$\def\frac{\dfrac}$

$\def\D{\displaystyle}\newcommand{\vcol}[2]{\begin{array}{|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}\newcommand{\vicol}[2]{\begin{array}{|c|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}\newcommand{\viicol}[2]{\begin{array}{|c|c|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}$

1 (CIE 2012, s, paper 11, question 7)

The table shows values of variables $\D x$ and $\D y.$

$\D \vicol{x& 1& 3& 6& 10& 14}{y& 2.5& 4.5& 0 &–20 &–56}$

(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = Ax + Bx^2,$ where $\D A$ and $\D B$ are constants. [4]

(ii) Use your graph to find the value of $\D A$ and of $\D B.$ [4]

2 (CIE 2012, s, paper 22, question 7)

The table shows experimental values of variables $\D x$ and $\D y.$

$\D \vcol{x& 5& 30& 150& 400}{y& 8.9& 21.9& 48.9& 80.6}$

(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. [4]

(ii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [4]

3 (CIE 2012, w, paper 11, question 10)

The table shows values of the variables $\D x$ and $\D y.$

$\D\vicol{ x^{\circ}& 10& 30 &45 &60 &80}{ y &11.2& 16 &19.5& 22.4& 24.7}$

(i) Using the graph paper below, plot a suitable straight line graph to show that, for 10° $\D \le x\le $ 80°, $\D \sqrt{y} = A \sin x + B,$ where $\D A$ and $\D B$ are positive constants. [4]

(ii) Use your graph to find the value of $\D A$ and of $\D B.$ [3]

(iii) Estimate the value of $\D y$ when $\D x = 50.$ [2]

(iv) Estimate the value of $\D x$ when $\D y = 12.$ [2]

4 (CIE 2012, w, paper 22, question 8)

(i) Express $\D y$ in terms of $\D x,$ giving your answer in the form $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. [5]

Another method of drawing a straight line graph for the relationship $\D y = ax^b,$ found in part (i), involves plotting $\D \lg x$ on the horizontal axis and $\D \lg(y^2)$ on the vertical axis. For this straight line graph what is

(ii) the gradient, [1]

(iii) the intercept on the vertical axis? [1]

5 (CIE 2012, w, paper 23, question 9)

The table shows experimental values of two variables $\D x$ and $\D y.$

$\D \vcol{ x& 1& 2& 3& 4}{y& 9.41 &1.29& – 0.69& – 1.77}$

It is known that $\D x$ and $\D y$ are related by the equation $\D y = \frac{a}{x^2}+bx,$ where $\D a$ and $\D b$ are constants.

(i) A straight line graph is to be drawn to represent this information. Given that $\D x^2y$ is plotted on the vertical axis, state the variable to be plotted on the horizontal axis. [1]

(ii) On the grid opposite, draw this straight line graph. [3]

(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [3]

(iv) Estimate the value of $\D y$ when $\D x$ is 3.7. [2]

6 (CIE 2013, s, paper 11, question 2)

Variables $\D x$ and $\D y$ are such that $\D y= Ab^x,$ where $\D A$ and $\D b$ are constants. The diagram shows the graph of $\D \ln y$ against $\D x,$ passing through the points (2, 4) and (8, 10). Find the value of $\D A$ and of $\D b.$ [5]

7 (CIE 2013, s, paper 22, question 1)

Variables $\D x$ and $\D y$ are such that when $\D \sqrt{y}$ is plotted against $\D x^2$ a straight line graph passing through the points (1, 3) and (4, 18) is obtained. Express $\D y$ in terms of $\D x.$ [4]

8 (CIE 2013, w, paper 13, question 10)

The variables $\D s$ and $\D t$ are related by the equation $\D t= ks^n,$ where $\D k$ and $\D n$ are constants. The table below shows values of variables $\D s$ and $\D t.$

$\D \vcol{s& 2& 4& 6& 8}{t& 25.00& 6.25& 2.78& 1.56}$

(i) A straight line graph is to be drawn for this information with $\D \lg t$ plotted on the vertical axis. State the variable which must be plotted on the horizontal axis. [1]

(ii) Draw this straight line graph on the grid below. [3]

(iii) Use your graph to find the value of $\D k$ and of $\D n.$ [4]

(iv) Estimate the value of $\D s$ when $\D t = 4.$ [2]

9 (CIE 2013, w, paper 21, question 8)

The table shows experimental values of two variables $\D x$ and $\D y.$

$\D \vcol{x& 2 &4& 6& 8}{y& 9.6& 38.4& 105& 232}$

It is known that $\D x$ and $\D y$ are related by the equation $\D y= ax^3+ bx,$ where $\D a$ and $\D b$ are constants.

(i) A straight line graph is to be drawn for this information with $\D \frac{y}{x}$ on the vertical axis. State the variable which must be plotted on the horizontal axis. [1]

(ii) Draw this straight line graph on the grid below. [2]

(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ [3]

(iv) Estimate the value of $\D x$ for which $\D 2y = 25x.$ [2]

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

The table shows values of variables $\D V$ and $\D p.$

$\D \vcol{ V &10& 50& 100& 200}{p& 95.0& 8.5& 3.0& 1.1}$

(i) By plotting a suitable straight line graph, show that $\D V$ and $\D p$ are related by the equation $\D p = kV^n ,$

where $\D k$ and $\D n$ are constants. [4]

Use your graph to find

(ii) the value of $\D n,$ [2]

(iii) the value of $\D p$ when $\D V = 35.$ [2]

11 (CIE 2014, s, paper 13, question 10)

The table shows experimental values of $\D x$ and $\D y.$

$\D \vcol{x& 1.50 &1.75& 2.00& 2.25}{y& 3.9& 8.3 &19.5& 51.7}$

(i) Complete the following table.

$\D \vcol{x^2&\qquad &\qquad &\qquad &\qquad}{\lg y&&&&}$[1]

(ii) By plotting a suitable straight line graph on the graph paper, show that $\D x$ and $\D y$ are related by the equation $\D y= Ab^{x^2},$ where $\D A$ and $\D b$ are constants. [2]

(iii) Use your graph to find the value of $\D A$ and of $\D b.$ [4]

(iv) Estimate the value of $\D y$ when $\D x = 1.25.$ [2]

12 (CIE 2014, s, paper 22, question 10)

Two variables $\D x$ and $\D y$ are connected by the relationship $\D y = Ab^x ,$ where $\D A$ and $\D b$ are constants.

(i) Transform the relationship $\D y = Ab^x$ into a straight line form. [2]

An experiment was carried out measuring values of $\D y$ for certain values of $\D x.$ The values of $\D \ln y$ and $\D x$ were plotted and a line of best fit was drawn. The graph is shown on the grid below.

(ii) Use the graph to determine the value of $\D A$ and the value of $\D b,$ giving each to 1 significant figure. [4]

(iii) Find $\D x$ when $\D y = 220.$ [2]

13 (CIE 2014, w, paper 11, question 9)

The table shows experimental values of variables $\D x$ and $\D y.$

$\D \vicol{x& 2& 2.5& 3 &3.5& 4}{y& 18.8& 29.6& 46.9& 74.1 &117.2}$

(i) By plotting a suitable straight line graph on the grid below, show that $\D x$ and $\D y$ are related by the equation $\D y = ab^x ,$ where $\D a$ and $\D b$ are constants. [4]

(ii) Use your graph to find the value of $\D a$ and of $\D b.$ [4]

14 (CIE 2014, w, paper 23, question 6)

Variables $\D x$ and $\D y$ are such that, when $\D \ln y$ is plotted against $\D 3^x ,$ a straight line graph passing through (4, 19) and (9, 39) is obtained.

(i) Find the equation of this line in the form $\D \ln y= m3^x+ c,$ where $\D m$ and $\D c$ are constants to be found. [3]

(ii) Find $\D y$ when $\D x = 0.5.$ [2]

(iii) Find $\D x$ when $\D y = 2000.$ [3]

15 (CIE 2015, s, paper 21, question 10)

The relationship between experimental values of two variables, $x$ and $y$, is given by $y=A b^{x}$, where $A$ and $b$ are constants.

(i) By transforming the relationship $y=A b^{x}$, show that plotting $\ln y$ against $x$ should produce a straight line graph. $\quad[2]$

(ii) The diagram below shows the results of plotting $\ln y$ against $x$ for 7 different pairs of values of variables, $x$ and $y .$ A line of best fit has been drawn.

By taking readings from the diagram, find the value of $A$ and of $b$, giving cach value correct to significant figure. $[4]$

(iii) Estimate the value of $y$ when $x=2.5$. [2]

*****

$\begin{aligned}&\begin{aligned}\text{(i) }y=& A b^{x} \\\ln y=& \ln A+x \ln b \\\text { Choose } &(1.5,10),(4,7.6) \\\text { gradient } &=\frac{7.6-10}{4-1.5}=-0.96 \\\ln y-10 &=-0.96(x-1.5) \\\ln y &=-0.96 x+11.44\qquad \cdots (1) \\y &=e^{-0.96 x} \cdot e^{11.44} \\&=\left(-e^{0.96}\right)^{x} \cdot 92967 \\&=(0.38)^{x}(90000) \\\therefore A &=90,000, \quad b=0.4 . \\\text { (ii) When } x &=2.5, \text { by }(1) \\\ln y &=-0.96(2.5)+11.44=9 \\y &=e^{9}=8103\end{aligned}\end{aligned}$

**********end math solution********************

16 (CIE 2015, w, paper 11, question 7)

Two variables, $x$ and $y$, are such that $y=A x^{h}$, where $A$ and $b$ are constants. When $\ln y$ is plotted against $\ln x$, a straight line graph is obtained which passes through the points $(1.4,5.8)$ and $(2.2,6.0)$.

(i) Find the value of $A$ and of $b$. $[4]$

(ii) Calculate the value of $y$ when $x=5$. $[2]$

*****

$\begin{aligned}&y=A x^{b} \\&\ln y=\ln A+b \ln x \quad \ldots(1) \\&\text { Two points }(1.4,5.8),(2.2,6.0) \\&\text { gradient }=\frac{6.0-5.8}{2.2-1.4}=\frac{1}{4} \\&\text { line equation } \\&\qquad \begin{aligned}\ln y-6 &=\frac{1}{4}(\ln x-2.2) \\\ln y &=\frac{1}{4} \ln x+5.45\end{aligned} \\&\text { compare with equation (1), } \\&\qquad \begin{array}{l}b=\frac{1}{4}, \ln A=5.45 \Rightarrow A=e^{5.45} \\\text { When } x=5,-y=e^{5.45} \cdot 5^{\frac{1}{4}}=348\end{array}\end{aligned}$

**********end math solution********************

17 (CIE 2015, w, paper 23, question 11)

The trees in a certain forest are dying because of an unknown virus.

The number of trees, $N$, surviving $t$ years after the onset of the virus is shown in the table below.

$\begin{array}{|c|c|c|cc|c|c|}\hline \text{t }& 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text{N } & 2000 & 1300 & 890 & 590 & 395 & 260 \\\hline\end{array}$

The relationship between $N$ and $t$ is thought to be of the form $N=A b^{-t}$.

(i) Transform this relationship into straight line form.

(ii) Using the given data, draw this straight line on the grid below. $[3]$

(iii) Use your graph to estimate the value of $A$ and of $b$.

If the trees continue to die in the same way, find

(iv) the number of trecs surviving after 10 years,

(v) the number of years taken until there are only 10 trees surviving. [2]

*****

$\begin{aligned}\text{(i) }&\log N=\log A-t \log b\\\text{(ii) }&\begin{array}[t]{|c|c|c|c|c|c|c|}\hline t & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \log N & 3.30 & 3.11 & 2.95 & 2.77 & 2.60 & 2.41 \\\hline \end{array}\end{aligned}$

Choose two points $(1,3.30)$ and $(6,2.415)$.

$\begin{aligned}\text{(iii) }&\text { gradient }=-\log b=\frac{2.415-3.3}{5} \rightarrow b=1.5\\&\text { intercept }=\log A=3.47 \rightarrow A=2950\\\text{(iv) }&t=10 \rightarrow N=\frac{2950}{1.5^{10}}=51\\\text{(v) }&N=10 \rightarrow 1.5^{t}=295 \rightarrow t=\frac{\log 295}{\log 1.5}\\&=14 \text { years }\end{aligned}$

**********end math solution********************

18 (CIE 2016, march, paper 12, question 8)

The variables $x$ and $y$ are such that when $\lg y$ is plotted against $\lg x$ the straight line graph shown above is obtained.

(i) Given that $y=A x^{b}$, find the value of $A$ and of $b$.

(ii) Find the value of $\lg y$ when $x=100$. $[2]$

(iii) Find the value of $x$ when $y=8000$. $[2]$

19 (CIE 2016, s, paper 12 , question 8)

Variables $x$ and $y$ are such that when $\lg y$ is plotted against $x^{2}$, the straight line graph shown above is obtained.

(i) Given that $y=A b^{x^{2}}$, find the value of $A$ and of $b$. $[4]$

(ii) Find the value of $y$ when $x=1.5$.

(iii) Find the positive value of $x$ when $y=2$.

20 (CIE 2016, w, paper 11, question 11)

The variables $x$ and $y$ are such that when $\ln y$ is plotted against $x$, a straight line graph is obtained. This line passes through the points $x=4, \ln y=0.20$ and $x=12, \ln y=0.08$.

(i) Given that $y=A b^{x}$, find the value of $A$ and of $b$. [5]

(ii) Find the value of $y$ when $x=6$. [2]

(iii) Find the value of $x$ when $y=1.1$. [2]

21 (CIE 2016, w, paper 13, question 7)

The variables $x$ and $y$ are such that when $\ln y$ is plotted against $\frac{1}{x}$ the straight line graph shown above is obtained.

(i) Given that $y=A \mathrm{e}^{\frac{6}{x}}$, find the value of $A$ and of $b$.

(ii) Find the value of $y$ when $x=0.32$.

(iii) Find the value of $x$ when $y=20$.

22 (CIE 2017, s, paper 21, question 10)

The table shows values of the variables $t$ and $P$.

$\begin{array}{|c|c|c|c|c|}\hline\text{t } & 1 & 1.5 & 2 & 2.5 \\\hline\text{P} & 4.39 & 8.33 & 15.8 & 30.0 \\\hline\end{array}$

(i) Draw the graph of $\ln P$ against $t$ on the grid below.

(ii) Use the graph to estimate the value of $P$ when $t=2.2$.

(iii) Find the gradient of the graph and state the coordinates of the point where the graph meets the vertical axis.

(iv) Using your answers to part (iii), show that $P=a b^{t}$, where $a$ and $b$ are constants to be found. $[3]$

(v) Given that your equation in part (iv) is valid for values of $t$ up to 10, find the smallest value of $t$, correct to 1 decimal place, for which $P$ is at least 1000 .

23 (CIE $2017, \mathrm{~s}$, paper 23 , question 3)

Variables $x$ and $y$ arc such that when $\sqrt[3]{y}$ is plotted against $\frac{1}{x}$, a straight line graph passing through the points $(0.2,5)$ and $(1,13)$ is obtained. Express $y$ in terms of $x$

24 (CIE 2017, w, paper 11, question 4)

When $\lg y$ is plotted against $x^{2}$ a straight line is obtained which passes through the points $(4,3)$ and $(12,7)$

(i) Find the gradient of the linc. [1]

(ii) Use your answer to part (i) to express $\lg y$ in terms of $x$. [2]

(iii) Hence express $y$ in terms of $x$, giving your answer in the form $y=A\left(10^{b x^{2}}\right)$ where $A$ and $b$ are constants.

25 (CIE 2017, w, paper 12, question 5)

When $\lg y$ is plotted against $x$, a straight linc is obtained which passes through the points $(0.6,0.3)$ and $(1.1,0.2)$

(i) Find $\lg y$ in terms of $x$. $[4]$

(ii) Find $y$ in terms of $x$, giving your answer in the form $y=A\left(10^{6 x}\right)$, where $A$ and $b$ are constants.

26 (CIE $2017, \mathrm{w}$, paper 13 , question 6 )

When $\ln y$ is plotted against $x^{2}$ a straight line is obtained which passes through the points $(0.2,2.4)$ and $(0.8,0.9)$

(i) Express $\ln y$ in the form $p x^{2}+q$, where $p$ and $q$ are constants. [3]

(ii) Hencc cxpress $y$ in terms of $z$, where $z=\mathrm{c}^{x^{2}}$. [3]

27 (CIE 2018, march, paper 12, question 9)

The table shows values of the variables $x$ and $y$.

$\begin{array}{|c|c|c|c|c|c|}\hline\text{x} & 2 & 4 & 6 & 8 & 10 \\\text{y} & 736 & 271 & 100 & 37 & 13 \\\hline\end{array}$

The relationship between $x$ and $y$ is thought to be of the form $y=A \mathrm{c}^{b x}$, where $A$ and $b$ arc constants.

(i) Transform this relationship into straight line form.

(ii) Hence, by plotting a suitable graph, show that the relationship $y=A \mathrm{e}^{h x}$ is correct.

(iii) Use your graph to find the value of $A$ and of $b$.

(iv) Estimate the value of $x$ when $y=500$.

(v) Estimate the value of $y$ when $x=5$.

28 (CIE 2018, s, paper 12, question 3) The variables $x$ and $y$ are such that when $\mathrm{e}^{y}$ is plotted against $x^{2}$, a straight line graph passing through the points $(5,3)$ and $(3,1)$ is obtained. Find $y$ in terms of $x .$

29 (CIE 2018, s, paper 21, question 8)

An experiment was carried out recording values of $y$ for certain values of $x$. The variables $x$ and $y$ are thought to be connected by the relationship $y=a x^{n}$, where $a$ and $n$ are constants.

(i) Transform the relationship $y=a x^{n}$ into straight line form.

The values of $\ln y$ and $\ln x$ were plotted and a line of best fit drawn. This is shown in the diagram below.

(ii) Use the graph to find the value of $a$ and of $n$, stating the coordinates of the points that you use. [3]

(iii) Find the value of $x$ when $y-50$.

**Answers**

1. (i) $\D y/x = A + Bx$

$\D \vicol{x& 1& 3& 6& 10& 14}{y/x& 2.5& 1.5& 0& -2& -4}$

(ii) $\D B = -0.5;A = 3$

2. (i) $\D \ln y = \ln a + b \ln x$

(ii) $\D b = 0.5; a = 4$

(iii) 32 to 49

3. (i) $\D \vicol{\sin x& 0.17& 0.5& 0.71& 0.87& 0.98}{\sqrt{y}& 3.35& 4 &4.42& 4.73& 4.97}$

(ii) $\D A = 2;B = 3$

(iii) $\D y = 20.5$

(iv) $\D x = 14.5$

4. (i) $\D y = 1000\sqrt{x}$

(ii) $\D m = 1$

(iii) $\D c = 6$

5. (i) $\D x^3$

(ii) $\D \vcol{x^3& 1& 8& 27& 64}{x^2y& 9.41 &5.16& -6.21& -28.32}$

(iii) $\D a = 10; b = -0.6$

(iv) $\D -1.48$

6. $\D b = e;A = e^2$

7. $\D y = (5x^2 - 2)^2$

8. (i) $\D \lg s$

(ii) $\D \vcol{\lg s& 0.3 &0.6& 0.78& 0.9}{\lg t& 1.4& 0.8& 0.44& 0.19}$

(iii) $\D n = -2; k = 100$

(iv) $\D s = 4.9$

9. (i) $\D x^2$

(ii) $\D \vcol{x^2& 4& 16& 36& 64}{\frac{y}{x}& 4.8& 9.6& 17.5& 29}$

(iv) $\D 4.8$

10. (i)

(ii) $\D n = 1.5$

(iii) $\D 15$

11. (i) $\D \vcol{x^2& 2.25& 3.06& 4& 5.06}{\lg y& 0.59& 0.92 &1.29& 1.71}$

(ii)

(iii) $\D b = 2.5;A = 0.5$

(iv) $\D 2.1$

12. (i) $\D \log y = \log A + x \log b$

(ii) $\D 0.5$ (iii) $\D 4.4$

13. (i) $\log y=\log a+x\log b$

$\vicol{x&2&2.5&3&3.5&4}{\lg y&1.27&1.47&1.67&1.87&2.07}$

(ii) $\D b = 2.5; a = 3$

14. (i) $\D \ln y = 4(3^x) + 3$

(ii) $\D y = 20500$

(iii) $\D x = 0.127$

15. (i) $\ln y=\ln A+x \ln b$

(ii) $A=90000, b=0.4$

(iii) $y=e^{9}$

16. (i) $b=0.25, A=e^{5.45}$

(ii) $y=348$

17. (i) $\log N=\log A-t \log b$

$\viicol{t&1&2&3&4&5&6}{\log N&3.30&3.11&2.95&2.77&2.60&2.41}$

(ii)

(iii) $b=1.5, A=2950$

(iv) 51 (v) 14

18. (i) $b=1.2, A=27.5$

(ii)lg $y=3.84$

(iii) $x=10^{2.05}=113$

19. (i) $b=0.617, A=8.71$

(ii) $y=2.93($ iii $) x=1.74$

20. (i) $b=0.985, A=1.30$

(ii) $y=1.19,($ iii $) x=11$

21. (i) $b=-0.8, A=110$

(ii) $y=9$, (iii) $x=0.47$

22. (i) $\begin{array}[t]{c||cccc}\text{t} & 1 & 1.5 & 2 & 2.5\\\hline \mathrm{ln} \mathrm{P} & 1.48 & 2.12 & 2.76 & 3.4\end{array}$

(ii) 18 to $22.2$

(iii) $1.25 \leq m \leq 1.34$

$(0, \mathrm{c})$ with $.1 \leq c \leq .3$

(iv) $\ln P=1.28 t+.2$

(v) $5.3$

23. $y=(10 / x+3)^{3}$

24. (i) $1 / 2$

(ii) $\lg y=\frac 12 x^2+1$

(iii) $y=10^{x^{2} / 2+1}$

25. (i)lg $y=0.42-0.2 x$

(ii) $y=2.63\left(10^{-0.2 x}\right)$

26. (i) $\ln y=-2.5 x^{2}+2.9$

(ii) $y=18.2 z^{-5 / 2}$

27. (i) $\ln y=\ln A+b x$

(ii)

(iii) $b=-.45$ to $-.55, A=1900$ to 2100

(iv) 2.2 to 3

(v)155 to 175

28. $y=\ln \left(x^{2}-2\right)$

29. (i) $\ln y=\ln a+n \ln x$

(ii) $\mathrm{n}=-.2$ to $-.3, \mathrm{a}=110$

(iii) 22

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