WAEC SSCE - General Mathematics - 1997 (Objective)

Which of the following statements is/are true when two straight lines intersect?
I. Adjacent angles are equal II. Vertically opposite angles are equal III. Adjacent angles are supplementary

II only

I and III only

II and III only

I, II and III

I and II

Answer Details

When two straight lines intersect, they form four angles at the point of intersection. - Vertically opposite angles are opposite to each other and are equal in measure. This is true and corresponds to statement II. - Adjacent angles are two angles that share a common vertex and a common side. They are not necessarily equal, but their sum is equal to 180 degrees. This is true and corresponds to statement III. - Statement I is not true because adjacent angles need not be equal in measure. For example, two adjacent angles in a scalene triangle are not equal in measure. Therefore, the correct answer is II and III only.

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Read lesson note on Angles (WAEC)

Angles And Intercepts On Parallel Lines

Angles

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Question 5 Report

The graph is the cumulative frequency curve for the weight distribution of 100 workers in a factory. Which of the points P,Q,R,S and T indicates the median weight?

Answer Details

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Algebraic Expressions

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Question 6 Report

Two sides of a triangle are perpendicular. If the two sides are 8cm and 6cm, calculate correct to the nearest degree, the smallest angle of the triangle.

35^o

36^o

37^o

38^o

53^o.

Answer Details

If two sides of a triangle are perpendicular, then those sides are the legs of a right-angled triangle. We can use Pythagoras' theorem to find the hypotenuse of this right-angled triangle which will be the longest side of the original triangle. Using Pythagoras' theorem: a² + b² = c² where a and b are the lengths of the legs and c is the length of the hypotenuse. In this case, a = 6cm and b = 8cm. So, 6² + 8² = c² 36 + 64 = c² 100 = c² c = 10cm So, the longest side of the triangle is 10cm. Now, we can use trigonometry to find the smallest angle of the triangle. sin A = opposite / hypotenuse where A is the smallest angle of the triangle. We know that the opposite side to A is 6cm (the shorter leg). So, sin A = 6 / 10 sin A = 0.6 Using a calculator or a trigonometric table, we can find the angle whose sine is 0.6. A = 36.87^o (rounded to the nearest degree) Therefore, the smallest angle of the triangle is approximately 37^o. So, the answer is (c) 37^o.

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Angles And Intercepts On Parallel Lines

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Question 7 Report

A chord is 5cm from the centre of a circle of diameter 26cm. Find the length of the chord

16cm

18cm

24cm

25.51cm

51.1cm

Answer Details

To find the length of the chord, we can use the Pythagorean theorem.

First, we can find the radius of the circle, which is half of the diameter:

radius = 26cm / 2 = 13cm

The chord divides the circle into two parts, each with its own radius, as shown below:

    O
   / \
  /   \
 /     \
--------  
  R    R

We can draw a line from the center of the circle to the midpoint of the chord, which will bisect the chord and form a right angle with the chord:

    O
   / \
  /   \
 /  .  \
--------  
  R    R

The line from the center of the circle to the midpoint of the chord is also the perpendicular bisector of the chord, and so it divides the chord into two equal segments.

Let the length of each of these segments be x. Then, we can use the Pythagorean theorem to find x:

x^2 + 5^2 = 13^2
x^2 = 169 - 25
x^2 = 144
x = 12

Therefore, the length of the chord is twice the length of one of the segments:

length of chord = 2x = 2(12) = 24 cm

So the correct option is (c) 24cm.

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Circles

Coordinate Geometry

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Question 8 Report

A and B are two sets. The number of elements in A∪B is 49, the number in A is 22 and number in B is 34. How many elements are in A ∩ E?

105

Answer Details

To find the number of elements in the intersection of A and B, we can use the formula: |A ∪ B| = |A| + |B| - |A ∩ B| We are given |A ∪ B| = 49, |A| = 22, and |B| = 34. Substituting these values into the formula gives: 49 = 22 + 34 - |A ∩ B| Simplifying, we get: |A ∩ B| = 22 + 34 - 49 = 7 Therefore, there are 7 elements in the intersection of A and B. The correct answer is 7.

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Sets

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Question 9 Report

Which of the following equations has its roots as 4 and -5?

x² + 4x - 20 = 0

x² + x + 20 = 0

x² - x + 20 = 0

x² + x - 20 = 0

x² - x - 20 = 0

Answer Details

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Quadratic Equations

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Question 10 Report

Simplify log₃9 + log₃15 - log₃5

log₃19

log₃

Answer Details

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Indices, Logarithms And Surds

Logarithms

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Question 11 Report

The eleventh term of an AP, is 25 and its first term is -3. Find its common difference.

1⁹/₁₀

2¹/₅

2³/₁₁

2¹/₂

2⁴/₅

Answer Details

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Question 12 Report

Express (0.0425 / 2.5) as a fraction

17/10,000

17/1,000

17/250

17/100

17/10

Answer Details

To express (0.0425 / 2.5) as a fraction, we need to simplify the expression as much as possible. First, we can convert the decimal 0.0425 to a fraction by putting the decimal numerator over 1 and multiplying both the numerator and denominator by 10000 to get 425/10000. Then we can simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 25, to get 17/400. Finally, we can simplify further by dividing both the numerator and denominator by 4, giving us a final answer of 17/1000. Therefore, the correct option is:

17/1000

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Number Bases

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Question 13 Report

Given that \(\frac{1}{2} \log_{10} P = 1\), find the value of P.

10⁴

10³

10²

10¹

10^-1

Answer Details

The equation \(\frac{1}{2} \log_{10} P = 1\) can be simplified by multiplying both sides by 2: \(\log_{10} P = 2\). We know that \(\log_{10} P = x\) is equivalent to \(10^x = P\). So, in this case, \(10^2 = P\), which means that P is equal to 100. Therefore, the correct answer is 10².

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Logarithms

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Question 14 Report

Two points P and Q are on longitude 67°W. Their latitudes differ by 90°. Calculate their distance apart in terms of π. (Take radius of the earth = 6400km).

6400πkm

6400km/π

3200πkm

3200km

3200km/π

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Coordinate Geometry

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Question 15 Report

Divide 3.6721 by 4

0.9180

1.4180

1.1680

1.9180

Answer Details

To divide 3.6721 by 4, we can use long division. The first step is to place the dividend, 3.6721, under the division symbol, and the divisor, 4, outside the division symbol. We then ask, how many times does 4 go into 3? Since 4 is greater than 3, we cannot divide 3 by 4, so we bring down the next digit, 6, and ask how many times 4 goes into 36. The answer is 9, so we write 9 above the 6, and subtract 36 - 36 to get 0. We then bring down the next digit, 7, and ask how many times 4 goes into 0. Since 4 is greater than 0, we cannot divide 0 by 4, so we add a decimal point and a 0, and ask how many times 4 goes into 7. The answer is 1 with a remainder of 3, so we write 1 above the 7 and bring down the next digit, 2. We add a decimal point to the quotient and ask how many times 4 goes into 32. The answer is 8, so we write 8 above the 2, and subtract 32 - 32 to get 0. We then bring down the next digit, 1, and ask how many times 4 goes into 1. Since 4 is greater than 1, we cannot divide 1 by 4, so we add a decimal point and a 0, and ask how many times 4 goes into 10. The answer is 2 with a remainder of 2, so we write 2 above the 1, and bring down the final digit, 0. We add a decimal point to the quotient and ask how many times 4 goes into 20. The answer is 5, so we write 5 above the 0, and subtract 20 - 20 to get 0. Therefore, the final answer is 0.9180. So, 3.6721 divided by 4 is 0.9180.

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Positive And Negative Integers, Rational Numbers

Modular Arithmetic

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Question 16 Report

Calculate the total surface area of a solid cone of slant height 15cm and base radius 8cm in terms of π

64πcm²

12Oπcm²

184πcm²

200πcm²

32Oπcm²

Answer Details

The total surface area of a cone is the sum of the curved surface area and the area of the base. The curved surface area of a cone can be found using the formula: πrl where r is the radius of the base and l is the slant height of the cone. The area of the base can be found using the formula: πr² where r is the radius of the base. Given the slant height of the cone as 15cm and base radius as 8cm, we can find the height of the cone using the Pythagorean theorem as follows: height² + 8² = 15² height² + 64 = 225 height² = 225 - 64 height² = 161 height = √161 Therefore, the height of the cone is √161 cm. Now we can find the curved surface area of the cone: πrl = π(8)(√161) = 8π√161 And we can find the area of the base: πr² = π(8)² = 64π The total surface area of the cone is the sum of the curved surface area and the area of the base: total surface area = curved surface area + area of base total surface area = 8π√161 + 64π total surface area = 8π(√161 + 8) Therefore, the total surface area of the cone is 8π(√161 + 8) cm², which is approximately equal to 184π cm². So the answer is (c) 184πcm².

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Coordinate Geometry

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Question 17 Report

A bag contains 3 white, 6 red, 5 blue identical balls. A ball is picked at random from bag. What is the probability that is either white or blue?

9/14

4/7

3/7

5/14

3/14

Answer Details

There are a total of 3+6+5=14 balls in the bag. The probability of selecting a white ball is 3/14 since there are 3 white balls in the bag. Similarly, the probability of selecting a blue ball is 5/14 since there are 5 blue balls in the bag. To find the probability of selecting either a white or a blue ball, we add the probabilities of selecting a white ball and a blue ball. However, we need to subtract the probability of selecting a ball that is both white and blue since we cannot count it twice. Since there are no balls that are both white and blue, we don't need to subtract anything. Therefore, the probability of selecting either a white or a blue ball is: 3/14 + 5/14 = 8/14 = 4/7 So, the answer is 4/7.

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Probability

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Question 18 Report

P varies inversely as the square of W. When W = 4, P = 9. Find the value of P when W = 9

729/16

16/9

4/3

Answer Details

The problem states that P varies inversely as the square of W, which means that as W increases, P decreases, and vice versa. We can write this relationship as P = k/W^2, where k is the constant of proportionality. To solve for k, we can use the given information that when W = 4, P = 9. Plugging these values into the equation, we get: 9 = k/4^2 9 = k/16 k = 9 x 16 k = 144 Now that we know k, we can use the equation to find P when W = 9: P = 144/9^2 P = 144/81 P = 16/9 Therefore, the value of P when W = 9 is 16/9.

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Question 19 Report

O is the centre of the circle PQRS. PR and QS intersect at T POR is a diameter, ?PQT = 42^o and ?QTR = 64^o; Find ?QRT

68^o

64^o

42^o

32^o

22^o.

Answer Details

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Circles

Triangles And Polygons

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Question 20 Report

A bag contains red, black and green identical balls. A ball is picked and replaced. The table shows the result of 100 trials. Find the experimental probability of picking a green ball.

21/25

1/3

4/21

4/25

Answer Details

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Probability

Surds (radicals)

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Question 21 Report

In the triangle XYZ , XM is the altitude from X to YZ.XY = 13cm, XZ = 15cm and YM = 5cm. Find the length of YZ

9cm

12cm

14cm

19.85cm

23cm

Answer Details

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Triangles And Polygons

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Question 22 Report

If x varies over the set of real numbers, which of the following is illustrated in the diagram above?

-2 ? x < 3

-2 < x ? 3

-2 < x < 3

-2 ? x ? 3

x ? 3

Answer Details

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Variation

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Question 23 Report

Solve me equation: 2/3 (x + 5) = 1/4(5x - 3)

1¹/₇

1⁸/₂₃

4³/₇

Answer Details

To solve the given equation: 2/3(x + 5) = 1/4(5x - 3) We can simplify both sides of the equation by getting rid of the fractions. To do this, we can multiply both sides by the least common multiple of the denominators, which is 12: 12 * 2/3(x + 5) = 12 * 1/4(5x - 3) 8(x + 5) = 3(5x - 3) Now, we can distribute the terms on both sides: 8x + 40 = 15x - 9 We can then isolate the variable on one side by subtracting 8x from both sides: 40 = 7x - 9 Finally, we can isolate the variable x by adding 9 to both sides and then dividing by 7: 49/7 = x Simplifying, we get: x = 7 Therefore, the solution to the equation is x = 7.

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Solution Of Linear Equations

Algebraic Fractions

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Question 24 Report

Cos 65^o has the same value as

Sin 65^o

Cos 25^o

Cos 115^o

Cos 205^o

Cos 295^o

Answer Details

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Sine, Cosine And Tangent Of An Angle

Introductory Calculus

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Question 25 Report

Solve the inequality 3x - 8 ≥ 5x

x ≤ 4

x ≥ 1

x ≤ -4

x ≤ -1

x < 2

Answer Details

To solve the inequality 3x - 8 ≥ 5x, we need to isolate the variable x on one side of the inequality sign. First, we can simplify the inequality by subtracting 3x from both sides: 3x - 8 - 3x ≥ 5x - 3x Simplifying further, we get: -8 ≥ 2x To isolate x, we divide both sides by 2: -4 ≥ x This is the same as x ≤ -4. Therefore, the correct option is: - x ≤ -4

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Question 26 Report

What is the probability of having an odd number in a single throw of a fair die with the faces numbered 1,2,3,4,5,6?

5/6

2/3

1/2

1/3

1/6

Answer Details

There are six possible outcomes when a fair die is rolled, which are the numbers 1, 2, 3, 4, 5, and 6. Since three of these numbers are odd (1, 3, and 5), and the die is fair (meaning that each outcome is equally likely), the probability of rolling an odd number is 3/6, which simplifies to 1/2. Therefore, the answer is 1/2.

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Probability

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Question 27 Report

The interior angles of a pentagon are 126^o, 114^o, y, 92^o and 83^o. Find the value of y.

85^o

95^o

105^o

115^o

125^o

Answer Details

The sum of the interior angles of a pentagon is given by the formula (n-2) × 180^o where n is the number of sides of the polygon. In this case, n=5 since it's a pentagon. So, sum of interior angles = (5-2) × 180^o = 540^o We know that the other four angles measure 126^o, 114^o, 92^o, and 83^o. Therefore, 126^o + 114^o + y + 92^o + 83^o = 540^o Simplifying the above equation, we get: y = 125^o Therefore, the value of y is 125^o.

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Question 28 Report

The diagram above is a triangle XYZ whose area is 23.5cm². XY =1Ocm and YZ = 8cm, calculate ? correct to the nearest degree.

34^o

35^o

36^o

37^o

40^o

Answer Details

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Question 29 Report

lf sin θ= \(\frac{-1}{2}\), find all the values of θ between 0° and 360°.

120^o,240^o

120^o180^o

210^o,300^o

210^o,330^o

300^o,360^o

Answer Details

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Sine, Cosine And Tangent Of An Angle

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Question 30 Report

If P = {3, 5, 6} and Q = {4, 5, 6} then P∩Q equals

{3, 4}

{4, 5}

{4, 6}

{5, 6}

{3, 5}

Answer Details

P∩Q represents the intersection of sets P and Q, which means finding the common elements between them. The common elements between P and Q are 5 and 6, so P∩Q equals {5, 6}. Therefore, the correct answer is (D) {5, 6}.

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Sets

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Question 31 Report

If two triangles are similar, which of the following is true? Their

corresponding sides are equal

corresponding angles are equal

corresponding altitudes are equal

areas are equal

perimeters are equal

Answer Details

If two triangles are similar, their corresponding angles are equal. This is a fundamental property of similar triangles. When two triangles have the same shape but different sizes, they are said to be similar. The corresponding sides of similar triangles are proportional, which means that if one side of a triangle is multiplied by a certain number, then all the corresponding sides of the other triangle must be multiplied by the same number. However, the sides are not necessarily equal. Similarly, the altitudes and areas of the triangles are also proportional, but not necessarily equal. Therefore, the correct answer is that the corresponding angles of similar triangles are equal.

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Triangles And Polygons

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Question 32 Report

Find the radius of a circle in which an arc of length 44cm subtends angle 200° at the centre of the circle. [Take π = 22/7]

3.9cm

12.6cm

25.2cm

38.4cm

124.5cm

Answer Details

In a circle, there are 360 degrees at the center. If an arc subtends an angle of 200 degrees at the center of the circle, then the length of that arc can be found as follows: Length of arc = (angle/360) x 2πr where angle is in degrees, r is the radius of the circle and π = 22/7 Substituting the given values we get: 44 = (200/360) x 2 x (22/7) x r 44 = (5/9) x (44/7) x r r = (44 x 7 x 9) / (5 x 44) r = 12.6 cm Therefore, the radius of the circle is 12.6 cm. Answer is correct.

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Circles

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Question 33 Report

The cross-section of a prism is a right-angled triangle 3cm by 4cm by 5cm. The height of the prism is 8cm. Calculate its volume

48cm³

6Ocm³

96cm³

120cm³

24Ocm³

Answer Details

To find the volume of the prism, we need to multiply the area of the base by the height. Since the cross-section of the prism is a right-angled triangle, the area of the triangle is given by: Area = 1/2 x base x height = 1/2 x 3cm x 4cm = 6cm^2 The height of the prism is given as 8cm. Therefore, the volume of the prism is given by: Volume = Area of base x height = 6cm^2 x 8cm = 48cm^3 Hence, the volume of the prism is 48cm^3. Therefore, the correct option is (a) 48cm^3.

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Question 34 Report

The table shows the scores of a group of students in a test. If the average score is 3.5, find the value of x

Answer Details

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Probability

Measures Of Location

Statistics

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Question 35 Report

The length of the two parallel sides of trapezium are 6cm and 10cm and the perpendicular distance between them is 5cm. Find the area of the trapezium

21cm²

4Ocm²

8Ocm²

150cm²

300cm²

Answer Details

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Volumes

Areas

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Question 36 Report

The angle of elevation of a point T on a tower from a point U on the horizontal ground is 30^o. If TU = 54m, how high is T above the horizontal ground?

108m

72m

46.3m

31.2m

27m

Answer Details

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Read lesson note on Angles (WAEC)

Bearings

Angles Of Elevation And Depression

Angles

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Question 37 Report

Make q the subject of the relation t = √^pq/_r - r²

q =

rt²/p - r³

q =

t²/p - r²

q =

rt/p - r³

q =

p - r³/rt²

Q = rt²(p - r³ )

Answer Details

To make q the subject of the formula, we need to isolate q on one side of the equation. Starting with: t = √^pq/_r - r² 1. Add r^2 to both sides of the equation: t + r^2 = √^pq/_r 2. Square both sides of the equation to eliminate the radical: (t + r^2)^2 = pq/r 3. Multiply both sides of the equation by r: r(t + r^2)^2 = pq 4. Divide both sides of the equation by p: q = r(t + r^2)^2 / p Therefore, the answer is: q = rt² / (p - r³)

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Solution Of Linear Equations

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Question 38 Report

TR is the tangent to the circle PQR with centre O. Find the size of ?PRT

44^o

46^o

60^o

74^o

118^o

Answer Details

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Circles

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Question 39 Report

PQRS and GHRS are parallelograms on the same base SR and between the same parallel straight line PH and SR. Which of the following is true?

? PGS = ? QHR

Parallelogram PQRS = ? QHR

Trapezium PQXS = ? QHR

? PGS = Parallelogram GHRS

Parallelogram PQRS = Parallelogram GHRS

Answer Details

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Triangles And Polygons

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Question 40 Report

From the diagram above, which of the following is true?

a +b + x =180

a = b + x

a ? b = 180 - x

a + b = x + 180

a ? b = x - 180.

Answer Details

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Algebraic Expressions

Angles And Intercepts On Parallel Lines

Statistics

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Question 41 Report

Find the common ratio in the GP: log 3, log 9, log 81 ..........

Answer Details

In a geometric progression (GP), the ratio between any two successive terms is constant. Let's find the ratio between the second and first terms of the given GP: log 9 / log 3 = log (3^2) / log 3 = 2 log 3 / log 3 = 2 Similarly, the ratio between the third and second terms is: log 81 / log 9 = log (9^2) / log 9 = 2 log 9 / log 9 = 2 Since the ratio between any two successive terms is the same, the common ratio of the given GP is 2. Therefore, the answer is 2.

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Sequence And Series

Logarithms

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Question 42 Report

Which of the following is not necessarily sufficient for the construction of a triangle?

Three sides

Two sides and an included angle

Two sides and a right angle

Two sides and an angle

Two angles and a side.

Answer Details

To construct a triangle, we need to satisfy the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Based on this theorem, the option that is not necessarily sufficient for the construction of a triangle is option (C) - two sides and a right angle. This is because having two sides and a right angle does not guarantee that the remaining side satisfies the triangle inequality theorem. For example, if the two sides are very short, the remaining side (hypotenuse) will also be short, and the sum of the lengths of the two shorter sides will be less than the length of the remaining side, violating the triangle inequality theorem.

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Euclidean Geometry

Triangles And Polygons

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Question 43 Report

Solve the equations: 4x -y = 11; 5x + 2y = 4.

x = 6, y = 13

x = -2, y = -3

x = -2, y = 3

x = 2, y = -3

x = 2, y = 3.