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| 1. | Find $\displaystyle\lim _{x \rightarrow 5} \dfrac{x^{2}+6 x-55}{x^{2}-2 x-15}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-10 x+25}{2 x^{2}-x-6}.$ $\mbox{ (3 marks)}$
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| 2. | Find $\displaystyle\lim _{x \rightarrow-2} \dfrac{3 x^{2}+4 x-4}{x^{2}+3 x+2}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{4 x^{2}-10 x+15}{2 x^{2}-3 x-5}.$ $\mbox{ (3 marks)}$
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| 3. | Evaluate $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-7 x+1}{x^{2}-2}$, and $\displaystyle\lim _{x \rightarrow 5} \dfrac{2 x^{2}-14 x+20}{x^{2}-25}.$ $\mbox{ (3 marks)}$
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| 4. | Calculate $\displaystyle\lim _{x \rightarrow-5} \dfrac{x^{2}-25}{x^{3}+5 x^{2}}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-25}{x^{3}+5 x^{2}}.$ $\mbox{ (3 marks)}$
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| 5. | Find $\displaystyle\lim _{x \rightarrow 0}\left[\left(2 x+\dfrac{1}{x^{2}}\right)-\left(x+\dfrac{1}{x}\right)^{2}\right], \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(3 x+2)(5 x-7)}{(2 x-3)^{2}}.$ $\mbox{ (3 marks)}$
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| 6. | Calculate $\displaystyle\lim _{x \rightarrow-2} \dfrac{x^{3}+8}{x^{2}-4}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{2 x}-\sqrt{a}}{\sqrt{2 x}+\sqrt{a}}.$ $\mbox{ (3 marks)}$
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| 7. | Calculate $\displaystyle\lim _{x \rightarrow-1} \dfrac{x^{3}+1}{x+1}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}.$ $\mbox{ (3 marks)}$
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| 8. | Calculate $\displaystyle\lim _{x \rightarrow 4} \dfrac{x^{3}-64}{x-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}.$ $\mbox{ (3 marks)}$
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| 9. | Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}+3 x-10}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{2 x^{2}-1}{x^{3}-1}.$ $\mbox{ (3 marks)}$
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| 10. | Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x-2}{\sqrt{x}-\sqrt{2}}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(x-4)(x-5)}{(x+1)(x-7)}.$ $\mbox{ (3 marks)}$
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| 11. | Evaluate $\displaystyle\lim _{x \rightarrow 6} \dfrac{3 x-18}{\sqrt{2 x-3}-\sqrt{x+3}}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{(1-2 x)(3+x)}{(x-2)^{2}}.$ $\mbox{ (3 marks)}$
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| 12. | Given that $y=(1+x) e^{3 x}$, prove that $\dfrac{d^{2} y}{d x^{2}}-6 \dfrac{d y}{d x}+9 y=0.$ $\mbox{ (5 marks)}$
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| 13. | Given that $y=(1-x) e^{2 x}$, prove that $\dfrac{d^{2} y}{d x^{2}}=4 \dfrac{d y}{d x}-4 y.$ $\mbox{ (5 marks)}$
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| 14. | Given that $y=x \sin x$, prove that $x \dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+x y+2 \sin x=0.$ $\mbox{ (5 marks)}$
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| 15. | If $x \cos y=\sin x$, prove that $\dfrac{d y}{d x}=\dfrac{\cos y(\cos y-\cos x)}{\sin x \sin y}.$ $\mbox{ (5 marks)}$
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| 16. | If $y=\sin ^{2} x$, show that $\dfrac{d^{2} y}{d x^{2}}+4 y-2=0.$ $\mbox{ (5 marks)}$
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| 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 $b.$ $\mbox{ (5 marks)}$
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| 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 $\dfrac{d^{2} z}{d x^{2}}+36 z=18.$ $\mbox{ (5 marks)}$
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| 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 $\dfrac{d^{2} z}{d x^{2}}+36 z=18.$ $\mbox{ (5 marks)}$
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| 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 $\dfrac{d^{2} z}{d x^{2}}+16 z=8.$ $\mbox{ (5 marks)}$
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| 21. | Differentiate $f(x)=\dfrac{1}{5 \cos x}$ and $g(x)=e^{5 x} \ln (\sqrt{5 x-1})$ with respect to $x.$ $\mbox{ (5 marks)}$
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| 22. | If $y=A \cos (\ln x)+B \sin (\ln x)$, where $A$ and $B$ are constants, show that $x^{2} y^{\prime \prime}+x y^{\prime}+y=0.$ $\mbox{ (5 marks)}$
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| 23. | Find the equation of the normal line to the curve $y=x^{2}-3 x+2$ at the point where $x=3.$ $\mbox{ (5 marks)}$
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| 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.$\mbox{ (5 marks)}$
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| 25. | Find the equations of the tangent and the normal to the curve $y=2 e^{3 x}$ at the point where $x=0.$ $\mbox{ (5 marks)}$
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| 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 $x=1.$ $\mbox{ (5 marks)}$
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| 27. | Find the equation of the tangent line to the curve $2 x^{2}+3 y^{2}=2 x y+23$ at the point $(2,3).$ $\mbox{ (5 marks)}$
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| 28. | Find the stationary points of the curve $y=x^{3}-3 x^{2}-9 x+10$ and determine their nature.$\mbox{ (5 marks)}$
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| 29. | Find the stationary points of the curve $y=x^{2}(3-x)$ and determine their nature.$\mbox{ (5 marks)}$
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| 30. | What is the smallest perimeter possible for a rectangle of area $16 \mathrm{ft}^{2}$ ? $\mbox{ (5 marks)}$
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| 31. | What is the smallest perimeter possible for a rectangle of area $25 \mathrm{~m}^{2} ?$
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| 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.$\mbox{ (5 marks)}$
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| 33. | Using $y=\sqrt{x}$, find the approximate value of $\sqrt{26}.$ $\mbox{ (5 marks)}$
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