2014 General Mathematics WAEC SSCE (School Candidates) May/June

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Objective Test Questions

1. Simplify: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\tfrac{2}{5}-6\tfrac{2}{3}+3}
   1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\tfrac{4}{15}}
   2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\tfrac{11}{15}}
   3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7\tfrac{4}{1 5}}
   4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7\tfrac{11}{15 }}
2. If 23_x = 32₅, find the value of x.
   1. 7
   2. 6
   3. 5
   4. 4
3. The volume of a cube is 512 cm³. Find the length of its side.
   1. 6 cm
   2. 7 cm
   3. 8 cm
   4. 9 cm
   The bar chart below shows the scores of some students in a test. Use it answer questions 4 and 5

   

4. How many students took the test?
   1. 18
   2. 19
   3. 20
   4. 22
5. If one student is selected at random, find the probability that he/she scored at most 2 marks.
   1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{11}{18}}
   2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{11}{20}}
   3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{7}{22}}
   4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{5}{19}}
6. Simplify: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{12}(\sqrt{48}-\sqrt{3})}
   1. 18
   2. 16
   3. 14
   4. 12
7. Which of the following number lines represents the solution to the inequality: -9 ≤Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{2}{3}x} < 5

   1. 

   2. 

   3. 

   4. 

8. 

   In the diagram, the value of x + y = 220°. Find the value of n.
   1. 20°
   2. 40°
   3. 60°
   4. 80°
9. Given that x > y and 3 < y, which of the following is/are true? I. y > 3 II. x < 3 III. x > y > 3
   1. I only
   2. I and II only
   3. I and III only
   4. I, II and III
10. Three quarters of a number added to two and a half of that number gives 13. Find the missing number.
    1. 4
    2. 5
    3. 6
    4. 7
11. If X = {0, 2, 4, 6}, Y = {1, 2, 3, 4} and Z = {1, 3} are subsets of U = {x: 0 ≤ x ≤ 6}, find x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cap} (Y Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup} Z).
    1. {0, 2, 6}
    2. {1, 3}
    3. {0, 6}
    4. { }
12. Find the truth set of the equation x² = 3(2x + 9)
    1. {x: x = 3, x = 9}
    2. {x: x = -3, x = -9}
    3. {x: x = 3, x = -9}
    4. {x: x = -3, x = 9}
13. The coordinates of points P and Q are (4, 3) and (2, -1) respectively. Find the shortest distance between P and Q
    1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\sqrt{2}}
    2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\sqrt{5}}
    3. $5{\sqrt {2}}$
    4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\sqrt{5}}
14. Make u the subject of the formula, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\tfrac{m}{2g}(v^2-u^2)}
    1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sqrt{v^2-\tfrac{2Eg}{m}}}
    2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sqrt{\tfrac{v^2}{m}-\tfrac{2Eg}{4}}}
    3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sqrt{v-\tfrac{2Eg}{m }}}
    4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sqrt{\tfrac{2v^2Eg}{m}}}

15. 

    In the diagram, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} QPT = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} PTS = 90°, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} PQR = 110° and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} TSR = 20°. Find the size of the obtuse angle QRS.
    1. 140°
    2. 130°
    3. 120°
    4. 110°
16. If x varies inversely as y and y varies directly as z, what is the relationship between x and z?
    1. x Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varpropto} z
    2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \varpropto \tfrac{1}{z }}
    3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \varpropto z^2}
    4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \varpropto z^\tfrac{1}{2 }}
17. Find the gradient of the line joining the points (2, -3) and (2, 5)
    1. 0
    2. 1
    3. 2
    4. undefined
18. If (x - a) is a factor of bx - ax + x² - ab, find the other factor.
    1. (x + b)
    2. (x - b)
    3. (a + b)
    4. (a - b)
19. The table shows the distribution of the height of plants in a nursery. Calculate the mean heights of the plants.

    Height	2	3	4	5	6
    Frequency	2	4	5	3	1

    1. 3.8
    2. 3.0
    3. 2.8
    4. 2.3

20. 

    In the diagram, PQR is a straight line, (m + n) = 120° and (n + r) = 100°. Find (m + r)
    1. 110°
    2. 120°
    3. 140°
    4. 160°

    

    In the diagram, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{SR}} is parallel to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{UW}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} WVT = x°, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} VUT = y°, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} RSV = 45° and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} VTU = 20°. Use this diagram to answer questions 21 and 22.
21. Find the value of x
    1. 20
    2. 45
    3. 65
    4. 135
22. Calculate the value of y
    1. 20
    2. 25
    3. 45
    4. 65
23. The area of a sector of a circle with a diameter 12 cm is 66 cm². If the sector is folded to form a cone, calculate the radius of the base of the cone. [Take π = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{22}{7}} ]
    1. 3.0 cm
    2. 3.5 cm
    3. 7.0 cm
    4. 7.5 cm
24. A chord, 7 cm long, is drawn in a circle with radius 3.7 cm. Calculate the distance of the chord from the centre of the circle.
    1. 0.7 cm
    2. 1.2 cm
    3. 2.0 cm
    4. 2.5 cm
25. Which of the following is a measure of dispersion?
    1. Range
    2. Percentile
    3. Median
    4. Quartile
26. A box contains 13 currency notes, all of which are either ₦50 or ₦20 notes. The total value of the currency notes is ₦530. How many ₦50 notes are in the box?
    1. 4
    2. 6
    3. 8
    4. 9

    

    The graph above is for the relation y = 2x² + x - 1. Use it to answer questions 27 and 28.
27. What are the coordinates of the point S?
    1. (1, 0, 2)
    2. (1, 0, 4)
    3. (1, 2, 0)
    4. (1, 4, 0)
28. Find the minimum value of y.
    1. 0.00
    2. -0.65
    3. -1.25
    4. -2.10
29. A ship sails x km due east to a point E and continues x km due north to F. Find the bearing of F from the starting point.
    1. 045°
    2. 090°
    3. 135°
    4. 225°
30. If x : y = 3 : 2 and y : z = 5 : 4, find the value of x in the ratio x : y : z
    1. 8
    2. 10
    3. 15
    4. 20
31. A trader bought sachet water for GH¢ 55.00 per dozen and sold them at 10 for GH¢ 50.00. Calculate, correct to 2 decimal places, his percentage gain.
    1. 8.00%
    2. 8.30%
    3. 9.09%
    4. 10.00%

32. 

    In the figure, PQ is a tangent to the circle at R and UT is parallel to PQ. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} TRQ = x°, find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ang} URT in terms of x.
    1. 2x°
    2. (90 - x)°
    3. (90 + x)°
    4. (180 - 2x)°
33. Given that cos x = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{12}{13}} , evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1-\tan x}{\tan x}}
    1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{5}{13}}
    2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{5}{7}}
    3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{7}{5}}
    4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{13}{5}}
34. Approximate 0.0033780 to 3 significant figures.
    1. 338
    2. 0.338
    3. 0.00338
    4. 0.003
35. Simplify: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\tfrac{8^2\times4^n+1}{2^{2n}\times16}}}
    1. 16
    2. 8
    3. 4
    4. 1
36. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{2}{x-3}-\tfrac{3}{x-2}} is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{P}{(x-3)(x-2)}} , find P.
    1. −x − 5
    2. −(x + 3)
    3. 5x − 13
    4. 5 − x
37. Subtract Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2}(a-b-c)} from the sum of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2}(a-b+c)} and ${\tfrac {1}{2}}(a+b-c)$
    1. ${\tfrac {1}{2}}(a+b+c)$
    2. ${\tfrac {1}{2}}(a-b-c)$
    3. ${\tfrac {1}{2}}(a-b+c)$
    4. ${\tfrac {1}{2}}(a+b-c)$
38. A man's eye level is 1.7 m above the horizontal ground and 13 m from a vertical pole. If the pole is 8.3 m high, calculate, correct to the nearest degree, the angle of elevation of the top of the pole from his eyes.
    1. 33°
    2. 32°
    3. 27°
    4. 26°
39. A chord subtends an angle of 120° at the centre of a circle of radius 3.5 cm. Find the perimeter of the minor sector containing the chord. [Take π = ${\tfrac {22}{7}}$]
    1. $14{\tfrac {1}{3}}$cm
    2. $12{\tfrac {5}{6}}$cm
    3. $8{\tfrac {1}{7}}$cm
    4. $7{\tfrac {1}{3}}$cm
40. In parallelogram, PQRS, ${\overline {QR}}$ is produced to M such that |QR| = |RM|. What fraction of the area of PQMS is the area of PRMS?
    1. ${\tfrac {1}{4}}$
    2. ${\tfrac {1}{3}}$
    3. ${\tfrac {2}{3}}$
    4. ${\tfrac {3}{4}}$

41. 

    Determine the value of m in the diagram
    1. 80°
    2. 90°
    3. 110°
    4. 150°
42. In a cumulative frequency graph, the lower quartile is 18 years while the 60yh percentile is 48 years. What percentage of the distribution is at most 18 years or greater than 48 years.
    1. 15%
    2. 35%
    3. 65%
    4. 85%
43. If a number is selected at random from each of the sets P = {1, 2, 3} and Q = {2, 3, 5}, find the probability that the sum of the numbers is prime.
    1. ${\tfrac {5}{9}}$
    2. ${\tfrac {4}{9}}$
    3. ${\tfrac {1}{3}}$
    4. ${\tfrac {2}{9}}$

44. 

    In the diagram, O is the centre of the circle, ${\overline {PR}}$ is a tangent to the circle at Q and $\angle$SOQ = 86°. Calculate the value of $\angle$SQR
    1. 43°
    2. 47°
    3. 54°
    4. 86°
45. If log 5.957 = 0.7750, find log ${\sqrt[{3}]{0.0005957}}$
    1. ${\bar {4}}$.1986
    2. ${\bar {2}}$.9250
    3. ${\bar {1}}$.5917
    4. ${\bar {1}}$.2853
46. The probability of an event P happening is ${\tfrac {1}{5}}$ and that of event Q is ${\tfrac {1}{4}}$. If the events are independent, what is the probability that neither of them happens?
    1. ${\tfrac {4}{5}}$
    2. ${\tfrac {3}{4}}$
    3. ${\tfrac {3}{5}}$
    4. ${\tfrac {1}{20}}$
47. Each exterior angle of a polygon is 30°. Calculate the sum of the interior angles.
    1. 540°
    2. 720°
    3. 1080°
    4. 1800°
48. Find the number of terms in the Arithmetic Progression (A.P) 2, -9, -20, ..., -141.
    1. 11
    2. 12
    3. 13
    4. 14
49. In what modulus is it true that 9 − 8 = 5
    1. mod 10
    2. mod 11
    3. mod 12
    4. mod 13
50. The radii of the base of two cylindrical tins, P and Q are r and 2r respectively. If the water level in P is 10 cm high, what would be the height of the same quantity of water in Q?
    1. 2.5 cm
    2. 5.0 cm
    3. 7.5 cm
    4. 20.0 cm

Theory

Section A

1. 1. Without using tables or calculator, simplify: ${\tfrac {0.6\times 32\times 0.004}{1.2\times 0.008\times 0.16}}$ leaving the answer in standard form (scientific notation).

   2. 

      In the diagram, ${\overline {EF}}$ is parallel to ${\overline {GH}}$. If $\angle$AEF = 3x°, $\angle$ABC = 120° and $\angle$CHG = 7x°, find the value of $\angle$GHB.
2. 1. Simplify $3\surd {75}-\surd {12}+\surd {108}$, leaving the answer in surd form (radicals).
   2. If 124_n = 232_five, find _n.
3. 1. Solve the simultaneous equations: ${\begin{matrix}{\tfrac {1}{x}}+{\tfrac {1}{y}}=5\\{\tfrac {1}{y}}-{\tfrac {1}{x}}=1\end{matrix}}$
   2. A man drives from Ibadan to Oyo, a distance of 48 km in 45 minutes. If he drives at 72 km/h where the surface is good and 48 km/h where it is bad, find the number of kilometres of good surface.

4. 1. 

      In the diagram, O is the centre of the circle radius r cm and $\angle$XOY = 90°. If the area of the shaded part is 504 cm². calculate the value of r. [Take π = ${\tfrac {22}{7}}$].
   2. Two isosceles triangles PQR and PQS are drawn on opposite sides of a common base PQ. If $\angle$PQR = 66° and $\angle$PSQ = 109°. calculate the value of $\angle$RQS.
5. A building contractor tendered for two independent contracts, X and Y. The probabilities that he will win contract X is 0.5 and not win contract Y is 0.3.What is the probability that he will win:
   1. both contracts;
   2. exactly one of the contracts;
   3. neither of the contracts?

Section B

6. 1. If ${\tfrac {3}{2p-{\tfrac {1}{2}}}}={\tfrac {\tfrac {1}{3}}{{\tfrac {1}{4}}p+1}}$, find p
   2. A television set was marked for sale at GH¢ 760.00 in order to make a profit of 20%. The television set was actually sold at a discount at 5%. Calculate, correct to 2 significant figures, the actual percentage profit.
7. 1. Copy and complete the table of values for the relation y = 2 sin x + 1

   x	0°	30°	60°	90°	120°	150°	180°	210°	240°	270°
   y	1.0				2.7			0.0	-0.7

   2. Using scales of 2 cm to 30° on the x-axis and 2 cm to 1 unit on the y-axis, draw the graph of y = 2 sin x + 1 for 0°≤ x ≤ 270°.
   3. Use the graph to find the values of x for which sin x = ${\tfrac {1}{4}}$
8. 1. Copy and complete the following table for multiplication modulo 11.

      $\otimes$	1	5	9	10
      1	1	5	9	10
      5	5
      9	9
      10	10

      Use the table to:
      1. evaluate (9 $\otimes$ 5) $\otimes$ (10 $\otimes$ 10);
      2. find the truth set of I. 10 $\otimes$ m = 2, II. n $\otimes$ n = 4.
   2. When a fraction is reduced to its lowest term, it is equal to ${\tfrac {3}{4}}$. The numerator of the fraction when doubled would be greater than the denominator. Find the fraction.

9. 1. 

      In the Venn diagram, P Q and R are subsets of the universal set U. If n(U) = 125, find:
      1. the value of x;
      2. n(P$\cup$Q$\cup$R').

   2. 

      In the diagram, O is the centre of the circle. If WX is parallel to YZ and $\angle$WXY = 50°, find the value of:
      1. $\angle$WYZ,
      2. $\angle$YEZ
10. 1. Solve (x−2) (x−3) = 12

    2. 

       In the diagram, M and N are the centres of two circles of equal radii 7 cm. The circles intercept at P and Q. If $\angle$PMQ = $\angle$PNQ = 60°, calculate, correct to the nearest whole number, the area of the shaded portion. [Take π = ${\tfrac {22}{7}}$]

11. Scores	1	2	3	4	5	6
    Frequency	2	5	13	11	9	10

    The table shows the distribution of outcomes when a die is thrown 50 times. Calculate the:
    1. mean deviation of the distribution;
    2. probability that a score selected at random is at least a 4.
12. 1. Given that 5 cos (x + 8.5)° − 1 = 0, 0°≤x≤90°, calculate, correct to the nearest degree, the value of x.
    2. The bearing of Q from P is 150° and the bearing of P from R is 015°. If Q and R are 24 km and 32 km respectively from P:
       1. represent this information in a diagram;
       2. calculate the distance between Q and R, correct to two decimal places;
       3. find the bearing of R from Q, correct to the nearest degree.
13. 1. Two functions, f and g, are defined by f : x $\rightarrow$ 3x + 2 where x is a real number.
       1. If f (x − 1)− 7 = 0, find the values of x.
       2. Evaluate: ${\tfrac {f(-{\tfrac {1}{2}})-g(3)}{f(4)-g(5)}}$
    2. An operation $(*)$ is defined on the set R, of real numbers, by m $(*)$ n = ${\tfrac {-n}{m^{2}+1}}$, where m, n $\in$ R. If −3, −10 $\in$ R, show whether or not $(*)$ is commutative.