CIE Calculus (Additional Mathematics -2018)

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 21, question 4)

(i) Find $\D \frac{d}{dx}(x^2\ln x).$ [2]

(ii) Hence, or otherwise, find $\D \displaystyle\int x\ln x dx.$ [3]


2 (CIE 2012, w, paper 12, question 11either)

A curve is such that $\D y = \frac{5x^2}{1+x^2}.$

(i) Show that $\D \frac{dy}{dx}=\frac{kx}{(1+x^2)^2},$ where k is an integer to be found. [4]

(ii) Find the coordinates of the stationary point on the curve and determine the nature of this

stationary point. [3]

(iii) By using your result from part (i), find $\D \displaystyle\int \frac{x}{(1+x^2)^2} dx$  and hence evaluate $\D \displaystyle\int_{-1}^{2}\frac{x}{(1+x^2)^2}dx.$[4]


3 (CIE 2012, w, paper 13, question 11or)

(i) Given that $\D y =\frac{3e^{2x}}{1+e^{2x}},$ show that $\D \frac{dy}{dx}=\frac{Ae^{2x}}{(1+e^{2x})^2},$  where $\D A$ is a constant to be found. [4]

(ii) Find the equation of the tangent to the curve $\D y = \frac{3e^{2x}}{1+e^{2x}}$  at the point where the curve

crosses the y-axis. [3]

(iii) Using your result from part (i), find $\D \displaystyle\int \frac{e^{2x}}{(1+e^{2x})^2}dx$  and hence evaluate $\D \displaystyle\int_{0}^{\ln 3}\frac{e^{2x}}{(1+e^{2x})^2}dx$ [4]


4 (CIE 2012, w, paper 22, question 7)

(i) Find $\D \frac{d}{dx} (\tan 4x).$ [2]

(ii) Hence find $\D \displaystyle\int (1 + \sec^2 4x) dx.$ [3]

(iii) Hence show that $\D \displaystyle\int_{-\frac{\pi}{16}}^{\frac{\pi}{16}}

(1 + \sec^2 4x) dx = k(\pi +4),$ where $\D k$ is a constant to be found. [2]


5 (CIE 2013, s, paper 12, question 10)

(a) (i) Find $\D \displaystyle\int \sqrt{2x-5}dx.$ [2]

(ii) Hence evaluate $\D \displaystyle\int_{3}^{15}\sqrt{2x-5}dx.$ [2]

(b) (i) Find $\D \frac{d}{dx}(x^3\ln x).$ [2]

(ii) Hence find $\D \displaystyle\int x^2 \ln xdx.$ [3]


6 (CIE 2013, s, paper 22, question 11)

A curve has equation $\D y = 3x +\frac{1}{(x-4)^3}.$

(i) Find $\D \frac{dy}{dx}$ and $\D \frac{d^2y}{dx^2}.$ [4]

(ii) Show that the coordinates of the stationary points of the curve are (5, 16) and (3, 8). [2]

(iii) Determine the nature of each of these stationary points. [2]

iv) Find $\D \displaystyle\int \left(3x+\frac{1}{(x-4)^3}\right)dx.$ [2]

(v) Hence find the area of the region enclosed by the curve, the line $\D x = 5,$ the x-axis and the line $\D x = 6 .$ [2]


7 (CIE 2013, w, paper 11, question 9)

(a) Differentiate $\D 4x^3 \ln(2x +1)$  with respect to x. [3]

(b) (i) Given that $\D y=\frac{2x}{\sqrt{x+2}},$  show that $\D \frac{dy}{dx}=\frac{x+4}{(\sqrt{x+2})^3}.$ [4]

(ii) Hence find $\D \displaystyle\int \frac{5x+20}{(\sqrt{x+2})^3}dx.$ [2]

(iii) Hence evaluate $\D \displaystyle\int_{2}^{7}\frac{5x+20}{(\sqrt{x+2})^3}dx.$ [2]


8 (CIE 2014, s, paper 11, question 5)

(i) Given that $\D y= e^{x^2},$  find $\D \frac{dy}{dx}.$ [2]

(ii) Use your answer to part (i) to find $\D \displaystyle\int xe^{x^2} dx.$ [2]

(iii) Hence evaluate $\D \displaystyle\int_{0}^{2}xe^{x^2}dx.$ [2]


9 (CIE 2014, s, paper 23, question 10)

(i) Given that $\D y=\frac{2x}{\sqrt{x^2+21}},$  show that $\D \frac{dy}{dx}=\frac{k}{\sqrt{(x^2+21)^3}},$  where $\D k$ is a constant to be found. [5]

(ii) Hence find $\D \displaystyle\int \frac{6}{\sqrt{(x^2+21)^3}}dx$ and evaluate $\D \displaystyle\int_{2}^{10}\frac{6}{\sqrt{(x^2+21)^3}}dx.$ [3]


10 (CIE 2014, w, paper 13, question 8)

(i) Given that $\D f(x) = x \ln x^3 ,$ show that $\D f'(x) = 3(1+\ln x).$  [3]

(ii) Hence find $\D \displaystyle\int (1+\ln x)dx.$ [2]

(iii) Hence find $\D \displaystyle\int_{1}^{2}\ln x dx$  in the form $\D p + \ln q,$ where $\D p$ and $\D q$ are integers. [3]


11 (CIE 2014, w, paper 21, question 8)

(i) Given that $\D y=\frac{x^2}{2+x^2},$ show that $\D \frac{dy}{dx}=\frac{kx}{(2+x^2)^2},$ where $\D k$ is a constant to be found. [3]

(ii) Hence find $\D \displaystyle\int \frac{x}{(2+x^2)^2}dx.$ [2]



12 (CIE $2015, \mathrm{w}$, paper 23 , question 10)

(i) Given that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\mathrm{e}^{2-x^{2}}\right)=k x \mathrm{e}^{2-x^{2}}$, state the value of $k$. $[1]$

(ii) Using your result from part (i), find $\displaystyle\int 3 x \mathrm{e}^{2-x^{2}} \mathrm{~d} x$.

(iii) Hence find the area enclosed by the curve $y=3 x \mathrm{e}^{2-x^{2}}$, the $x$ -axis and the lines $x=1$ and $x=\sqrt{2}, \quad$ [2]

(iv) Find the coordinates of the stationary points on the curve $y=3 x \mathrm{e}^{2-x^{2}}$. [4]


13 (CIE 2016 , march, paper 12, question 10)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(x(2 x-1)^{\frac{3}{2}}\right)$. [3]

(ii) Hence, show that $\displaystyle\int x(2 x-1)^{\frac{1}{2}} \mathrm{~d} x=\frac{(2 x-1)^{\frac{3}{2}}}{15}(p x+q)+c$, where $c$ is a constant of integration, and $p$ and $q$ are integers to be found.

(iii) Hence find $\displaystyle\int_{0.5}^{1} x(2 x-1)^{\frac{1}{2}} \mathrm{~d} x$. [2]


14 (CIE 2016, s, paper 11, question 5) Do not use a calculator in this question.

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{e}^{4 x}}{4}-x \mathrm{e}^{4 x}\right)=p x \mathrm{e}^{4 x}$, where $p$ is an integer to be found. $[4]$

(ii) Hence find the exact value of $\displaystyle\int_{0}^{1 \mathrm{n} 2} x \mathrm{e}^{4 \mathrm{x}} \mathrm{d} x$, giving your answer in the form $a \ln 2+\frac{b}{c}$, where $a, b$ and $c$ are integers to be found. $[4]$


15 (CIE 2016, w, paper 13, question 6)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(\ln \left(3 x^{2}-11\right)\right)$.

(ii) Hence show that $\displaystyle\int \frac{x}{3 x^{2}-11} \mathrm{~d} x=p \ln \left(3 x^{2}-11\right)+c$, where $p$ is a constant to be found, and $c$ is a constant of integration.

(iii) Given that $\displaystyle\int_{2}^{a} \frac{x}{3 x^{2}-11} \mathrm{~d} x=\ln 2$, where $a>2$, find the value of $a$. $[4]$


16 (CIE $2016, \mathrm{w}$, paper 21 , question 8) The function $\mathrm{f}(x)$ is given by $\mathrm{f}(x)=\frac{3 x^{3}-1}{x^{3}+1}$ for $0 \leqslant x \leqslant 3$.

(i) Show that $\mathrm{f}^{\prime}(x)=\frac{k x^{2}}{\left(x^{3}+1\right)^{2}}$, where $k$ is a constant to be determined. [3]

(ii) Find $\displaystyle\int \frac{x^{2}}{\left(x^{3}+1\right)^{2}} \mathrm{~d} x$ and hence evaluate $\displaystyle\int_{1}^{2} \frac{x^{2}}{\left(x^{3}+1\right)^{2}} \mathrm{~d} x$.

(iii) Find $\mathrm{f}^{-1}(x)$, stating its domain.


17 (CIE 2017, march, paper 22, question 9)

(a) Find $\displaystyle\int \mathrm{e}^{2 x+1} \mathrm{~d} x$.

(b) (i) Given that $y=\frac{x}{\ln x}$, find $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

(ii) Hence find $\displaystyle\int\left(\frac{1}{\ln x}-\frac{1}{(\ln x)^{2}}+\frac{1}{x^{2}}\right) \mathrm{d} x$. [3]


18 (CIE 2017 , s, paper 11 , question 9)

(i) Show that $5+4 \tan ^{2}\left(\frac{x}{3}\right)=4 \sec ^{2}\left(\frac{x}{3}\right)+1$. $[1]$

(ii) Given that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\tan \left(\frac{x}{3}\right)\right)=\frac{1}{3} \sec ^{2}\left(\frac{x}{3}\right)$, find $\displaystyle\int \sec ^{2}\left(\frac{x}{3}\right) \mathrm{d} x$. $[1]$


(iii) The diagram shows part of the curve $y=5+4 \tan ^{2}\left(\frac{x}{3}\right)$. Using the results from parts (i) and (ii), find the exact area of the shaded region enclosed by the curve, the $x$ -axis and the lines $x=\frac{\pi}{2}$ and $x=\pi .$


19 (CIE 2017, s, paper 12 , question 11) The curve $y=\mathrm{f}(x)$ passes through the point $\left(\frac{1}{2}, \frac{7}{2}\right)$ and is such that $\mathrm{f}^{\prime}(x)=\mathrm{e}^{2 x-1}$.

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

(ii) Find the value of $x$ for which $f^{\prime \prime}(x)=4$, giving your answer in the form $a+b \ln \sqrt{2}$, where $a$ and $b$ are constants.


20 (CIE 2017, s, paper 13 , question 9 )

It is given that $\displaystyle\int_{-k}^{k}\left(15 e^{5 x}-5 e^{-5 x}\right) \mathrm{d} x=6$.

(i) Show that $\mathrm{e}^{5 k}-\mathrm{e}^{-5 k}=3$. $[5]$

(ii) Hence, using the substitution $y=\mathrm{e}^{5 k}$, or otherwise, find the value of $k$ $[3]$


21 (CIE 2017, s, paper 13, question 10) It is given that $y=(10 x+2) \ln (5 x+1)$.

(i) Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$. [4]

(ii) Hence show that $\displaystyle\int \ln (5 x+1) \mathrm{d} x=\frac{(a x+b)}{5} \ln (5 x+1)-x+c$, where $a$ and $b$ are integers and $c$ is a constant of integration. [3]

(iii) Hence find $\displaystyle\int_{0}^{\frac{1}{5}} \ln (5 x+1) \mathrm{d} x$, giving your answer in the form $\frac{d+\ln f}{5}$, where $d$ and $f$ are integers. $[2]$


22 (CIE 2017, s, paper 22, question 5)

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left[0.4 x^{5}(0.2-\ln 5 x)\right]=k x^{4} \ln 5 x$, where $k$ is an integer to be found. [2]

(ii) Express $\ln 125 x^{3}$ in terms of $\ln 5 x$. $[1]$

(iii) Hence find $\displaystyle\int\left(x^{4} \ln 125 x^{3}\right) \mathrm{d} x$. [2]


23 (CIE $2017, \mathrm{w}$, paper 13 , question 2) A curve is such that its gradient at the point $(x, y)$ is given by $10 \mathrm{e}^{5 x}+3$. Given that the curve passes though the point $(0,9)$, find the equation of the curve.


24 (CIE 2017, w, paper 21, question 5)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{5}{3 x+2}\right)$. [2]

(ii) Use your answer to part (i) to find $\displaystyle\int \frac{30}{(3 x+2)^{2}} \mathrm{~d} x$. [2]

(iii) Hence evaluate $\displaystyle\int_{1}^{2} \frac{30}{(3 x+2)^{2}} \mathrm{~d} x$. $[2]$


25 (CIE 2017, w, paper 22, question 9)

(i) Find $\frac{\mathrm{d}}{\mathrm{d} x}(x \ln x)$. $[2]$

(ii) Hence find $\displaystyle\int \ln x \mathrm{~d} x$. [2]

(iii) Hence, given that $k>0$, show that $\displaystyle\int_{k}^{2 k} \ln x \mathrm{~d} x=k(\ln 4 k-1)$. [4]


26 (CIE 2017, w, paper 23 , question 9)

(i) Show that $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\ln x}{x^{3}}\right)=\frac{1-3 \ln x}{x^{4}}$. [3]

(ii) Find the exact coordinates of the stationary point of the curve $y=\frac{\ln x}{r^{3}}$. $[3]$

(iii) Use the result from part (i) to find $\displaystyle\int\left(\frac{\ln x}{x^{4}}\right) \mathrm{d} x$. [4]


27 (CIE 2018, march, paper 12 , question 10)


The diagram shows the graph of $y=(x+2)^{2}(1-3 x)$. The curve has a minimum at the point $A$, a maximum at the point $B$ and intersects the $y$ -axis and the $x$ -axis at the points $C$ and $D$ respectively.

(i) Find the $x$ -coordinate of $A$ and of $B$. $[5]$

(ii) Write down the coordinates of $C$ and of $D$. [2]

(iii) Showing all your working, find the area of the shaded region. [5]


28 (CIE 2018, march, paper 22, question 6)

(i) Differentiate $1+\tan \left(\frac{x}{3}\right)$ with respect to $x$. [2]

(ii) Hence find $\displaystyle\int \sec ^{2}\left(\frac{x}{3}\right) \mathrm{d} x$. [2]


Answer

1. (i) $\D 2x \ln x + x$

(ii) $\D 0.5x^2 \ln x - x^2/4$

2. (i) $\D k = 10$

(ii) $\D (0,0),$ min

(iii) $\D \frac{x^2}{2(1+x^2)},$

$\D 0.15$

3. (i) $\D A = 6$

(ii) $\D 2y - 3 = 3x$

(iii) $\D \frac{e^{2x}}{2(1+e^{2x})}, 0.2$

4. (i) $\D 4 \sec^2 4x$

(ii) $\D x + \frac{1}{4}\tan 4x$

(iii) $\D k=1/8$

5. (a)(i) $\D \frac{1}{3}(2x - 5)^{3/2}$

(ii) $\D 124/3$

(b)(i) $\D x^2 + 3x^2 \ln x$

(ii) $\D \frac{1}{3}(x^3 \ln x - \frac{x^3}{3})$

6. (i) $\D y'= 3-3(x-4)^{-4}$

$\D y''=12(x-4)^{-5}$

(iii) $\D x = 5$, min, $\D x = 3,$ max

(iv) $\D \frac{3x^2}{2}-\frac{(x-4)^2}{2}$

(v) $\D 135/8$

7. (a) $\D 12x^2 \ln(2x+1)+8x^3/(2x+1)$

(b)(ii) $\D \frac{10x}{\sqrt{x+2}}$

(iii) $\D 40/3$

8. $\D 2xe^{x^2},0.5e^{x^2},26.8$

9. (i) $\D k = 42$

(ii) $\D \frac{8}{55}$

10. (ii) $\D x \ln x$

(iii) $\D -1 + \ln 4$

11. (i) $\D k = 4$

(ii) $\D \frac{x^2}{4(2+x^2)}$


12. (i) $k=-2$

(ii) $(-3 / 2) e^{2-x^{2}}$

(iii) $2.58$

(iv) $x=\pm 0.707, y=\pm 9.51$

13. (i) $3 x(2 x-1)^{1 / 2}+(2 x-1)^{3 / 2}$

(ii) $p=3, q=1($ iii $) 4 / 15$

14. (i) $-4 x e^{4 x}$

(ii) $4 \ln 2-15 / 16$

15. (i) $6 x /\left(3 x^{2}-11\right)$

(ii) $p=1 / 6$, (iii) $a=5$

16. (i) $k=12(\mathrm{ii}) 7 / 54$

(iii) $f^{-1}(x)=\sqrt[3]{\frac{x+1}{3-x}}$

$\mathrm{D}:-1 \leq x \leq 20 / 7$

17. (a). $5 e^{2 x+1}(+\mathrm{c})$

(bi) $(\ln x-1) /(\ln x)^{2}$

(bii) $x / \ln x-1 / x(+c)$

18. (ii) $3 \tan (\mathrm{x} / 3)$

(iii) $8 \sqrt{3}+\pi / 2$

19. (i) $f(x)=(1 / 2) e^{2 x-1}+3$

(ii) $x=1 / 2+\ln \sqrt{2}$

20. $k=0.239$

21. (i) $(10 x+2) \times \frac{5}{5 x+1}+10 \ln (5 x+1)$

(ii) $\frac{5 x+1}{5} \ln (5 x+1)-x$

(iii) $\frac{-1+\ln 4}{5}$

22. (ii) $3 \ln 5 x$

(iii) $-1.5\left(0.4 x^{3}(0.2-\ln 5 x)\right)$


23. $y=2 e^{5 x}+3 x+7$

24. (i) $15(3 x+2)^{-2}$

(ii) $-10 /(3 x+2)$ (iii) $3 / 4$

25. (i) $1+\ln x$

(ii) $x \ln x-x+(c)$

(iii) $k(\ln 4 k-1)$

26. (ii) $\left(e^{1 / 3}, 1 /(3 e)\right)$

(iii) $-1 /\left(9 x^{3}\right)-\ln x /\left(3 x^{3}\right)$

27. (i) $-2,-4 / 9$

(ii) $C(0,4), D(1 / 3,0)$

(iii) $0.744$

28. $(1 / 3) \sec ^{2}(x / 3)$

$3 \tan (x / 3)+c$


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