Sunday, December 23, 2018

Straight Line Equations (CIE)

$\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}}$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)

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]

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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