WAEC SSCE - General Mathematics - 1988

Evaluate log₁₀6 + log₁₀45 - log₁₀27 without using logarithm tables

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In the diagram above, PR is perpendicular from P to QS, PQ = 2cm, QR = 1cm and PR = RS. what is the size of the angle QPS?

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A group of students took a test and the following frequency table shows the scores above

The median score is

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A water tank is made with a right-angled trapezium - shaped vertical cross-section so that when it is placed on sloping ground the top of the tank is level. The dimension are given in the diagram above
By what number will you multiply your answer in 22 above to determine the capacity of the tank in liters?

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The volume of the tank in cubic meters is 3008 m^3, as we found in question 22. To convert this volume to liters, we need to multiply by 1000, since there are 1000 liters in one cubic meter: V_liters = V_meters^3 * 1000 V_liters = 3008 * 1000 V_liters = 3,008,000 liters Therefore, we need to multiply our answer from question 22 by 1000 to determine the capacity of the tank in liters. The closest answer choice is 1000, which is the correct answer.

Which of the regions P, Q, R, S and T satisfies the following 0 < y < 1, y < x + 2, x < 0

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Factorize 5y² + 2y - 3a²

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The histogram below shows the number of candidates, in thousands, obtaining given ranges of marks in a State examination. Find the total number of candidates that sat for the examination

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Find the 8th term of the A.P -3, -1, 1 ......

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In an arithmetic progression (A.P.), each term is obtained by adding a fixed constant (d) to the previous term. The common difference (d) between any two consecutive terms of an A.P. is constant. In this A.P., the first term is -3 and the common difference is 2, since each term is obtained by adding 2 to the previous term (-3 + 2 = -1, -1 + 2 = 1, and so on). To find the 8th term, we can use the formula: nth term = a + (n-1)d where nth term is the term we want to find (in this case, the 8th term), a is the first term, n is the position of the term we want to find, and d is the common difference. Substituting the given values, we get: 8th term = -3 + (8-1)*2 8th term = -3 + 14 8th term = 11 Therefore, the 8th term of the A.P. -3, -1, 1...... is 11.

In the diagram above, find the ratio |PQ|/|PR|

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If cos 60° = 1/2, which of the following angle has cosine of -1/2?

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Let's use the fact that cosine is an even function, meaning that cos(-x) = cos(x). Therefore, we can find the angle with cosine of -1/2 by finding the angle with cosine of 1/2 and then taking its negative. We know that cos(60°) = 1/2. The cosine function is positive in the first and fourth quadrants, so we need to look for angles in those quadrants where cos(x) = 1/2. In the first quadrant, the reference angle for which cosine is 1/2 is 60°. In the fourth quadrant, the reference angle is 360° - 60° = 300°. Taking the negative of these reference angles, we get that the angles with cosine of -1/2 are: - 60° - 180° = -120° - 300° - 180° = 120° Therefore, the answer is 120°.

The histogram below shows the number of candidates, in thousands, obtaining given ranges of marks in a State examination. How many candidates scored at most 30%?

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The number 186,470 was corrected to 186,000. Which of the following describe the degree of approximation made?
I. to the nearest hundred II. to the nearest thousand III. to 3 significant figures IV. to 4 significant figures

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A group of students took a test and the following frequency table shows the scores above
Find the mode

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A water tank is made with a right-angled trapezium - shaped vertical cross-section so that when it is placed on sloping ground the top of the tank is level. The dimension are given in the diagram above
If the tank is completely filled with water, how many cubic metres can it hold?

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Cotonou and Niamey are on the same line of longitude and Niamey is 7^o north of Cotonou. If the radius of the earth is 6400km, how far is Niamey north of Cotonou along the line of longitude, correct to the nearest kilometer? [Take π = 22/7]

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A ladder 9m long leans against a vertical wall, making an angle of 64° with the horizontal ground. calculate correct to one decimal place, how far the foot of the ladder is from the wall.

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Simplify 36\(^\frac{1}{2}\) x 64\(-^\frac{1}{3}\) x 5\(^0\)

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To simplify this expression, we can use the rules of exponents. - 36\(^{\frac{1}{2}}\) can be written as the square root of 36, which equals 6. - 64\(^{-\frac{1}{3}}\) can be written as the cube root of 64 raised to the power of -1, which equals 1/4. - 5\(^0\) equals 1. So, the expression becomes: 6 x 1/4 x 1 = 6/4 = 3/2 = 1.5. Therefore, the answer is 1¹/₂.

A group of students took a test and the following frequency table shows the scores above

The mean score is

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Find the product xy if, x, 3/2, 6/7, y are in G.P

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To find the product of x and y, we can use the formula for the nth term of a geometric progression: an = a1 * r^(n-1) where an is the nth term, a1 is the first term, r is the common ratio, and n is the number of terms. Since x, 3/2, 6/7, y are in geometric progression, we can write: 3/2 = x * r 6/7 = 3/2 * r y = 6/7 * r Solving for r in the second equation: 6/7 = 3/2 * r r = (6/7) / (3/2) r = 4/7 Substituting r in the first and third equations: 3/2 = x * (4/7) x = (3/2) / (4/7) x = 21/8 y = 6/7 * (4/7) y = 24/49 Therefore, the product xy is: xy = (21/8) * (24/49) xy = 9/7 Therefore, the answer is: 9/7.

Abidjan is 4^o west of Accra and on the same circle of latitude. If the radius of this circle of latitude is 6370km, how far is Abidjan west of Accra, correct to the nearest kilometer? [Take π = 22/7]

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Abidjan is 4^o west of Accra, and they are on the same circle of latitude. Since the circumference of a circle of radius r is given by 2πr, the distance between two points on the same circle of latitude is proportional to the angle between them. Since Abidjan is 4^o west of Accra, the distance between them is 4/360 times the circumference of the circle of latitude. Using the formula for the circumference of a circle, we have: C = 2πr = 2 x 22/7 x 6370 = 40,014 km (approx.) Therefore, the distance between Accra and Abidjan is: 4/360 x 40,014 km = 445 km (approx.) Hence, the answer is 445km, correct to the nearest kilometer.

In the diagram above, O is the centre of the circle through points L, M and N, if ?MLN = 74o and ?MNL = 39o, calculate ?LON.

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Evaluate (101.5)² - (100.5)²

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We can use the identity for the difference of squares: a^2 - b^2 = (a + b)(a - b) In this case, we have: (101.5)^2 - (100.5)^2 = [(101.5 + 100.5)(101.5 - 100.5)] = (202)(1) = 202 Therefore, the answer is option (D) 202.

Which of the following is a correct method for constructing an angle of 60^o at Q?

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The correct method for constructing an angle of 60^o at Q is option II only, which involves using a compass to draw an arc from point Q, cutting the line segment PQ at point R. Then, a new arc is drawn from point R, cutting the previous arc at point S. The line segment QS passing through Q and S forms a 60^o angle with the line segment PQ. Option I only involves bisecting an angle of 120^o formed by the line segments PQ and PR, which would result in an angle of 60^o at Q only if the original angle was 120^o. However, the original angle is not given in the question. Option III only involves using a ruler to measure a distance of 3 cm from Q on the line segment PQ, which would not necessarily result in an angle of 60^o at Q.

A rope of length 18m is used to form a sector of a circle of radius 3.5m on a school playing field. What is the size of the angle of the sector, correct to the nearest degree?

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If sinθ = 3/5 find tanθ for 0 < θ < 90^o

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To find the value of tanθ, we can use the relationship: tanθ = sinθ / cosθ We are given sinθ = 3/5. To find cosθ, we can use the identity: sin^2θ + cos^2θ = 1 Substituting sinθ = 3/5, we get: (3/5)^2 + cos^2θ = 1 9/25 + cos^2θ = 1 cos^2θ = 16/25 cosθ = ±4/5 Since 0 < θ < 90°, we know that cosθ is positive. Therefore, cosθ = 4/5. Substituting sinθ = 3/5 and cosθ = 4/5 into the equation for tanθ, we get: tanθ = sinθ / cosθ tanθ = (3/5) / (4/5) tanθ = 3/4 Therefore, the value of tanθ for sinθ = 3/5 is 3/4.

If Q = (all perfect squares less than 30) and P = (all odd numbers from 1 to 10). Find Q ∩ P.

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The set Q contains all perfect squares less than 30. These are 1, 4, 9, 16, and 25. The set P contains all odd numbers from 1 to 10. These are 1, 3, 5, 7, and 9. The symbol ∩ means "intersection," which represents the common elements in two or more sets. To find Q ∩ P, we need to identify the elements that are present in both sets Q and P. Looking at the sets Q and P, we can see that the only element they have in common is 9. Therefore, Q ∩ P = {9}. Therefore, the answer is (1, 9).

What is the probability of having an odd number in a single toss of a fair die?

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A fair die has six faces numbered from 1 to 6. Each face has an equal chance of showing up when the die is rolled. To find the probability of rolling an odd number, we need to count the number of odd faces on the die, which are 1, 3, and 5. Since there are three odd faces out of the six possible faces, the probability of rolling an odd number is: Probability of rolling an odd number = Number of odd faces / Total number of faces Probability of rolling an odd number = 3 / 6 Probability of rolling an odd number = 1/2 Therefore, the probability of having an odd number in a single toss of a fair die is 1/2.

In the diagram above, O is the center of the circle, BOC is a diameter and angle ADC is 37o
What is ACB?

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Since BOC is a diameter, then angle BAC is 90 degrees (as it is an inscribed angle that subtends a diameter). Also, angle ADC is 37 degrees (given). Therefore, angle ACB can be found using the fact that the angles in a triangle sum to 180 degrees. Thus: angle ACB = 180 - angle BAC - angle ADC = 180 - 90 - 37 = 53 degrees Therefore, the answer is 53 degrees.

A student told to draw the graph of y = x² + 4x - 6. He is then told to draw a liner graph on the same axis such that the intersection of the two graphs will give the solutions to the equation x² + 4x - 7 = 0, What is the equation of the linear graph he needs to draw?

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Express the product of 0.06 and 0.09 in standard form

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To multiply two numbers in standard form, we multiply the decimal parts of the numbers and add the exponents of 10. So, to find the product of 0.06 and 0.09, we multiply 6 and 9 to get 54, and count the total number of decimal places in the factors, which is 4. Therefore, we express the product in standard form as 5.4 x 10^-3. Answer: 5.4 x 10^-3

Calculate the value of x in the diagram above

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A sales girl gave a change of N1.15 to a customer instead of N1.25. Calculate her percentage error

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The sales girl gave a change of N1.15 instead of N1.25, which means that she made an error of N0.10. To find the percentage error made by the sales girl, we can use the formula: Percentage error = (Error / Actual value) x 100% In this case, the error is N0.10 and the actual value is N1.25. Substituting these values into the formula, we get: Percentage error = (0.10 / 1.25) x 100% Percentage error = 0.08 x 100% Percentage error = 8% Therefore, the percentage error made by the sales girl is 8%.

What is the difference in longitude between P (lat. 50°N. long. 50°W) and Q (lat.50°N, long. 150°W)?

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To find the difference in longitude between P (50°W) and Q (150°W), we subtract the longitude of P from the longitude of Q: Longitude of Q - Longitude of P = 150°W - 50°W = 100° Therefore, the answer is 100°.

The surnames of 40 children in a class arranged in alphabetical order. 16 of the surnames begins with O and 9 of the surname begins with A, 14, of the letters of the alphabet do not appear as the first letter of a surname

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There are a total of 40 children in the class. The probability of choosing a child with a surname beginning with O is 16/40, and the probability of choosing a child with a surname beginning with A is 9/40. However, we have to be careful not to double-count the surnames that begin with both O and A. Therefore, we need to subtract the probability of choosing a child with a surname beginning with both O and A, which is the intersection of the two events. The intersection is the number of surnames that begin with both O and A, which is zero in this case because it is not given that any surname starts with both O and A. So the probability of choosing a child with a surname beginning with either O or A is: P(O or A) = P(O) + P(A) - P(O and A) P(O or A) = 16/40 + 9/40 - 0 P(O or A) = 25/40 Simplifying, we get: P(O or A) = 5/8 Therefore, the probability of choosing a child with a surname beginning with either O or A is 5/8. Answer: (A) 5/8.

A cone is made from a sector of circle of radius 14cm and angle of 90^o. What is the area of the curved surface of the cone? (Take x = 22/7)

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Points P and Q respectively 24m north and 7m east point R. Calculate |PQ| in meters

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We can use the Pythagorean theorem to find the distance between points P and Q. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, we can consider P, Q, and R as the three vertices of a right-angled triangle, with PQ as the hypotenuse. Since P is 24 meters north of R and Q is 7 meters east of R, we can draw a right-angled triangle with vertical side 24 meters and horizontal side 7 meters. Then, using the Pythagorean theorem, we have: |PQ| = sqrt(24^2 + 7^2) |PQ| = sqrt(576 + 49) |PQ| = sqrt(625) |PQ| = 25 meters Therefore, the distance between points P and Q is 25 meters. Hence the answer is 25.

Points P and Q respectively 24m north and 7m east point R. What is the bearing of Q from P to the nearest whole degree?

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Simplify (27^1/3)²

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The expression (27^(1/3))^2 means we take the cube root of 27 first, which is 3, and then we square the result. (27^(1/3))^2 = 3^2 = 9 Therefore, the answer is 9.

Find the quadratic equation whose roots are x = -2 or x = 7

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If the roots of a quadratic equation are given, we can express the quadratic equation in factored form. For the given problem, the roots are x = -2 or x = 7. This means that the factors of the quadratic equation are (x + 2) and (x - 7). Multiplying these factors, we get: (x + 2)(x - 7) = x² - 5x - 14 = 0 Therefore, the quadratic equation whose roots are x = -2 or x = 7 is x² - 5x - 14 = 0.

IN the diagram above, LK/PQ, reflex angle KLM = 241° and ?QPM = 89°. What is the value of ?LMP?

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The surnames of 40 children in a class arranged in alphabetical order. 16 of the surnames begins with O and 9 of the surname begins with A, 14, of the letters of the alphabet do not appear as the first letter of a surname

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In a certain school, the lesson periods for each week are as itemised below: English 10, Mathematics 7, Biology 3, Statistics 4, Ibo 3, others 9. Draw a pie chart to illustrate this information.

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The number of items produced by a company over a five- year period is given below:

(i) Plot a bar chart for this information; (ii) What is the average production for the five- year period.

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The table below is for the relation \(y = 2 + x - x^{2}\)

(a) Using a scale of 2cm to 1 unit on each axis, draw the graph of the relation in the interval \(-2 \leq x \leq 3\).

(b) From your graph, find the greatest value of y and the value of x for which this occurs.

(c) Using the same scale and axes, draw the graph of \(y = 1 - x\)

(d) Use your graphs to solve the equation \(1 + 2x - x^{2} = 0\)

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The table below shows the frequency distribution of the marks of 800 candidates in an examination.

(a) (i) Construct a cumulative frequency table ; (ii) Draw the Ogive ; (iii) Use your ogive to determine the 50th percentile.

(b) The candidates that scored less than 25% are to be withdrawn from the institution, while those that scored than 75% are to be awarded scholarship. Estimate the number of students that will be retained, but will not enjoy the award.

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Illustrate the following on graph paper and shade the region which satisfies all the three inequalities at the same time :

\(- x + 5y \leq 10 ; 3x - 4y \leq 8\) and \(x > -1\).

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(a) Given that \(\sin \alpha = 0.3907\), use tables to find the value of : (i) \(\tan \alpha\) ; (ii) \(\cos \alpha\).

(b) A ladder of length 4.5m leans against a vertical wall making an angle of 50° with the horizontal ground. If the bottom of a window is 4m above the ground, what is the distance between the top of the ladder and the bottom of the window? (Answer correct to the nearest cm)

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(a) Prove that the sum of the angles in a triangle is two right angles.

(b) In a triangle LMN, the side NM is produced to P and the bisector of < LNP meets ML produced at Q. If < LMN = 46°, and < MLN = 80°, calculate < LQN, stating clearly your reasins.

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A sector of a circle of radius 7cm subtending an angle of 270° at the centre of the circle is used to form a cone.

(a) Find the base radius of the cone.

(b) Calculate the area of the base of the cone to the nearest square centimetre.

[Take \(\pi = \frac{22}{7}\)]

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Two towns K and Q are on the parallel of latitude 46°N. The longitude of town K is 130°W and that of town Q is 103°W. A third town P also on latitude 46°N is on longitude 23°E, Calculate:

(i) the length of the parallel of latitude 46°N, to the nearest 100km;

(ii) the distance between K and Q, correct to the nearest 100km;

(iii) the distance between Q and P measured along the parallel of latitude, to the nearest 10km.

[Take \(\pi = 3.142\); Radius of the earth = 6400km]

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Using a ruler and a pair of compasses only, 

(a) construct (i) a triangle ABC such that |AB| = 5cm, |AC| = 7.5cm and < CAB = 120° ; (ii) the locus \(L_{1}\) of points equidistant from A and B ; (iii) the locus \(L_{2}\) of points equidistant from Ab and AC, which passes through the triangle ABC.

(b) Label the point P where \(L_{1}\) and \(L_{2}\) intersect; 

(c) Measure |CP|.

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In the diagram, < PQR = < PSQ = 90°, |PS| = 9 cm, |SR| = 16 cm and |SQ| = x cm.

(a) Find the value of x using a trigonometric ratio.

(b) Calculate : (i) the size of < QRS to the nearest degree; (ii) |PQ|.

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(a) The subsets A, B and C of a universal set are defined as follows :

A = {m, a, p, e} ; B = {a, e, i, o, u} ; C = {l, m, n, o, p, q, r, s, t, u}. List the elements of the following sets.

(i) \(A \cup B\) ; (ii) \(A \cup C\) ; (iii) \(A \cup (B \cap C)\).

(b) Out of the 400 students in the final year in a Senior Secondary School, 300 are offering Biology and 190 are offering Chemistry.

(i) How many students are offering both Biology and Chemistry, if only 70 students are offering neither Biology nor Chemistry? (ii) How many students are offering at least one of Biology or Chemistry?

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