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)

The variables $\D x$ and $\D y$ are related in such a way that when $\D \lg y$ is plotted against $\D \lg x$ a straight line graph is obtained as shown in the diagram. The line passes through the points (2, 4) and (8, 7).

(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 each value correct to significant figure.[4]

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

### 16 (CIE 2015, w, paper 11, question 7) Two variables, $x$ and $y$, are such that $y=A x^{b}$, 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]

### 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.

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

### 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.

### (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 trees surviving after 10 years,

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

### 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 $b$.$[5]$

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

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

### 19 (CIE $2016, \mathrm{~s}$, 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 $x=1.5$.[2]

### (iii) Find the positive value of $x$ when $y=2$.[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 $y=A \mathrm{e}^{\frac{b}{x}}$, find the value of $A$ and of $b$.[4]

### (ii) Find the value of $y$ when $x=0.32$.[2]

### (iii) Find the value of $x$ when $y=20$.[2]

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

### The table shows values of the variables $t$ and $P$,

### (i) Draw the graph of $\ln P$ against $t$ on the grid below.[2]

### (ii) Use the graph to estimate the value of $P$ when $t=2.2$.[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 $f$ correct to 1 decimal place, for which $P$ is at least 1000 .

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

### Variables $x$ and $y$ are 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 line.

### (ii) Use your answer to part (i) to express lgy in terms of $x$.

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

### 25 (CIE 2017, w, paper 12, question 5) When $\lg y$ is plotted against $x$, a straight line is obtained which passes through the points $(0.6,0.3)$ and $(1,1,0.2)$.

### (i) Find $\lg y$ in terms of $x$.

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

### 26 (CIE 2017, 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) Hence express $y$ in terms of $z$, where $z=\mathrm{e}^{x^{2}}$.[3]

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

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

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

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

### (i) Transform this relationship into straight line form. [1]

### (ii) Hence, by plotting a suitable graph, show that the relationship $y=A \mathrm{e}^{\mathrm{hx}}$ is correct.$[2]$

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

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

### (v) Estimate the value of $y$ when $x=5$.$[2]$

### 28 (CIE $2018, \mathrm{~s}$, 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.$[2]$

### 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 $y=50$.$[2]$

### 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)

### (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) $\log N=\log A-t \log b$

### (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{tabular}{l|llll}

### $t$ & 1 & $1.5$ & 2 & $2.5$

### \end{tabular}

### \begin{tabular}{l|ll}

### $\ln \mathrm{P}$ & $1.48$ & $2.12$

### \end{tabular}

### (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) $\lg y=1 / 2 x^{2}+1$

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

### 25. (i)\operatorname{lg} y = 0 . 4 2 - 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) $\ln y=\ln A+b x$

### lny

### (ii)

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

### (iv) $2.2$ to3

### (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

### 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)

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

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

(ii) $\D y = 20500$

(iii) $\D x = 0.127$

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