R$ and $O S$, respectively, of the pyramid $O P Q R S$. The pyramids $O P Q R S$ and $O A B C D$ are similar.

(i) Find an expression for $A D$ in terms of $h$ and hence show that the volume $V$ of the cuboid is given by $V=\frac{h^{3}}{4}-4 h^{2}+16 h$ units $^{3}$.

(ii) Given that $h$ can vary, find the value of $h$ for which $V$ is a maximum.$[4]$

31 $(\mathrm{CIE} 2015, \mathrm{w}$, paper 11, question 5)

Variables $x$ and $y$ are such that $y=(x-3) \ln \left(2 x^{2}+1\right)$

(i) Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ when $x=2$.$[4]$

(ii) Hence find the approximate change in $y$ when $x$ changes from 2 to $2.03$$[2]$

32 (CIE $2015, \mathrm{w}$, paper 11, question 8)

Find the equation of the tangent to the curve $y=\frac{2 x-1}{\sqrt{x^{2}+5}}$ at the point where $x=2$.[7]

33 (CIE 2015, w, paper 13, question 5)

Find the equation of the normal to the curve $y=5 \tan x-3 \quad$ at the point where $x=\frac{\pi}{4}, \quad[5]$

34 (CIE 2015, w, paper 21, question 6) 

The figure shows part of the graph of $y=a+b \sin c x$.

(i) Find the value of cach of the integers $a, b$ and $c$.$[3]$

Using your values of $a, b$ and $c$ find

(ii) $\frac{\mathrm{d} y}{\mathrm{~d} x}$,$[2]$

(iii) the equation of the normal to the curve at $\left(\frac{\pi}{2}, 3\right)$.[3]

35 $(\mathrm{CIE} 2015, \mathrm{w}$, paper 21, question 7$)$ 

A cone, of height $8 \mathrm{~cm}$ and base radius $6 \mathrm{~cm}$, is placed over a cylinder of radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$ and is in contact with the cylinder along the cylinder's upper rim. The arrangement is symmetrical and the diagram shows a vertical cross-section through the vertex of the cone.

(i) Use similar triangles to express $h$ in terms of $r$.

(ii) Hence show that the volume, $V \mathrm{~cm}^{3}$, of the cylinder is given by $V=8 \pi r^{2}-\frac{4}{3} \pi r^{3} .$ $[1]$

(iii) Given that $r$ can vary, find the value of $r$ which gives a stationary value of $V$. Find this stationary value of $V$ in terms of $\pi$ and determine its nature. $[6]$

36 (CIE 2016, march, paper 22, question 11) A curve has equation $y=\frac{x}{x^{2}+1}$

(i) Find the coordinates of the stationary points of the curve. $[5]$

(ii) Show that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{p x^{3}+q x}{\left(x^{2}+1\right)^{3}}$, where $p$ and $q$ are integers to be found, and determine the nature of the stationary points of the curve. $[5]$

37 (CIE 2016, s, paper 11, question 3)

Find the equation of the normal to the curve $y=\ln \left(2 x^{2}-7\right)$ at the point where the curve crosses the positive $x$-axis. Give your answer in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers. [5]

38 (CIE 2016, s, paper 12, question 9) A curve passes through the point $\left(2,-\frac{4}{3}\right)$ and is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=(3 x+10)^{-\frac{1}{2}}$.

(i) Find the equation of the curve.$[4]$

The normal to the curve, at the point where $x=5$, meets the line $y=-\frac{5}{3}$ at the point $P$.

(ii) Find the $x$-coordinate of $P$.$[6]$

39 (CIE 2016, s, paper 12, question 10) 

The diagram shows a badge, made of thin sheet metal, consisting of two semi-circular pieces, centres $B$ and $C$, each of radius $x \mathrm{~cm}$. They are attached to each other by a rectangular piece of thin sheet metal, $A B C D$, such that $A B$ and $C D$ are the radii of the semi-circular pieces and $A D=B C=y \mathrm{~cm}$.

(i) Given that the area of the badge is $20 \mathrm{~cm}^{2}$, show that the perimeter, $P \mathrm{~cm}$, of the badge is given by $P=2 x+\frac{40}{x}$

(ii) Given that $x$ can vary, find the minimum value of $P_{+}$justifying that this value is a minimum. $[5]$

40 (CIE 2016, s, paper 21, question 12) A curve has equation $y=\frac{2 x-5}{x-1}-12 x$

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Find $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$

(iii) Find the coordinates of the stationary points of the curve and determine their nature

41 (CIE 2016, s, paper 22, question 2) Variables $x$ and $y$ are related by the equation $y=\frac{5 x-1}{3-x}$.

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$, simplifying your answer.

(ii) Hence find the approximate change in $x$ when $y$ increases from 9 by the small amount $0.07 .$

42 (CIE 2016, w, paper 11, question 5)

(i) Find the equation of the normal to the curve $y=\frac{1}{2} \ln (3 x+2)$ at the point $P$ where $x=-\frac{1}{3}$. [4]

The normal to the curve at the point $P$ intersects the $y$-axis at the point $Q .$ The curve $y=\frac{1}{2} \ln (3 x+2)$ intersects the $y$-axis at the point $R$

(ii) Find the area of the triangle $P Q R .$

43 (CIE $2016, \mathrm{w}$, paper 13 , question 10 )

A curve $y=\mathrm{f}(x)$ is such that $\mathrm{f}^{\prime}(x)=6 x-8 \mathrm{e}^{2 x}$.

(i) Given that the curve passes through the point $P(0,-3)$, find the equation of the curve.[5] 

The normal to the curve $y=\mathrm{f}(x)$ at $P$ meets the line $y=2-3 x$ at the point $Q$.

(ii) Find the area of the triangle $O P Q$, where $O$ is the origin.$[5]$

44 (CIE $2016, \mathrm{w}$, paper 21, question 5$)$

The curve with equation $y=x^{3}+2 x^{2}-7 x+2$ passes through the point $A(-2,16)$. Find

(i) the equation of the tangent to the curve at the point $A$,$[3]$

(ii) the coordinates of the point where this tangent meets the curve again.$[5]$

45 (CIE 2016, w, paper 21, question 7) 

(i) Show that the area, $A \mathrm{~cm}^{2}$, of the trapezium $P Q R S$ is given by $A=(7+x) \sqrt{9-x^{2}}$.[2]

(ii) Given that $x$ can vary, find the stationary value of $A$.$[7]$

46 (CIE $2016, \mathrm{w}$, paper 23, question 3) 

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\sin x}{1+\cos x}\right)=\frac{1}{1+\cos x}$. $[4]$

(ii) The diagram shows part of the graph of $y=\frac{1}{1+\cos x} .$ Use the result from part (i) to find the area enclosed by the graph and the lines $x=0, x=2$ and $y=0$.

47 (CIE 2016, w, paper 23, question 7) 

In this question all lengths are in metres.

A conical tent is to be made with height $h$, base radius $r$ and slant height $0.5 h+2$, as shown in the diagram.

(i) Show that $r^{2}=2 h+4-0.75 h^{2}$.$[2]$

The volume of the tent, $V$, is given by $\frac{1}{3} \pi r^{2} h$.

(ii) Given that $h$ can vary find, correct to 2 decimal places, the value of $h$ which gives a stationary value of $V_{.} \quad[5]$

(iii) Determine the nature of this stationary value.$[2]$

48 (CIE 2017, march, paper 12, question 8) 

A curve is such that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=4 \sin 2 x$. The curve has a gradient of 5 at the point where $x=\frac{\pi}{2}$.

(i) Find an expression for the gradient of the curve at the point $(x, y)$. The curve passes through the point $P\left(\frac{\pi}{12},-\frac{1}{2}\right)$.

(ii) Find the equation of the curve.

(iii) Find the equation of the normal to the curve at the point $P$, giving your answer in the form $y=m x+c$, where $m$ and $c$ are constants correct to 3 decimal places.[3]

49 (CIE 2017, march, paper 22, question 12) 

The diagram shows a shape made by cutting an equilateral triangle out of a rectangle of width $x \mathrm{~cm}$.

The perimeter of the shape is $20 \mathrm{~cm}$.

(i) Show that the area, $A \mathrm{~cm}^{2}$, of the shape is given by

(ii) Given that $x$ can vary, find the value of $x$ which produces the maximum area and calculate this maximum area. Give your answers to 2 significant figures. [4]

50 (CIE 2017, s, paper 11, question 1)

The line $y=k x-5$, where $k$ is a positive constant, is a tangent to the curve $y=x^{2}+4 x$ at the point $A$.

(i) Find the exact value of $k$.

(ii) Find the gradient of the normal to the curve at the point $A$, giving your answer in the form $a+b \sqrt{5}$, where $a$ and $b$ are constants.

51 (CIE 2017, s, paper 11, question 7) Show that the curve $y=\left(3 x^{2}+8\right)^{\frac{5}{3}}$ has only one stationary point. Find the coordinates of this stationar\} point and determine its nature. $[8]$

52 (CIE 2017, s, paper 13, question 11) A curve has equation $y=6 x-x \sqrt{x}$.

(i) Find the coordinates of the stationary point of the curve.$[4]$

(ii) Determine the nature of this stationary point.$[2]$

(iii) Find the approximate change in $y$ when $x$ increases from 4 to $4+h$, where $h$ is small.

53 (CIE $2017, \mathrm{~s}$, paper 21, question 1) Find the equation of the curve which passes through the point $(2,17)$ and for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=4 x^{3}+1$. [4]

54 (CIE 2017, w, paper 11, question 7)

(i) Write $\ln \left(\frac{2 x+1}{2 x-1}\right)$ as the difference of two logarithms.$[1]$

A curve has equation $y=\ln \left(\frac{2 x+1}{2 x-1}\right)+4 x$ for $x>\frac{1}{2}$.

(ii) Using your answer to part (i) show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{a x^{2}+b}{4 x^{2}-1}$, where $a$ and $b$ are integers.[4]

(iii) Hence find the $x$-coordinate of the stationary point on the curve.[2]

(iv) Determine the nature of this stationary point.$[2]$

55 (CIE 2017, w, paper 13, question 8) It is given that $y=(x-4)(3 x-1)^{\frac{5}{3}}$

(i) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=(3 x-1)^{\frac{1}{5}}(A x+B)$, where $A$ and $B$ are integers to be found.[5]

(ii) Hence find, in terms of $h$, where $h$ is small, the approximate change in $y$ when $x$ increases from 3 to $3+h$[3]

56 (CIE 2017,w, paper 22 , question 6) The volume of a closed cylinder of base radius $x \mathrm{~cm}$ and height $h \mathrm{~cm}$ is $500 \mathrm{~cm}^{3}$.

(i) Express $h$ in terms of $x$.

(ii) Show that the total surface area of the cylinder is given by $A=2 \pi x^{2}+\frac{1000}{x} \mathrm{~cm}^{2}$.

(iii) Given that $x$ can vary, find the stationary value of $A$ and show that this value is a minimum.

57 (CIE 2017, w, paper 22, question 7)

The gradient of the normal to a curve at the point with coordinates $(x, y)$ is given by $\frac{\sqrt{x}}{1-3 x}$.

(i) Find the equation of the curve, given that the curve passes through the point $(1,-10)$.[5]

(ii) Find, in the form $y=m x+c$, the equation of the tangent to the curve at the point where $x=4 . \quad$ [4]

58 (CIE 2018, march, paper 22, question 12) The volume, $V$, and surface area, $S$, of a sphere of radius $r$ are given by $V=\frac{4}{3} \pi r^{3}$ and $S=4 \pi r^{2}$ respectively.

The volume of a sphere increases at a rate of $200 \mathrm{~cm}^{3}$ per second. At the instant when the radius of the sphere is $10 \mathrm{~cm}$, find

(i) the rate of increasc of the radius of the sphere,

(ii) the rate of increase of the surface area of the sphere.

59 (CIE 2018, s, paper 21, question 12) 

In this question all lengths are in metres.

A water container is in the shape of a triangular prism. The diagrams show the container and its cross-section. The cross-section of the water in the container is an isosceles triangle $A B C$, with angle $A B C=$ angle $B A C=30^{\circ}$. The length of $A B$ is $x$ and the depth of water is $h$. The length of the container is 5 .

(i) Show that $x=2 \sqrt{3} h$ and hence find the volume of water in the container in terms of $h .$[3]

(ii) The container is filled at a rate of $0.5 \mathrm{~m}^{3}$ per minute. At the instant when $h$ is $0.25 \mathrm{~m}$, find

(a) the rate at which $h$ is increasing,

(b) the rate at which $x$ is increasing.[2]

Answers

$1. (i)y = -3x + 4,$
(ii)(-2,10)

2. (i)$\D 2e^{2x} - 2e^{-2x}$

(ii)$\D y = \frac{5}{2} (iii)-7.25$

3. (i) $\D6\sqrt{4x+1} (ii)30p$

4. x = 8; h = 4

5. (ii)A = 2;B = 1

(iii)x = 0; y = -:5, max

6. (i)3,(ii)0.5

7. A(18; 0);B(0; 9);M(9; 4:5)

8. 6.8

9. (ii)x = 60; T = 92/3

10. (i)y = 1 - 3x

(ii)(0; 1)

(iii)y + 2 = (x - 1)/3

11. (0,-4.61)

12. -0:8

13. (ii) $x=16, A=384$ max

14. $y=.5 x+10$

15. (i) $A=2($ ii $) x=-2, y=e^{-4}$

16. (i) $k=2$ (ii) $-256 p$

17. (ii) $x=10, L=210, \min$

18. $(-2,58),(6,200)$

19. (ii) $0.128$ (ii) $2.51$

20. $\mathrm{A}=1395,1390$

21. (ii) $\mathrm{V}=1024$

22. $4 y=3 x-22$

23. $r=7.5, A=56.25$

24. (i) $x \leq 2, x \geq 4$

(ii) $y=(1 / 3) x+17,(0,17)$

25. (ii) $y=11 x-12$

(iii) $y=2 x^{3}-4 x^{2}+3 x+4$

26. $x=2, y=12$

27. (i) $(2.5,12) \min$

$(1.5,4) \max$

(ii) $y=(-1 / 3)(x-3)+13 ; x=2.23$

28. (i) $y=0.91 x-1.1$

(ii). $5(\ln 3)^{3}$

29. $49 \pi^{2} / 64$

30. (i) $(8-h) / 2$ (ii) $8 / 3$

31. $1.31,0.0393$

32. $9 y=4 x+1$

33. $10 y+x-20-\pi / 4$

34. (i) $a=3, b=2, c=4$

(ii) $8 \cos 4 x$

(iii) $y=-x / 8+3.2$

35. (i) $h=4(6-r) / 3$

(iii) $r=4, V=128 \pi / 3$

36. (i) $(1,0.5), \max$

$(-1,-0.5), \min$

37. $x+8 y-2=0$

38. (i) $y=(2 / 3)(3 x+10)^{1 / 2}-4$

(ii) $x=5.2$

39. (ii) $x=2 \sqrt{5}, P=8 \sqrt{5}$

40. (i) $3(x-1)^{-2}-12$

(ii) $-6(x-1)^{-3}$

(iii) $(0.5,2)$ min $(1.5,-22)$ max

41. (i) $14 /(3-x)^{2}$

(ii) $0.005$

42. (i) $y=(-2 / 3)(x+1 / 3)$

(ii) $0.0948$

43. (i) $3 x^{2}-4 e^{2 x}+1$

(ii) $2.4$

44. (i) $y=-3 x+10$

(ii) $x=2, y=4$

45. $x=1, A=16 \sqrt{2}$

46. (ii) $1.56$

47. $h=2.49$, max

48. (i) $\frac{d y}{d x}=3-2 \cos 2 x$

(ii) $y=3 x-\sin 2 x-\frac{\pi}{4}$

(iii) $y=-0.789 x-0.294$

49. (ii) $x=2.6, A=13$

50. (i) $4+2 \sqrt{5}$

(ii) $1-\sqrt{5} / 2$

51. $(0,32)$

52. (i) $x=16, y=32$

(ii) $\max$

(iii) $3 \overline{\text { h }}$

53. $y=x^{4}+x-1$

54. (i) $\ln (2 x+1)-\ln (2 x-1)$

(ii) $\frac{16 x^{2}-8}{4 x^{2}-1}$

(iii) $1 / \sqrt{2}$

(iv) minimum

55. (i) $(3 x-1)^{2 / 3}(8 x-21)$

(ii) $12 h$

56. (i) $h=500 /\left(\pi x^{2}\right)$

(ii) $(4.3,349)$

57. (i) $y=2 x^{3 / 2}-2 x^{1 / 2}-10$

(ii) $y=5.5 x-20$

58. $0.159,40$

59. ii $0.115,2 / 5$