CIE (Differentation, Additional Mathematics -2018)

 


$\def\D{\displaystyle}\def\frac{\dfrac}$

1 (CIE 2012, s, paper 12, question 5)

Differentiate the following with respect to $\D x.$

(i) $\D (2-x^2)\ln (3x+1)$ [3]

(ii) $\D \frac{4-\tan 2x}{5x}$ [3]


2 (CIE 2012, s, paper 21, question 7)

Given that $\D f(x)=x^2-\frac{648}{\sqrt{x}},$ find the value of $\D x$ for which $\D f''(x)=0.$ [6]



3 (CIE 2012, w, paper 11, question 5)

Given that $\D y=\frac{x^2}{\cos 4x},$ find

(i) $\D \frac{dy}{dx}$, [3]

(ii) the approximate change in $\D y$ when $\D x$ increases from $\D \frac{\pi}{4}$to $\D \frac{\pi}{4}+p$, where $\D p$ is small. [2]


4 (CIE 2013, s, paper 22, question 7)

Differentiate with respect to $\D x$,

(i) $\D (3-5x)^{12}$ [2]

(ii)$\D x^2\sin x$ [2]

(iii)$\D \frac{\tan x}{1+e^{2x}}$ [4]


5 (CIE 2014, s, paper 21, question 7)

Given that a curve has equation $\displaystyle y=\frac{1}{x}+2\sqrt{x},$ where $\D x>0,$ find

(i) $\D \frac{dt}{dx},$ [2]

(ii) $\D \frac{d^2y}{dx^2}.$ [2]


Hence, or otherwise, find

(iii) the coordinates and nature of the stationary point of the curve. [4]


6 (CIE 2014, s, paper 21, question 10)

Find $\D \frac{dy}{dx}$ when

(i) $\D y=\cos 2x  \sin\left(\frac{x}{3}\right),$ [4]

(ii) $\D y=\frac{\tan x}{1+\ln x}.$ [4]


7 (CIE 2014, s, paper 22, question 7)

Differentiate with respect to $\D x$

(i) $\D x^4e^{3x},$ [2]

(ii) $\D \ln(2+\cos x),$ [2]

(iii) $\D \frac{\sin x}{1+\sqrt{x}}.$ [3]


8 (CIE 2015, w, paper 23, question 3)

(a) Given that $\D y=\frac{x^3}{2-x^2},$ find $\D \frac{dy}{dx}.$ [3]

(b) Given that $\D  y=x\sqrt{4x+6}, $ show that $\D \frac{dy}{dx}= \frac{k(x+1)}{\sqrt{4x+6}}$ and state the value of $\D k.$ [3]


9 (CIE 2016, march, paper 22, question 1)

Two variables $\D x$ and $\D y$ are such that $\D y=\frac{5}{\sqrt{x-9}}$ for $\D x>9$.

(i) Find an expression for $\D \frac{dy}{dx}.$ [2]

(ii) Hence, find the approximate change in $\D y$ as $\D x$ increases from 13 to $\D 13+h$, where $\D h$ is small. [2]


10 (CIE 2016, s, paper 12, question 6)

Show that $\D \frac{d}{dx}(e^{3x} \sqrt{4x+1})$ can be written in the form $\D \frac{e^{3x}(px+q)}{\sqrt{4x+1}},$ where $p$ and $q$ are integers to be found. [5]


11 (CIE 2017, s, paper 11, question 10)

(a) Given that $\D y=\frac{e^{3x}}{4x^2+1},$ find $\D \frac{dy}{dx}.$ [3]

(b) Variables $x,y,$ and $t$ are such that $\D y=4\cos \left(x+\frac{\pi}{3}\right)+ 2\sqrt{3}\sin\left(x+\frac{\pi}{3}\right)$ and $\D \frac{dy}{dt}=10.$

(i) Find the value of $\D \frac{dy}{dx}$ when $\D x=\frac{\pi}{2}.$ [3]

(ii) Find the value of $\D\frac{dx}{dt}$ when $\D x=\frac{\pi}{2}.$ [2]


12 (CIE 2017, s, paper 12, question 2)

It is given that $\D \frac{(5x^2+4)^{\frac{1}{2}}}{x+1}.$ Showing all your working, find the exact value of $\D \frac{dy}{dx}$ when $\D x = 3.$ [5]


13 (CIE 2017, s, paper 21, question 3)

The variables $\D x$ and $\D y$ are such that $\D y = \ln(x^2 + 1).$

(i) Find an expression for $\D \frac{dy}{dx}.$ [2]

(ii) Hence, find the approximate change in $\D y$ when $\D x $ increases from 3 to $\D 3 + h,$ where $h$ is small. [2]


14 (CIE 2017, s, paper 23, question 7)

Differentiate with respect to $\D x$.

(i) $\D (1+4x)^{10}\cos x,$

(ii) $\D\frac{e^{4x-5}}{\tan x}.$


15 (CIE 2017, w, paper 12, question 4)

Given that $\D y= \frac{\ln(3x^2+2)}{x^2+1},$ find the value of $\D \frac{dy}{dx}$ when $\D x = 2,$ giving your answer as $\D a + b \ln 14,$ where $a$ and $b$ are fractions in their simplest form. [6]


16 (CIE 2017, w, paper 21, question 7)

Find $y$ in terms of $x$, given that $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=6 x+\frac{2}{x^{3}}$ and that when $x=1, y=3$ and $\frac{\mathrm{d} y}{\mathrm{~d} x}=1$.


17 (CIE 2017, w, paper 21, question 12)

(i) Differentiate $(\cos x)^{-1}$ with respect to $x$.

(ii) Hence find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ given that $y=\tan x+4(\cos x)^{-1}$.

(iii) Using your answer to part (ii) find the values of $x$ in the range $0 \leqslant x \leqslant 2 \pi$ such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=4$. [6]


18 (CIE 2017, w, paper 22, question 10)

(i) Without using a calculator, solve the equation $6 c^{3}-7 c^{2}+1=0$. $[5]$ It is given that $y=\tan x+6 \sin x$.

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

(iii) If $\frac{\mathrm{d} y}{\mathrm{~d} x}=7$ show that $6 \cos ^{3} x-7 \cos ^{2} x+1=0$. [2]

(iv) Hence solve the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=7$ for $0 \leqslant x \leqslant \pi$ radians. [2]


19 (CIE 2018 , march, paper 12 , question 2)

A curve has equation $y=4+5 \sin 3 x$.

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

(ii) Hence find the equation of the tangent to the curve $y=4+5 \sin 3 x \quad$ at the point where $x=\frac{\pi}{3}$. [3]


20 (CIE 2018, march, paper 12 , question 4 ) It is given that $y=\frac{\ln \left(4 x^{2}-1\right)}{x+2}$.

(i) Find the values of $x$ for which $y$ is not defined.

[2]

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

(iii) Hence find the approximate increase in $y$ when $x$ increases from 2 to $2+h$, where $h$ is small. [2]


21 (CIE 2018, s, paper 11, question 11) 


The diagram shows part of the graph of $y=16 x+\frac{27}{x^{2}}$, which has a minimum at $A$.

(i) Find the coordinates of $A$. [4]

The points $P$ and $Q$ lie on the curve $\quad y=16 x+\frac{27}{x^{2}}$ and have $x$ -coordinates 1 and 3 respectively.

(ii) Find the area enclosed by the curve and the line $P Q$. You must show all your working. [6]


$22(\mathrm{CIE} 2018, \mathrm{~s}$, paper 12, question 6$)$

Find the coordinates of the stationary point of the curve $y=\frac{x+2}{\sqrt{2 x-1}}$.


$23(\mathrm{CIE} 2018, \mathrm{~s}$, paper 21, question 2

The variables $x$ and $y$ are such that $y=\ln (3 x-1)$ for $x>\frac{1}{3}$.

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

(ii) Hence find the approximate change in $x$ when $y$ increases from $\ln (1.2)$ to $\ln (1.2)+0.125$.


24 (CIE 2018, s, paper 21, question 7)

Differentiate with respect to $x$

(i) $4 x \tan x$ [2]

(ii) $\frac{\mathrm{e}^{3 x+1}}{x^{2}-1}$.


25 (CIE 2018, s, paper 22, question 9)

(i) Differentiate $x^{4}(\sqrt{\sin x})$ with respect to $x$.

(ii) Hence find $\int\left(x+\frac{x^{4} \cos x}{\sqrt{\sin x}}+8 x^{3}(\sqrt{\sin x})\right) \mathrm{d} x$.


Answers


1.(i) $\D\frac{3(2-x^2)}{3x+1}-2x\ln(3x+1)$

(ii) $\D \frac{-10x\sec^22x+5\tan2x-20}{25x^2}$

2. $\D x=9$

3. (i) $\D \frac{2x\cos4x+4x^2\sin4x}{\cos^24x}$

(ii) $\D \frac{-\pi}{2}p$

4. (i) $\D -60(3-5x)^{11}$

(ii) $\D x^2\cos x+2x\sin x$

(iii)$\D \frac{(1+e^{2x})\sec^2x-\tan x(2e^{2x})}{(1+e^{2x})^2}$

5. (i)$\D \frac{-1}{x^2}+\frac{1}{\sqrt{x}}$

(ii)$\D \frac{2}{x^3}-\frac{1}{2x^{\frac{3}{2}}}$

(iii) $\D (1,3)$, min

6. (i) $\D \cos\frac{x}{3}\frac{\cos2x}{3} -2\sin(2x)\sin(\frac{x}{3})$

(ii) $\D \frac{(1+\ln x)\sec^2x-\frac{\tan x}{x}}{(1+\ln x)^2}$

7. (i) $\D 4x^3e^{3x}+ 3x^4e^{3x}$

(ii) $\D \frac{-\sin x}{2+\cos x}$

(iii) $\frac{(1+\sqrt{x})\cos x-\sin x(\frac{1}{2\sqrt{x}})}{(1+\sqrt{x})^2}$

8. (a)$\D \frac{(2-x^2)3x^2+2x^4}{(2-x^2)^2}$

(b) $\D k=6$

9. (i) $\D\frac{-5}{2(x-9)^{3/2}}$

(ii) $\D -0.3125h$

10. $\D p=12,q=5$

11. (i) (a) $\D \frac{3e^{3x}}{4x^2+1}- \frac{8xe^{3x}}{(4x^2+1)^2}$

(b) $\D -5,-2$

12. 11/112

13 (i) $\D \frac{2x}{x^2+1}$

(ii) $\D 0.6 h$

14(i) $\D (1+4x)^{10}(-\sin x)+40 (1+4x)^9\cos x$

(ii) $\D \frac{e^{4x-5}(4\tan x)-\sec^2x}{\tan x}$

15. $\D \frac{6}{35}-\frac{4}{25}\ln 14$

16. $y=x^{3}+1 / x-x+2$

17. (i) $\sin x /\left(\cos ^{2} x\right)$

(ii) $\sec ^{2} x+4 \sin x /\left(\cos ^{2} x\right)$

(iii) $\pi / 6,5 \pi / 6$

18. (i) $1,1 / 2,-1 / 3$

(ii) $\sec ^{2} x+6 \cos x$

$(\mathrm{iv}) 0,1.05,1.91$

19. (i) $15 \cos 3 x$

(ii) $y=-15 x+5 \pi+4$

20. (i) $-2,-.5 \leq x \leq .5$

(ii) $\frac{(x+2) 8 x}{4 x^{2}-1}-\ln \left(4 x^{2}-1\right)$

(iii) $4 / 15-\ln (15) / 16$

21. $x=\frac{3}{2}, y=36, \mathrm{~A}=12$

22. $x=3, y=\sqrt{5}$

23. (i) $3 /(3 \mathrm{x}-1)$

(ii) $0.05$

24. (i) $4 \tan x+4 x \sec ^{2} x$

(ii) $\frac{\left(x^{2}-1\right)\left(3 e^{3 x+1}\right)-2 x e^{3 x+1}}{\left(x^{2}+1\right)^{2}}$

25. (i) $4 x^{3} \sqrt{\sin x}$

$+\left(x^{4} / 2\right)\left(\cos x(\sin x)^{-1 / 2}\right) / 2$

(ii) $x^{2} / 2+2 x^{4} \sqrt{\sin x}[+c]$



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