Calculus (Myanmar Exam Board)

Group (2015-2019) $\def\frac{\dfrac}$

1. (2015/Myanmar /q6 )
Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}+3 x-10}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{2 x}-\sqrt{a}}{\sqrt{2 x}+\sqrt{a}}$. $\quad(3$ marks $)$

2. (2015/Myanmar /q15a )
If $y=\ln (\cos 2 x)$, prove that $\dfrac{d^{2} y}{d x^{2}}+\left(\dfrac{d y}{d x}\right)^{2}+4=0$. $\quad(5$ marks $)$

3. (2015/Myanmar /q15b )
Determine the turning point on the curve $y=2 {x}^{3}+3 x^{2}-12 x+7$ and state whether it is a maximum or a minimum. Then sketch the graph of the curve. (5 marks)

4. (2015/FC /q6 )
Find $\displaystyle\lim _{x \rightarrow 5} \dfrac{x^{3}-125}{5-x}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{4 x^{2}-10 x+15}{2 x^{2}-3 x-5}, \quad$ (3 marks)

5. (2015/FC /q15a )
If $y=A \cos (\ln x)+B \sin (\ln x)$, where $A$ and $B$ are constants, show that $x^{2} y^{\prime \prime}+x y^{\prime} =0 . \quad \quad$ (5 marks)

6. (2015/FC /q15b )
Given that $x+y=5$; calculate the minimum value of $x^{2}+x y+y^{2}$. (5 marks)

7. (2016/Myanmar /q6 )
Differentiate $y=\dfrac{1}{x}$ with repsect to $x$ from the first principles. (5 marks)

8. (2016/Myanmar /q15a )
If $y=(3+4 x) e^{-2 x}$, then prove that $\dfrac{d^{2} y}{d x^{2}}+4 \dfrac{d y}{d x}+4 y=0$. (5 marks)

9. (2016/Myanmar /q15b )
Find the minimum value of the sum of a positive number and its reciprocal. (5 marks)

10. (2016/FC /q6 )
Find the value of $a$ and $b$ for which $\dfrac{d}{d x}\left[\dfrac{\sin x}{2+\cos x}\right]=\dfrac{3 a+b \cos x}{(2+\cos x)^{2}}$. (5 marks)

11. (2016/FC /q15a )
Find the coordinates of the points on the curve $x^{2}-y^{2}=3 x y-39$ at which the tangents are (i) parallel (ii) perpendicular to the line $x+y=1$. (5 marks)

12. (2016/FC /q15b )
Find the stationary points on the curve $y=27+12 x+3 x^{2}-2 x^{3}$ and determine the nature of these points. (5 marks)

13. (2017/Myanmar /q6 )
Calculate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{\sqrt{x+2}-2}$ and $\displaystyle\lim _{x \rightarrow 0} \dfrac{\cos x-1}{\sin ^{2} x}$. (5 marks)
Q6(a) Solution

14. (2017/Myanmar /q15a )
Find the equation of the tangent line to the curve $x^{3}+y^{3}-9 x y=0$ at the point $(3,2)$. (5 marks)
Q15(a) Solution

15. (2017/Myanmar /q15b )
Find the two positive numbers whose sum is 82 and whose product is as large as possible. (5 marks)
Q15(b) Solution

16. (2017/FC /q6 )
Find $\displaystyle\lim _{x \rightarrow 2} \dfrac{2^{2 x}-5\left(2^{x}\right)+4}{2^{x}-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-16}{x^{4}-4 x^{3}}$. $(3 \mathrm{marks})$

17. (2017/FC /q15a )
Show that the point $\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)$ lies on the curve $x \sin 2 y=y \cos 2 x$. Then find the equations of tangent and normal to the curve at the point $\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)$. (5 marks)

18. (2017/FC /q15b )
If $y=A \cos \left(\ln \dfrac{x}{2}\right)+B \sin \left(\ln \dfrac{x}{2}\right)$, where $A$ and $B$ are constants, show that $\mathrm{x}^{2} \mathrm{y}^{\prime \prime}+\mathrm{x} \mathrm{y}^{\prime}+\mathrm{y}=0$. (5 marks)

19. (2018/Myanmar /q6 )
Differentiate $f(x)=1-2 x^{2}$ with respect to $x$ at $x=2$ from the first principles. (3 marks)
Click for Solution

20. (2018/Myanmar /q15a )
Given that $x y=\sin x$, prove that $\dfrac{d^{2} y}{d x^{2}}+\dfrac{2}{x} \dfrac{d y}{d x}+y=0$. (5 marks)
Click for Solution

21. (2018/Myanmar /q15b )
Find the approximate change in the "alume of a sphere when its radius increases fron $2 \mathrm{~cm}$ to $2.05 \mathrm{~cm}$. (5 marks)
Click for Solution

22. (2018/FC /q6 )
Evaluate $\displaystyle\lim _{x \rightarrow 1} \dfrac{x^{4}-1}{x^{3}-1}$ and $\displaystyle\lim _{x \rightarrow 0} \dfrac{\dfrac{1}{x-1}+\dfrac{1}{x+1}}{x}$ (3 marks)

23. (2018/FC /q15a )
Given that $y=\sin (\sin x)$, prove that $\dfrac{d^{2} y}{d x^{2}}+\tan x \dfrac{d y}{d x}+y \cos ^{2} x=0$. (5 marks)

24. (2018/FC /q15b )
Show that the point $(0, \pi)$ lies on the cirve $x^{2} \cos ^{2} y=\sin y$. Then find the equations of tangent and normal to the curve at the point $(0,\pi)$. (5 marks)

25. (2019/Myanmar /q5b )
Differentiate $\mathrm{x}^{3}+2 \mathrm{x}$ with respect to $\mathrm{x}$ from the first principle. (3 marks)Click for Solution

26. (2019/Myanmar /q13b )
If $y=\operatorname{In}\left(\sin ^{3} 2 x\right)$, then prove that $3 \dfrac{d^{2} y}{d x^{2}}+\left(\dfrac{d y}{d x}\right)^{2}+36=0 . \quad$ (5 marks) Click for Solution

27. (2019/Myanmar /q14b )
Find the normals to the curve $x y+2 x-y=0$ that are parallel to the line $2 x+y=0$. (5 marks) Click for Solution

28. (2019/FC /q5b )
Differentiate $\mathrm{x}^{2}-3 \mathrm{x}$ with respect to $\mathrm{x}$ from the first principles. (3 marks) Click for Solution 5(b)


29. (2019/FC /q13b )
If $y=\operatorname{In}\left(\sin ^{3} 2 x\right)$, then prove that $3 \dfrac{d^{2} y}{d x^{2}}+\left(\dfrac{d y}{d x}\right)^{2}+36=0$. (5 marks) Click for Solution 13(b)


30. (2019/FC /q14b )
Show that the point $\left(1, \dfrac{\pi}{2}\right)$ lies on the curve $2 \mathrm{xy}+\pi \sin \mathrm{y}=2 \pi$. Then find the equations of tangent and normal to the curve at the point $\left(1, \dfrac{\pi}{2}\right)$. (5 marks) Click for Solution 14(b)

Answer (2015-2019)

1.  $\dfrac{12}{7},1$

2.  Prove 

3.  $(-2,27)$ maximum point, (1,0) minimum point

4.  $-75,2$

5.  Prove 

6.  $\dfrac{75}{4}$

7.  $\dfrac{d y}{d x}=-\dfrac{1}{x^{2}}$

8.   Prove 

9.  2

10.  $a=\dfrac{1}{3}, b=2$

11.   (i) $(1,5),(-1,-5)$ (ii) No tangent 

12.  $(2,47) \max ;(-1,20) \min$

13.  $48,-\dfrac{1}{2}$

14.   

15.  $x=41, y=41$

16.  3,0

17.  $y-2x=0,8y+4x=5\pi$ 

18.  Prove

19.  $f^{\prime}(2)=-8$

20.  Prove 

21.  $0.8 \pi $

22.  $\dfrac{4}{3},-2$

23.   Prove 

24.  $y=\pi, x=0$

25.  $3x^2+2$

26.  Proof

27.  $2x+y=\pm 3$

28.  $2x-3$

29.  Prove

30.  Tangent equation $y-\dfrac{\pi}{2}=-\dfrac{\pi}{2}(x-1),$ 

Normal equation $y-\dfrac{\pi}{2}=\dfrac{2}{\pi}(x-1)$

Group (2014)

1. Evaluate $\displaystyle\lim _{x \rightarrow 4} \dfrac{x^{2}-x-12}{x^{2}-11 x+28}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{3 x^{2}-x+2}{x^{2}+1}$. (3 marks)

2. Calculate $\displaystyle\lim _{x \rightarrow 3} \dfrac{2 x^{2}-18}{x^{4}-27 x}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{2 x^{2}-18}{x^{4}-27 x}$. (3 marks)

3. Calculate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{4}-16}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}$. (3 marks)

4. Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{2^{2 x}-5\left(2^{x}\right)+4}{2^{x}-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-4}{(x+3)(x-4)}$. (3 marks)

5. Calculate $\displaystyle\lim _{x \rightarrow-1} \dfrac{\dfrac{1}{x}+1}{2 x+2}$ and $\displaystyle\lim _{x \rightarrow \infty}(\sqrt{7+x}-\sqrt{x})$. (3 marks)

6. Evaluate $\displaystyle\lim _{x \rightarrow \infty} \dfrac{1-x}{x^{2}-2 x+1}$ and $\displaystyle\lim _{x \rightarrow 3} \dfrac{\sqrt{x}-\sqrt{3}}{3-x}$. (3 marks)

7. Find $\displaystyle\lim _{x \rightarrow 3} \dfrac{\sqrt{x+6}-3}{x-3}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{2^{x}+2^{-x}}{2^{x}-2^{-x}}$. (3 marks)

8. Calculate $\displaystyle\lim _{x \rightarrow 1} \dfrac{x-1}{\sqrt{x^{2}+3}-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{2^{x}+2^{-x}}{2^{x}-2^{-x}}$. (3 marks)

9. Calculate $\displaystyle\lim _{x \rightarrow 3} \dfrac{\sqrt{x+6}-3}{x-3}$ and $\displaystyle\lim _{n \rightarrow \infty} \dfrac{1+2+3+\ldots+n}{n^{2}}$. (3 marks)

10. Calculate $\displaystyle\lim _{x \rightarrow 0} \dfrac{(1+x)^{\dfrac{1}{3}}-(1-x)^{\dfrac{1}{3}}}{x}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{p}+x^{p-1}+1}{x^{q}+3^{9-2}+2}$ if $p<q$. (3 marks)

11. Given that $y=\dfrac{3 e^{2 x}}{2 x+1}$, find the value of $k$ for which $\dfrac{d y}{d x}=\dfrac{k x y}{2 x+1}$. (5 marks)

12. Find $\dfrac{d y}{d x}$ at $x=1$ if $y=\dfrac{3 x^{2}-2 x}{2 x^{3}+3} .$ Find also $\dfrac{d z}{d x}$ if $z=3^{2 x} \tan 4 x$. (5 marks)

13. Differentiate $\cos ^{2} \sqrt{x^{2}+1}$ and $\ln \left(e^{\sin 2 x}+1\right)$ with respect to $x$. (5 marks)

14. Find $\dfrac{d y}{d x}$ if $x+\sin y=\cos (x y)$ and if $e^{5 x} \ln \sqrt{5 x-1}$. (5 marks)

15. If $y=\dfrac{2 x^{2}+3}{x}$, then prove that $x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}=y$. (5 marks)

16. If $y=\dfrac{\sin x}{x}$, prove that $\dfrac{d^{2} y}{d x^{2}}+\dfrac{2}{x} \dfrac{d y}{d x}+y=0$. (5 marks)

17. Given that $y=\sin (\sin \theta)$, prove that $\dfrac{d^{2} y}{d \theta^{2}}+\tan \theta \dfrac{d y}{d \theta}+y \cos ^{2} \theta=0$. (5 marks)

18. Given that $x y=\sin x$, prove that $\dfrac{d^{2} y}{d x^{2}}+\dfrac{2}{x} \dfrac{d y}{d x}+y=0$. (5 marks)

19. If $y \cos x=e^{x}$, show that $\dfrac{d^{2} y}{d x^{2}}-2 \tan x \dfrac{d y}{d x}-2 y=0$. (5 marks)

20. If $y=\ln (\cos 2 x)$, prove that $\dfrac{d^{2} y}{d x^{2}}+\left(\dfrac{d y}{d x}\right)^{2}+4=0$. (5 marks)

21. If $y=\ln (\sin 3 x)$, prove that $\dfrac{d^{2} y}{d x^{2}}+\left(\dfrac{d y}{d x}\right)^{2}+9=0$. (5 marks)

22. Given that the gradient of the curve $y=a x^{2}-b x+3$ at the point $(2,7)$ is 8 . Find the values of $a$ and $b$. (5 marks)

23. Find the equation of the normal to the curve $y=(2 x+a)^{3}$ at the point where $y=a^{3}$. (5 marks)

24. Find the equation of the normal line to the curve $y=\dfrac{6}{1-2 x}$ at the poin wiere $x=2$. (5 marks)

25. Find the value of $k$ for which $y=2 x+k$ is a normal to $y=2 x^{2}-3$. (5 marks)

26. Find the equation of the normal to the curve $x y-2 x=y+3$ at the point where the curve meets the X-axis. (5 marks)

27. Show that the equation of the tangent to the curve $x^{2}+x y+y=0$ at the point $(a, b)$ is $x(2 a+b)+y(a+1)+b=0$. (5 marks)

28. Determine the turning points on the curve $y=x^{3}-4 x^{2}-3 x+18$ State whether each of these points is a maximum or a minimum. (5 marks)

29. Find the stationary point of the curve $y=3-(2 x-1)^{4}$ and determine its nature. (5 marks)

30. Find the value of $x$ between 0 and $\dfrac{\pi}{2}$ for which the curve $y=e^{x} \cos x$ has a stationary point. Determine whether it is a maximum or a minimum point. (5 marks)

31. If a piece of string, 200 feet long is made to enclose a rectangle, show that the enclose area is the greatest when the rectangle is a square. (5 marks)

32. Given that the volume of a solid cylinder of radius $r \mathrm{~cm}$ is $250 \pi \mathrm{cm}^{3}$, find the value of $r$ for which the total surface area of the solid is minimum. (5 marks)

Answer (2014)

1. $-\dfrac{7}{3}, 3$

2. $\dfrac{4}{27}, 0$

3. $32,-1$

4. 3,1

5. $-\dfrac{1}{2}, 0$

6. $0 .-\dfrac{\sqrt{3}}{6}$

7. $\dfrac{1}{6}, 1$

8. 2,1 

9. $\dfrac{1}{6}, \dfrac{1}{2}$

10. $\dfrac{2}{3}, 0$

11. $k=4$

12. $\dfrac{14}{25}, \dfrac{d z}{d x}=9^{x} \ln 9 \tan 4 x+9^{x} \cdot \sec ^{2} 4 x .4$

13. $\dfrac{e^{\sin 2 x}}{e^{\sin 2 x}+1} \cdot 2 \cos 2 x,-2 \cos \sqrt{x^{2}+1} \cdot \sin \sqrt{x^{2}+1} \cdot \dfrac{x}{\sqrt{x^{2}+1}}$

14. $\dfrac{d y}{d x}=\dfrac{-1-y \sin (x y)}{x \sin (x y)+\cos y}, \dfrac{d y}{d x}=5 e^{5 x} \ln \sqrt{5 x-1}+\dfrac{5 e^{5 x}}{2(5 x-1)}$

15. Prove

16. Prove

17. Prove

18. Prove

19. Prove

20. Prove

21. Prove

22. $a=3, b=4$

23. $6 a^{2} y+x=6 a^{5}$

24. $y=\dfrac{3}{4} x-\dfrac{7}{2}$

25. $k=-\dfrac{87}{32}$

26. $10 x-8 y+15=0$

27. Show

28. $\left(-\dfrac{1}{3}, \dfrac{500}{27}\right)$, maximum point $/(3,0)$, minimum point

29. $\left(\dfrac{1}{2}, 3\right)$, maximum $\quad$ 

30. $x=\dfrac{\pi}{4}$, The point is maximum

31. Prove

32. $r=\left(\dfrac{500}{4 \pi}\right)^{\dfrac{1}{3}} \mathrm{~cm}$

Group (2013)

1. Find $\displaystyle\lim _{x \rightarrow 2} \frac{x^{2}-2 x}{x^{2}-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{(x+4)(8 x-5)}{(2 x+1)(x-9)}$. (3 marks)

2. Calculate $\displaystyle\lim _{x \rightarrow 3} \frac{21-x-2 x^{2}}{3 x^{2}-7 x-6}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{3 x^{2}-2 x+5}{x^{2}-2}$. (3 marks)

3. Calculate $\displaystyle\lim _{x \rightarrow-3} \frac{x^{3}+27}{x^{2}-9}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{7+5 x-3 x^{2}}{(2 x-3)^{2}}$. (3 marks)

4. Find $\displaystyle\lim _{x \rightarrow 0}\left[\left(2 x+\frac{1}{x^{2}}\right)-\left(x+\frac{1}{x}\right)^{2}\right], \displaystyle\lim _{x \rightarrow \infty} \frac{(3 x+2)(5 x-7)}{(2 x-3)^{2}}$. (3 marks)

5. Calculate $\displaystyle\lim _{x \rightarrow 4} \frac{x^{2}-16}{x^{4}-4 x^{3}}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-16}{x^{4}-4 x^{3}}$. (3 marks)

6. Find $\displaystyle\lim _{x \rightarrow \infty} \frac{\sqrt{x}-\sqrt{5}}{\sqrt{x}+\sqrt{5}}$ and $\displaystyle\lim _{x \rightarrow-2} \frac{-x^{2}+2 x+8}{x+2}$. (3 marks)

7. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-\sqrt{6-x}}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{3}+5 x^{2}-x+1}{x^{2}+2 x-3}$. (3 marks)

8. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-\sqrt{6-x}}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{(x+4)(8 x-5)}{(x+1)(x-7)}$. (3 marks)

9. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{\sqrt{x}-\sqrt{2}}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{3 \sqrt{x}+\sqrt{2}}{\sqrt{x}-\sqrt{2}}$. (3 marks)

10. Evaluate $\displaystyle\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{1+2+4+\ldots+2^{x-1}}{2^{x}+1}$. (3 marks)

11. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{2}+1}{x+1}-x+1\right)$. (3 marks)

12. Differentiate $y=\frac{1}{x^{2}}$ with respect to $x$ at $x=2$ from the first principles. (5 marks)

13. Differentiate $y=\frac{1}{\sqrt{x}}$ with respect to $x$ at $x=2$ from the first principles. (5 marks)

14. Given that $y=x^{2}\left(x^{2}-3\right)^{7}$, find the numerical value of $\frac{d y}{d x}$ when $x=2$. (5 marks)

15. Calculate the gradient of the curve $y=x \sqrt{x+3}$ and find the coordinates of the point at which the gradient is 0. (5 marks)

16. The gradient of the curve $y=3 x^{2}+5 x-12$ is 17 at the point $P$. Calculate the coordinates of $P$. The curve cuts the $\mathrm{x}$-axis at $Q$ and $R$. Find the gradient of the curve at $Q$ and at $R$. (5 marks)

17. Given that $y=(1+x) e^{3 x}$, prove that $\frac{d^{2} y}{d x^{2}}-6 \frac{d y}{d x}+9 y=0$. (5 marks)

18. Find the derivatives of the functions $\cos ^{3} 2 x \cdot \ln \left(3-x^{2}\right)$ and $\frac{e^{2-x^{3}}}{\sqrt{5-3 x+7 x^{2}}}$. (5 marks)

19. Find $\frac{d y}{d x}$ if $y=\frac{3 x^{2}-8}{2 x-5}$. Find also $\frac{d z}{d x}$ if $x e^{z}+\ln (x z)=\sin x$. (5 marks)

20. Find the equations of normal line to the curve $y=x^{2}-5 x+6$ at the points where this curve cuts the $x$-axis. (5 marks)

21. Find the equations of the normal lines to the curve $y=x^{2}-2 x-8$ at the points where this curve cuts the $x$-axis. (5 marks)

22. Find the equation of the tangent line to the curve $3 x^{2}+2 y^{2}=2 x y+23$ at the point $(3,2)$. (5 marks)

23. Show that the equation of the tangent to the curve $y=(2 x+a)^{3}$ at the point where $y=a^{3}$ is $y=a^{2}(a+6 x)$. (5 marks)

24. If $y=3 x \sin 3 x+\cos 3 x$, show that $x \frac{d^{2} y}{d x^{2}}+9 x y=2 \frac{d y}{d x}$. (5 marks)

25. Given that $y=x \cos x$, show that $x \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+x y+2 \cos x=0$. (5 marks)

26. Determine the turning point on the curve $y=3 x^{2}-6 x+3$ and state whether it is a maximum or a minimum. (5 marks)

27. If $x+y=42$, find the maximum value of $x y$. (5 marks)

28. Given that $y=\sqrt{x}$, determine the approximate value for $\sqrt{101}$ by using approximation. (5 marks)

29. Find two positive numbers whose sum is 20 and whose product is as large as possible. (5 marks)

30. A rectangular field is surrounded by a fence on three of its sides and a straight hedge on the fourth side. If the area of the field is to be 11250 square metres, find the smallest possible length of the fence. (5 marks)

31. If the perimeter of a rectangle is $24 \mathrm{~m}$, show that the area is the greatest when this rectangle is a square and find the maximum area. (5 marks)

32. If the area of a rectangle is $49 \mathrm{~cm}^{2}$, show that the perime the smallest when this rectangle is a square and find the smallest perimeter. (5 marks)

33. Two positive numbers $x$ and $y$ vary in such a way that $x^{4} y=32$. Another number $z$ is defined by $z=x^{2}+y$. Find the values of $x$ and $y$ for which $z$ has a stationary value and show that this value of $z$ is a minimum. (5 marks)

Answer (1013)

1. $\frac{1}{2}, 4$

2. $-\frac{13}{11}, 3$

3. $-\frac{9}{2},-\frac{3}{4}$

4. $-2, \frac{15}{4} $

5. $\frac{1}{8}, 0$

6. $1,6 \quad$ 

7. $\frac{1}{2}, \infty$

8. $\frac{1}{2}, 8$

9. $6 \sqrt{2}, 3$

10. $\frac{1}{6}, 1$

11. $\frac{\sqrt{3}}{24}, 0$

12. $-\frac{1}{4}$

13. $-\frac{1}{4\sqrt{2}}$

14. 116

15. $(-2,-2)$

16. $(2,10) ;13 ;-13$

17. Show

18. $\frac{-2 x \cos ^{3} 2 x}{3-x^{2}}-6 \cos ^{2} 2 x \sin 2 x \ln \left(3-x^{2}\right)$ $\frac{e^{2-x^{3}}\left(-30 x^{2}+18 x^{3}-42 x^{4}-14 x+3\right)}{2\left(5-3 x+7 x^{2}\right)^{3 / 2}}$

19. $\frac{6 x^{2}-30 x+16}{(2 x-5)^{2}} ; \frac{\cos x-e^{z}-\frac{1}{x}}{x e^{z}+\frac{1}{z}}$

20. $x+y-3=0 ;y-x+2=0$

21. $6y+x-4=0;6y-x-2=0$

22. $y+7 x-23=0 \quad$ 

23. Show

24. Show

25. Show

26. $(1,0)$ is a turning point. $(1,0)$ is a minimum point.

27. $441$ 

28. 10.05

29. 10 and 10

30. 300 metres

31. 36 m$^2$

32. 28 cm

33. $x=2,y=2$

Group (2012)

$\quad\;$		$\,$
1.	Calculate $\displaystyle\lim _{x \rightarrow 1} \dfrac{x^{3}-1}{x-1}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{3 x^{2}-3 x+9}{x^{2}-2}$. (3 marks)
2.	Calculate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}-x-2}, \displaystyle\lim _{x \rightarrow \infty} \dfrac{x-2}{2 x^{2}+3 x-1}$. (3 marks)
3.	Find $\displaystyle\lim _{x \rightarrow 5} \dfrac{2 x^{2}-3 x-35}{x^{3}-25 x}, \displaystyle\lim _{x \rightarrow \infty} \dfrac{(3-2 x)(x-5)}{(5 x+1)(x-4)}$. (3 marks)
4.	Calculate $\displaystyle\lim _{x \rightarrow 3} \dfrac{x^{2}-9}{x^{3}-4 x^{2}+3 x}, \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-9}{x^{3}-4 x^{2}+3 x}$. (3 marks)
5.	Find the limits $\displaystyle\lim _{x \rightarrow \infty} \dfrac{(2 x+3)(x-4)}{3 x^{2}+2}$, and $\displaystyle\lim _{x \rightarrow 1} \dfrac{x(1-x)}{x^{3}-1}$. (3 marks)
6.	Calculate $\displaystyle\lim _{x \rightarrow 3} \dfrac{x^{3}-27}{3 x-x^{2}}$ and $\displaystyle\lim _{x \rightarrow \infty}(\sqrt{x+3}-\sqrt{x})$. (3 marks)
7.	Evaluate $\displaystyle\lim _{x \rightarrow 3} \dfrac{\sqrt{x+3}-\sqrt{9-x}}{x-3}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{4 x^{2}-2 x+7}{x^{2}-1}$. (3 marks)
8.	Find the limits: $\displaystyle\lim _{x \rightarrow 2} \dfrac{3 x(\sqrt{x}-\sqrt{2})}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{3 x}+\sqrt{5}}{\sqrt{3 x}-\sqrt{8}}$. (3 marks)
9.	Find $\displaystyle\lim _{x \rightarrow 0}\left[\left(2 x-\dfrac{1}{2 x}\right)^{2}-\left(\dfrac{1}{2 x}+4 x\right)^{2}\right], \displaystyle\lim _{x \rightarrow \infty} \dfrac{9 x^{2}-4}{27 x^{3}-8}$. (3 marks)
10.	Find $\displaystyle\lim _{x \rightarrow 1} \dfrac{\dfrac{1}{x}-1}{2 x-2}$ and $\displaystyle\lim _{t \rightarrow \infty}\left[\left(\dfrac{t}{t+1}\right)\left(\dfrac{t^{2}}{t+t^{2}}\right)\right]$. (3 marks)
11.	Calculate $\displaystyle\lim _{x \rightarrow 1} \dfrac{\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+1}{(x-1)^{2}}$ end $\displaystyle\lim _{x \rightarrow \infty} \dfrac{(3 x-1)(2 x+5)}{(x-3)(3 x+7)}$. (3 marks)
12.	Differentiate $y=x^{2}-5 x+4$ with respect to $x$ from the first principles. (5 marks)
13.	If $y=x^{2}+2 x+3$, show that $\left(\dfrac{d y}{d x}\right)^{2}+\left(\dfrac{d^{2} y}{d x^{2}}\right)^{3}=4 y$. (5 marks)
14.	If $y=3 x^{2}+4 x$, prove that $x^{2} \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}+2 y=0$. (5 marks)
15.	Find the values of $a$ and $b$ for which $\dfrac{d}{d x}\left(\dfrac{\sin x}{2+\cos x}\right)=\dfrac{3 a+b \cos x}{(2+\cos x)^{2}}$. (5 marks)
16.	Find the equations of the tangent and the normal line to the curve $y=x^{2}-3 x+2$ at the point where $x=3$. (5 marks)
17.	Find the equation of the tangent to the curve $x^{2}+x y+y=-2$ at the point where $x=2$. (5 marks)
18.	Find the stationary points of the curve $y=x^{2}(x-2)$ and determine their nature. (5 marks)
19.	If $y=3 x \sin 3 x-\cos 3 x$, find $x \dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+9 x y$. (5 marks)
20.	If $y=3 e^{\cos x}$, prove that $\dfrac{d^{2} y}{d x^{2}}=(\cot x-\sin x) \dfrac{d y}{d x}$. (5 marks)
21.	If $y=e^{2 x} \sin 3 x$, prove that $\dfrac{d^{2} y}{d x^{2}}-4 \dfrac{d y}{d x}+13 y=0$. (5 marks)
22.	If $y=e^{2 x} \cos 3 x$, prove that $\dfrac{d^{2} y}{d x^{2}}-4 \dfrac{d y}{d x}+3 y=0$. (5 marks)
23.	Given that $y=\dfrac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$, show that $\dfrac{d y}{d x}=1-y^{2}$. (5 marks)
24.	Given that $\dfrac{d}{d x}\left(e^{2 x} \tan 3 x\right)=e^{2 x} f(x)$, find $f(x)$ and $f^{\prime}(x)$. (5 marks)
25.	If $y=\ln \dfrac{x^{2}}{x^{2}+1}$, find the rate of change of $y$ with respect to $x$ at $x=2$. Find also $\dfrac{d z}{d x}$ if $x^{3}-4 x z+z^{2}=14$. (5 marks)
26.	The point $P(3,4)$ lies on the curve $y=3 x^{2}-12 x+3$. Find the intersection point of the normal to the curve at $P$ with the line $x+3=0$. (5 marks)
27.	Find the coordinates of the points on the curve $x^{2}-y^{2}=x y-5$ at which the tangents are parallel to the line $x+y=1$. (5 marks)
28.	Find two positive numbers whose product is 361 and whose sum is as small as possible. (5 marks)
29.	$x$ and $y$ are two positive numbers such that $x+y=20$. Find the minimum value of $2 x^{2}+3 y^{2}$. (5 marks)
30.	Find two positive numbers whose sum is 20 and whose product is as large as possible. (5 marks)
31.	A rectangular field is surrounded by a fence on three of its sides and a straight hedge on the fourth sides. If the length of the fence is 320 meters, find the maximum area of the field enclosed. (5 marks)
32.	A rectangular box has a square base of side $x \mathrm{~cm}$. If the sum of one side of the square and the height is $12 \mathrm{~cm}$, express the volume of the box in terms of $x$. Use this expression to determine the maximum volume of the box. (5 marks)
33.	What is the largest area possible for a right triangle whose hypotenuse is $8 \mathrm{~cm}$ long. (5 marks)

Answer (2012)

$\qquad$		$\,$
1.	3,3
2.	4,0
3.	$\dfrac{17}{50}, \dfrac{-2}{5}$
4.	1,0
5.	$\dfrac{2}{3}, \dfrac{-1}{3}$
6.	$-9,0$
7.	$\dfrac{\sqrt{6}}{6}, 4$
8.	$\dfrac{3 \sqrt{2}}{2}, 1$
9.	$-6,0$
10.	$\dfrac{-1}{2}, 1$
11.	$\dfrac{1}{9}, 2$
12.	$2 x-5$
13.	Prove
14.	Prove
15.	$a=\dfrac{1}{3}, b=2$
16.	$3 x-y-7=0 ; x+3 y-9=0$
17.	$2 x+3 y+2=0$
18.	$\operatorname{maximum}(0,0)$ and minimum $\left(\dfrac{4}{3}, \dfrac{-32}{27}\right)$
19.	$-12 \sin 3 x$
20.	Prove
21.	Prove
22.	Prove
23.	Prove
24.	$f(x)=3 \sec ^{2} 3 x+2 \tan 3 x, f(x)=18 \sec ^{2} 3 x \tan 3 x+6 \sec ^{2} 3 x$
25.	$\dfrac{1}{5}, \dfrac{4 z-3 x^{2}}{2 z-4 x}$
26.	$y=3 x^{2}-12 x+3$
27.	$(1,-3)$ and $(-1,3)$
28.	19 and 19
29.	480
30.	10 and 10
31.	$12800 \mathrm{~m}^{2} $
32.	$256 \mathrm{~cm}^{3}$
33.	16 cm$^2$

Group (2011)

$\quad\;\,$		$\,$
1.	Evaluate $\displaystyle\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x^{2}-11 x+28}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{3 x^{2}-x+2}{x^{2}+1}$. $\mbox{ (3 marks)}$
2.	Calculate $\displaystyle\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{4}-3 x^{3}}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-9}{x^{4}-3 x^{3}}$. $\mbox{ (3 marks)}$
3.	Find $\displaystyle\lim _{x \rightarrow 2} \frac{2 x-x^{2}}{x^{2}-3 x+2}, \displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x+1}{x^{2}-x+2}$. $\mbox{ (3 marks)}$
4.	Find $\displaystyle\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{2}-x-2}, \displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-x-2}{x^{3}-8}$. $\mbox{ (3 marks)}$
5.	Calculate $\displaystyle\lim _{x \rightarrow 5} \frac{x^{3}-125}{5-x}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{9 x^{2}-1}{x^{2}-x}$. $\mbox{ (3 marks)}$
6.	Calculate $\displaystyle\lim _{x \rightarrow 1} \frac{x^{3}-2 x^{2}}{x}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{(x+1)(x+2)}{x^{2}-4}$. $\mbox{ (3 marks)}$
7.	Find the limits $\displaystyle\lim _{x \rightarrow \infty} \frac{(x-3)(2 x+4)}{3 x^{2}-2}$, and $\displaystyle\lim _{x \rightarrow 0} \frac{x(1-x)}{x^{3}+3 x}$. $\mbox{ (3 marks)}$
8.	Find the limits $\displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{2}-5}{(3 x-2)(x+4)}$, and $\displaystyle\lim _{x \rightarrow 0} \frac{x(1-x)}{x^{3}+3 x}$. $\mbox{ (3 marks)}$
9.	Calculate $\displaystyle\lim _{x \rightarrow 5} \frac{2 x^{2}-14 x+20}{x^{2}-25}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{-2}-14 x^{-1}+20}{x^{-2}-25}$. $\mbox{ (3 marks)}$
10.	Calculate $\displaystyle\lim _{x \rightarrow 3} \frac{\sqrt{x}-\sqrt{3}}{x^{2}-3^{2}}, \displaystyle\lim _{x \rightarrow \infty} \frac{x^{3}-2}{2 x^{3}+3 x^{2}-1}$. $\mbox{ (3 marks)}$
11.	Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{5-x^{2}}{x^{2}-x}$. $\mbox{ (3 marks)}$
12.	Differentiate $y=2 \pi x+2 \cos \pi x$ and $y=\frac{x^{2}+\tan 3 x}{e^{x}}$ with respect to $x$. $\mbox{ (5 marks)}$
13.	If $x+\sin y=\cos (x y)$, find $\frac{d y}{d x}$. If $z=\sqrt{\frac{x^{2}+1}{x^{2}-1}}$, find $\frac{d z}{d x}$. $\mbox{ (5 marks)}$
14.	Find $\frac{d y}{d x}$ if $y=\frac{3 x+8}{2 x^{2}+5}$. Find also $\frac{d z}{d x}$ if $x+\cos z=\tan (x z)$. $\mbox{ (5 marks)}$
15.	Find the approximate change in the volume of a sphere when its radius increases from $2 \mathrm{~cm}$ to $2.05 \mathrm{~cm}$. $\mbox{ (5 marks)}$
16.	Using the derivative of a suitable function, find an approximate value of $\sqrt{143.5}$. $\mbox{ (5 marks)}$
17.	Given that $y=e^{3 x} \sin 2 x$, prove that $\frac{d^{2} y}{d x^{2}}-6 \frac{d y}{d x}+13 y=0$. $\mbox{ (5 marks)}$
18.	If $y=3 x \sin 3 x+\cos 3 x$, then prove that $x \frac{d^{2} y}{d x^{2}}+9 x y=2 \frac{d y}{d x}$. $\mbox{ (5 marks)}$
19.	Find $\frac{d y}{d x}$ if $y=(5+3 x) e^{-2 x}.$ Find also $\frac{d y}{d x}$ if $y=\frac{\sin 3 x}{\sqrt{x^{2}+1}}$. $\mbox{ (5 marks)}$
20.	Given that $x y=\sin x$, prove that $\frac{d^{2} y}{d x^{2}}+\frac{2}{x} \frac{d y}{d x}+y=0$. $\mbox{ (5 marks)}$
21.	Given that $x^{2}-y^{2}=5$, show that $y^{2} y^{\prime \prime}+x y^{\prime}=y$. $\mbox{ (5 marks)}$
22.	Given that the gradient of the curve $y=a x^{2}-b x+3$ at the point $(2,7)$ is 8. Find the values of $a$ and $b$. $\mbox{ (5 marks)}$
23.	Find the stationary points of the curve $y=x^{3}-3 x+2$ and determine their nature. $\mbox{ (5 marks)}$
24.	Find the stationary points of the curve $y=x^{3}(4-x)+5$ and determine their nature. $\mbox{ (5 marks)}$
25.	Given that $x+y=5$, calculate the maximum value of $2 x^{2}+x y-3 y^{2}$. $\mbox{ (5 marks)}$
26.	Find the minimum value of the sum of a positive number and its reciprocal. $\mbox{ (5 marks)}$
27.	Find the $x$-coordinate, for $0 < x< \frac{\pi}{2}$, of the stationary point on the curve $y=e^{\sqrt{3} x} \cos x$. $\mbox{ (5 marks)}$
28.	Given that $y=\frac{3 x^{2}+2}{x}$, prove that $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-y=0$. $\mbox{ (5 marks)}$
29.	Find the equations of the tangent and normal lines to the curve $y=x^{2}-5 x+6$ at the point $(1,2)$. $\mbox{ (5 marks)}$
30.	Find the equation of the normal line to the curve $y=x^{2}-3 x+2$ given that the gradient of the normal is $\frac{1}{2}$. $\mbox{ (5 marks)}$
31.	Show that the equation of the tangent to the curve $x^{2}+x y+y=0$ at the point $(a, b)$ is $x(2 a+b)+y(a+1)+b=0$. $\mbox{ (5 marks)}$

Answer (2011)

$\quad\;\,$		$\,$
1.	$-\frac{7}{3}, 3$
2.	$\frac{2}{9}, 0$
3.	$-2,2$
4.	4,0
5.	$-75,9$
6.	$-1,1$
7.	$\frac{2}{3}, \frac{1}{3}$
8.	$\frac{2}{3}, \frac{1}{3}$
9.	$\frac{3}{5},-\frac{4}{5}$
10.	$\frac{\sqrt{3}}{36}, \frac{1}{2}$
11.	$\frac 14,-1$
12.	$2 \pi(1-\sin (\pi x)) ; \frac{2 x+3 \sec ^{2} 3 x-x^{2}-\tan 3 x}{e^{x}}$
13.	$\frac{-y \sin (x y)-1}{\cos y+x \sin (x y)} ; \frac{-2 x \sqrt{x^{2}-1}}{\left(x^{2}-1\right)^{2} \sqrt{x^{2}+1}}$
14.	$\frac{-6 x^{2}-32 x+15}{\left(2 x^{2}+5\right)^{2}} ; \frac{1-z \sec ^{2}(x z)}{\sin z+x \sec ^{2}(x z)}$
15.	$0.8 \pi \mathrm{cm}^{3}$
16.	$11.979$
17.	Prove
18.	Prove
19.	$(-7-6 x) e^{-2 x} ; \frac{\left(3 x^{2}+3\right) \cos 3 x-x \sin 3 x}{\left(x^{2}+1\right) \sqrt{x^{2}+1}}$
20.	Prove
21.	Prove
22.	$a=3 ; b=4$
23.	$(1,0)$ minimum; $(-1,4)$ maximum
24.	$(0,5)$ inflexion; $(3,32)$ maximum
25.	$\frac{625}{8}$
26.	2
27.	$\frac{\pi}{3}$
28.	Prove
29.	$y+3 x-5=0 ; 3 y-x-5=0$
30.	$2 y-x-1=0$
31.	Prove

Group (2010)

$\quad\;\,$		$\,$
1.	Find $\displaystyle\lim _{x \rightarrow 5} \dfrac{x^{2}+6 x-55}{x^{2}-2 x-15}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-10 x+25}{2 x^{2}-x-6}.$ $\mbox{ (3 marks)}$
2.	Find $\displaystyle\lim _{x \rightarrow-2} \dfrac{3 x^{2}+4 x-4}{x^{2}+3 x+2}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{4 x^{2}-10 x+15}{2 x^{2}-3 x-5}.$ $\mbox{ (3 marks)}$
3.	Evaluate $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-7 x+1}{x^{2}-2}$, and $\displaystyle\lim _{x \rightarrow 5} \dfrac{2 x^{2}-14 x+20}{x^{2}-25}.$ $\mbox{ (3 marks)}$
4.	Calculate $\displaystyle\lim _{x \rightarrow-5} \dfrac{x^{2}-25}{x^{3}+5 x^{2}}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-25}{x^{3}+5 x^{2}}.$ $\mbox{ (3 marks)}$
5.	Find $\displaystyle\lim _{x \rightarrow 0}\left[\left(2 x+\dfrac{1}{x^{2}}\right)-\left(x+\dfrac{1}{x}\right)^{2}\right], \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(3 x+2)(5 x-7)}{(2 x-3)^{2}}.$ $\mbox{ (3 marks)}$
6.	Calculate $\displaystyle\lim _{x \rightarrow-2} \dfrac{x^{3}+8}{x^{2}-4}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{2 x}-\sqrt{a}}{\sqrt{2 x}+\sqrt{a}}.$ $\mbox{ (3 marks)}$
7.	Calculate $\displaystyle\lim _{x \rightarrow-1} \dfrac{x^{3}+1}{x+1}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}.$ $\mbox{ (3 marks)}$
8.	Calculate $\displaystyle\lim _{x \rightarrow 4} \dfrac{x^{3}-64}{x-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}.$ $\mbox{ (3 marks)}$
9.	Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}+3 x-10}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{2 x^{2}-1}{x^{3}-1}.$ $\mbox{ (3 marks)}$
10.	Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x-2}{\sqrt{x}-\sqrt{2}}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(x-4)(x-5)}{(x+1)(x-7)}.$ $\mbox{ (3 marks)}$
11.	Evaluate $\displaystyle\lim _{x \rightarrow 6} \dfrac{3 x-18}{\sqrt{2 x-3}-\sqrt{x+3}}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{(1-2 x)(3+x)}{(x-2)^{2}}.$ $\mbox{ (3 marks)}$
12.	Given that $y=(1+x) e^{3 x}$, prove that $\dfrac{d^{2} y}{d x^{2}}-6 \dfrac{d y}{d x}+9 y=0.$ $\mbox{ (5 marks)}$
13.	Given that $y=(1-x) e^{2 x}$, prove that $\dfrac{d^{2} y}{d x^{2}}=4 \dfrac{d y}{d x}-4 y.$ $\mbox{ (5 marks)}$
14.	Given that $y=x \sin x$, prove that $x \dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+x y+2 \sin x=0.$ $\mbox{ (5 marks)}$
15.	If $x \cos y=\sin x$, prove that $\dfrac{d y}{d x}=\dfrac{\cos y(\cos y-\cos x)}{\sin x \sin y}.$ $\mbox{ (5 marks)}$
16.	If $y=\sin ^{2} x$, show that $\dfrac{d^{2} y}{d x^{2}}+4 y-2=0.$ $\mbox{ (5 marks)}$
17.	If $y=\sin ^{2} x$ and $\dfrac{d^{2} y}{d x^{2}}+\dfrac{d y}{d x}=a \cos 2 x+b \sin 2 x$, where $a$ and $b$ are constants, find the value of $a$ and of $b.$ $\mbox{ (5 marks)}$
18.	If $y=\sin ^{2} 3 x$, prove that $\dfrac{d^{2} y}{d x^{2}}+36 y=18.$ By using this result show that, if $z=\cos ^{2} 3 x$, then $\dfrac{d^{2} z}{d x^{2}}+36 z=18.$ $\mbox{ (5 marks)}$
19.	If $y=\cos ^{2} 3 x$, prove that $\dfrac{d^{2} y}{d x^{2}}+36 y=18.$ By using this result show that, if $z=\sin ^{2} 3 x$, then $\dfrac{d^{2} z}{d x^{2}}+36 z=18.$ $\mbox{ (5 marks)}$
20.	If $y=\cos ^{2} 2 x$, prove that $\dfrac{d^{2} y}{d x^{2}}+16 y=8.$ By using this result show that, if $z=\sin ^{2} 2 x$, then $\dfrac{d^{2} z}{d x^{2}}+16 z=8.$ $\mbox{ (5 marks)}$
21.	Differentiate $f(x)=\dfrac{1}{5 \cos x}$ and $g(x)=e^{5 x} \ln (\sqrt{5 x-1})$ with respect to $x.$ $\mbox{ (5 marks)}$
22.	If $y=A \cos (\ln x)+B \sin (\ln x)$, where $A$ and $B$ are constants, show that $x^{2} y^{\prime \prime}+x y^{\prime}+y=0.$ $\mbox{ (5 marks)}$
23.	Find the equation of the normal line to the curve $y=x^{2}-3 x+2$ at the point where $x=3.$ $\mbox{ (5 marks)}$
24.	Find the equations of the normal lines to the curve $y=x^{2}-2 x-8$ at the points where this curve cuts the $x$-axis.$\mbox{ (5 marks)}$
25.	Find the equations of the tangent and the normal to the curve $y=2 e^{3 x}$ at the point where $x=0.$ $\mbox{ (5 marks)}$
26.	Find the equation of the tangent line to the curve $x^{2}+x y+y=5$ at the point where the curve cuts the line $x=1.$ $\mbox{ (5 marks)}$
27.	Find the equation of the tangent line to the curve $2 x^{2}+3 y^{2}=2 x y+23$ at the point $(2,3).$ $\mbox{ (5 marks)}$
28.	Find the stationary points of the curve $y=x^{3}-3 x^{2}-9 x+10$ and determine their nature.$\mbox{ (5 marks)}$
29.	Find the stationary points of the curve $y=x^{2}(3-x)$ and determine their nature.$\mbox{ (5 marks)}$
30.	What is the smallest perimeter possible for a rectangle of area $16 \mathrm{ft}^{2}$ ? $\mbox{ (5 marks)}$
31.	What is the smallest perimeter possible for a rectangle of area $25 \mathrm{~m}^{2} ?$
32.	A rectangular box has a square base of side $x \mathrm{~cm}.$ If the sum of one side of the square and the height is $12 \mathrm{~cm}$, express the volume of the box in terms of $x.$ Use this expression to determine the maximum volume of the box.$\mbox{ (5 marks)}$
33.	Using $y=\sqrt{x}$, find the approximate value of $\sqrt{26}.$ $\mbox{ (5 marks)}$

Answer (2010)

$\quad\;\,$		$\,$
1.	$2, \dfrac{1}{2}$
2.	2,8
3.	$1, \dfrac{3}{5}$
4.	$-\dfrac{2}{5}, 0 \quad$
5.	$-2, \dfrac{15}{4}$
6.	$-3,1$
7.	$3,-1$
8.	$48,-1$
9.	$\dfrac{12}{7}, 0$
10.	$2 \sqrt{2}, 1$
11.	$18,-2$
12.	Prove
13.	Prove
14.	Prove
15.	Prove
16.	Prove
17.	$a=2, b=1 \quad$
18.	Prove
19.	Prove
20.	Prove
21.	$f^{\prime}(x)=\dfrac{1}{5} \sec x \tan x, \quad g^{\prime}(x)=\dfrac{5}{2} e^{5 x}\left(\dfrac{1}{5 x-1}+\ln (5 x-1)\right)$
22.	Prove
23.	$x+3 y=9$
24.	$6 y-x=2 ; x+6 y=4$
25.	$y-6 x=2 ; 6 y+x=12$
26.	$2 x+y=4 \quad$
27.	$x+7 y=23 \quad$
28.	$(-1,15)$ maximum, $(3,-17)$ minimum
29.	$(0,0)$ minimum, $(2,4)$ maximum
30.	16 ft
31.	20 m
32.	$12 x^{2}-x^{3} ;256 \mathrm{~cm}^{3} \quad$
33.	$5.1$