SPM Practice 2 (Question 11 & 12)

Posted on May 30, 2020 by Myhometuition

Question 11 (3 marks):
Diagram 6 shows the graph of a straight line

$\frac{x^{2}}{y} against \frac{1}{x} .$

Diagram 11

Based on Diagram 6, express y in terms of x.

Solution:

$\begin{array}{l} m = \frac{4 - (- 5)}{6 - 0} = \frac{3}{2} \\ c = - 5 \\ Y = \frac{x^{2}}{y}, X = \frac{1}{x} \\ Y = m X + c \\ \frac{x^{2}}{y} = \frac{3}{2} (\frac{1}{x}) + (- 5) \\ \frac{x^{2}}{y} = \frac{3}{2 x} - 5 \\ \frac{x^{2}}{y} = \frac{3 - 10 x}{2 x} \\ \frac{y}{x^{2}} = \frac{2 x}{3 - 10 x} \\ y = \frac{2 x^{3}}{3 - 10 x} \end{array}$

Question 12 (3 marks):
The variables x and y are related by the equation

$y = x + \frac{r}{x^{2}}$ , where r is a constant. Diagram 8 shows a straight line graph obtained by plotting

$(y - x) against \frac{1}{x^{2}} .$

Diagram 12

Express h in terms of p and r.

Solution:

$\begin{array}{l} y = x + \frac{r}{x^{2}} \\ y - x = r (\frac{1}{x^{2}}) + 0 \\ Y = m X + c \\ m = r, c = 0 \\ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ r = \frac{5 p - 0}{\frac{h}{2} - 0} \\ \frac{h r}{2} = 5 p \\ h r = 10 p \\ h = \frac{10 p}{r} \end{array}$

SPM Practice 2 (Linear Law) – Question 1

Posted on May 30, 2020 by Myhometuition

Question 1 (10 marks):
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment. A straight line will be obtained when a graph of

$\frac{y^{2}}{x} against \frac{1}{x}$ is plotted.

(a) Based on Table 1, construct a table for the values of

$\frac{1}{x} and \frac{y^{2}}{x} .$

$\begin{array}{l} (b) Plot \frac{y^{2}}{x} against \frac{1}{x}, using a scale of 2 cm to 0 .1 unit on the \frac{1}{x} -axis \\ and 2cm to 2 units on the \frac{y^{2}}{x} -axis . \\ Hence, draw the line of best fit . \end{array}$

(c) Using the graph in 1(b)
(i) find the value of y when x = 2.7,
(ii) express y in terms of x.

Solution:
(a)

(b)

(c)(i)

$\begin{array}{l} When x = 2.7, \frac{1}{x} = 0.37 \\ From graph, \\ \frac{y^{2}}{x} = 5.2 \\ \frac{y^{2}}{2.7} = 5.2 \\ y = 3.75 \end{array}$

(c)(ii)

$\begin{array}{l} Form graph, y -intercept, c = –4 \\ gradient, m = \frac{16 - (- 4)}{0.8 - 0} = 25 \\ Y = m X + c \\ \frac{y^{2}}{x} = 25 (\frac{1}{x}) - 4 \\ y = \sqrt{25 - 4 x} \end{array}$

SPM Practice 3 (Linear Law) – Question 6

Posted on May 30, 2020 by Myhometuition

Question 6
The table below shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation

$\frac{a}{y} = \frac{b}{x} + 1$ , where k and p are constants.

(a) Based on the table above, construct a table for the values of

$\frac{1}{x}$ and

$\frac{1}{y}$ . Plot

$\frac{1}{y}$ against

$\frac{1}{x}$ , using a scale of 2 cm to 0.1 unit on the

$\frac{1}{x}$ - axis and 2 cm to 0.2 unit on the

$\frac{1}{y}$ - axis. Hence, draw the line of best fit.
(b) Use the graph from (b) to find the value of
(i) a,
(ii) b.

Solution
Step 1 : Construct a table consisting X and Y.

Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

Steps to draw line of best fit - Click here

Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph

Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice 3 (Linear Law) – Question 5

Posted on May 30, 2020 by Myhometuition

Question 5
The following table shows the corresponding values of two variables, x and y, that are related by the equation

$y = p k^{\sqrt{x}}$ , where p and k are constants.

(a) Plot

$\log_{10} y$ against

$\sqrt{x}$ . Hence, draw the line of best fit

(b) Use your graph in (a) to find the values of p and k.

Solution
Step 1 : Construct a table consisting X and Y.

Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

For steps to draw line of best fit - Click here

Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph

Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice 2 (Linear Law) – Question 4

Posted on May 30, 2020 by Myhometuition

Question 4
The table below shows the corresponding values of two variables, x and y, that are related by the equation

$y = q x + \frac{p}{q x}$ , where p and q are constants.

One of the values of y is incorrectly recorded.
(a) Using scale of 2 cm to 5 units on the both axis, plot the graph of xy against

$x^{2}$ . Hence, draw the line of best fit

(b) Use your graph in (a) to answer the following questions:
(i) State the values of y which is incorrectly recorded and determine its actual value.
(ii) Find the value of p and of q.

Solution
Step 1 : Construct a table consisting X and Y.

Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

Steps to draw line of best fit - Click here

(b) (i) State the values of y which is incorrectly recorded and determine its actual value.

Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph

Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice 2 (Question 8 – 10)

Posted on May 25, 2020 by Myhometuition

Question 8:
The variables x and y are related by the equation $y = \frac{p}{3^{x}}$ , where k is a constant.
Diagram below shows the straight line graph obtained by plotting $\log_{10} y$ against x.

Express the equation $y = \frac{p}{3^{x}}$ in its linear form used to obtain the straight line graph shown in Diagram above.
Find the value of p.

Solution:

Question 9:
Variable x and y are related by the equation $y^{2} = p x^{q}$ . When the graph lg y against lg x is drawn, the resulting straight line has a gradient of -2 and an vertical intercept of 0.5 . Calculate the value of p and of q.

Solution:

Question 10:
Variable x and y are related by the equation $y = \frac{c}{d - x}$ . When the graph y against xy is drawn the resulting line has gradient 0.25 and an intercept on the y-axis of 1.25. Calculate the value of c and of d.

Solution:

SPM Practice 2 (Question 6 & 7)

Posted on May 25, 2020 by Myhometuition

Question 6:
The variables x and y are related by the equation $y = p x^{3}$ , where p is a constant. Find the value of p and n.

Solution:

Question 7:
Diagram A shows part of the curve $y = a x^{2} + b x$ . Diagram B shows part of the straight line obtained when the equation is reduced to the linear form. Find
(a) the values of a and b,
(b) the values of p and q.

Diagram A

Diagram B

Solution:

SPM Practice 2 (Question 4 & 5)

Posted on May 25, 2020 by Myhometuition

Question 4:
The diagram shows part of the straight line graph obtained by plotting $\frac{y}{x}$ against $\sqrt{x}$ .

Given its original non-linear equation is $y = p x + q x^{\frac{3}{2}}$ . Calculate the values of p and q.

Solution:

Question 5:
The diagram below shows the graph of the straight line that is related by the equation $\frac{x}{y} = \frac{2}{x} + 3 x$ .

Find the values of p and k.

Solution:

SPM Practice 3 (Linear Law) – Question 3

Posted on May 17, 2020 by Myhometuition

Question 3
The table below shows the corresponding values of two variables, x and y, that are related by the equation

$y = 5 h x^{2} + \frac{k}{h} x$ , where h and k are constants.

(a) Using a scale of 2 cm to 1 unit on the x - axis and 2 cm to 0.2 units on the

$\frac{y}{x}$ – axis, plot the graph of

$\frac{y}{x}$ against x . Hence, draw the line of best fit.

(b) Use your graph in (a) to find the values of
(i) h,
(ii) k,
(iii) y when x = 6.

Solution
Step 1 : Construct a table consisting X and Y.

Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

Steps to draw line of best fit - Click here

Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph

Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

(b) (iii) find the value of y when x = 6.

SPM Practice 3 (Linear Law) – Question 2

Posted on May 17, 2020 by Myhometuition

Question 2 (10 marks):
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment.
The variables x and y are related by the equation

$y - \sqrt{h} = \frac{h k}{x}$ , where h and k are constants.

(a) Plot xy against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the xy-axis.
Hence, draw the line of best fit.

(b) Using the graph in 2(a), find
(i) the value of h and of k,
(ii) the correct value of y if one of the values of y has been wrongly recorded during the experiment.

Solution:
(a)

(b)

$\begin{array}{l} y - \sqrt{h} = \frac{h k}{x} \\ x y - \sqrt{h} x = h k \\ x y = \sqrt{h} x + h k \\ Y = m X + C \\ Y = x y, m = \sqrt{h}, C = h k \end{array}$

(b)(i)

$\begin{array}{l} m = \frac{36.5}{5.1} \\ \sqrt{h} = \frac{36.5}{5.1} \\ \sqrt{h} = 7.157 \\ h = 51.22 \\ C = - 4 \\ h k = - 4 \\ k = \frac{- 4}{h} \\ k = - \frac{4}{51.22} \\ k = - 0.0781 \end{array}$

(b)(ii)

$\begin{array}{l} x y = 21 \\ 3.5 y = 21 \\ y = \frac{21}{3.5} = 6.0 \\ Correct value of y is 6 .0 . \end{array}$