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