WAEC SSCE - General Mathematics - 1997

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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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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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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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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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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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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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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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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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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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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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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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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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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:

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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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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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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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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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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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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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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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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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(a) The 20th term. For an A.P., \(T_n=a+(n-1)d\).

\[T_6=a+5d=35,\qquad T_{13}=a+12d=77.\]

Subtracting the first from the second: \(7d=42\Rightarrow d=6\). Then \(a=35-5(6)=5\).

\[T_{20}=a+19d=5+19(6)=5+114=119.\]

(b) Shading the Venn diagram. The three circles \(P\) (top left), \(Q\) (top right) and \(R\) (bottom) divide \(U\) into eight regions.

(i) \(P\cap Q'\cap R\): shade the region lying inside both \(P\) and \(R\) but outside \(Q\). This is the overlap (lens) of the \(P\) and \(R\) circles, with the central piece that also lies in \(Q\) left unshaded.

(ii) \(P'\cap Q\cap R'\): shade the region lying inside \(Q\) only, that is, the part of the \(Q\) circle that is outside both \(P\) and \(R\).

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(a) Binary multiplication table. Multiply in base ten, then convert back to base two (e.g. \(11\times 11 = 3\times 3 = 9 = 1001_{two}\), \(11\times 101 = 3\times 5 = 15 = 1111_{two}\)).

\(\times\)	10	11	100	101
10	100	110	1000	1010
11	110	1001	1100	1111
100	1000	1100	10000	10100

(b) Convert \(11.011_{two}\) to base ten.

\[1\times 2 + 1\times 1 + 0\times\tfrac{1}{2} + 1\times\tfrac{1}{4} + 1\times\tfrac{1}{8} = 2 + 1 + 0.25 + 0.125 = \mathbf{3.375}\]

(c) Simplify.

\[\frac{9.6\times 10^{18}}{0.24\times 10^{5}} = \frac{9.6}{0.24}\times 10^{18-5} = 40\times 10^{13} = 4.0\times 10^{14}\]

So \(P = 4.0\) and \(m = 14\).

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(a) Evaluate \(\dfrac{15.05 \times \sqrt{0.00695}}{6.95 \times 10^{2}}\) with logarithms.

First \(\sqrt{0.00695}\): \(\log 0.00695 = \bar{3}.8420\), half of it \(= \bar{2}.9210\), so \(\sqrt{0.00695} = 0.08337\).

Numerator: \(\log 15.05 = 1.1775\); \(1.1775 + \bar{2}.9210 = 0.0985\), giving \(15.05 \times 0.08337 = 1.2547\).

Divide by \(695\): \(\log 695 = 2.8420\); \(0.0985 - 2.8420 = \bar{3}.2565\).

\[\text{answer} = 1.806 \times 10^{-3} \approx 0.00181\]

(b) Two boys and three girls; two are chosen at random from the five. Total ways \(= \binom{5}{2} = 10\).

(i) Both boys: \(\dfrac{\binom{2}{2}}{\binom{5}{2}} = \dfrac{1}{10}\).

(ii) Different sexes (one boy, one girl): \(\dfrac{2 \times 3}{10} = \dfrac{6}{10} = \dfrac{3}{5}\).

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(a) Given \(\sin x=\dfrac{5}{13}\) with \(0^\circ\le x\le 90^\circ\), evaluate \(\dfrac{\cos x-2\sin x}{2\tan x}\).

Using a \(5\)-\(12\)-\(13\) right triangle (adjacent \(=\sqrt{13^2-5^2}=\sqrt{144}=12\)):

\[\cos x=\frac{12}{13},\qquad \tan x=\frac{5}{12}.\]

Substitute:

\[\frac{\cos x-2\sin x}{2\tan x}=\frac{\dfrac{12}{13}-2\cdot\dfrac{5}{13}}{2\cdot\dfrac{5}{12}}=\frac{\dfrac{12-10}{13}}{\dfrac{10}{12}}=\frac{\dfrac{2}{13}}{\dfrac{5}{6}}.\]\[=\frac{2}{13}\times\frac{6}{5}=\mathbf{\frac{12}{65}}.\]

(b) \(NQ\) is the vertical height of the mountain on horizontal ground \(PRN\) (with \(R\) between \(P\) and \(N\)). The angle of elevation of the top \(Q\) is \(30^\circ\) from \(P\) and \(70^\circ\) from \(R\), and \(PR=500\text{ m}\).

(i) \(|QP|\). In triangle \(PQR\), \(\angle QPR=30^\circ\). Since \(\angle QRN=70^\circ\) is exterior to the triangle at \(R\), the interior angle is

\[\angle QRP=180^\circ-70^\circ=110^\circ,\]\[\angle PQR=180^\circ-30^\circ-110^\circ=40^\circ.\]

By the sine rule:

\[\frac{|QP|}{\sin\angle QRP}=\frac{|PR|}{\sin\angle PQR}\]\[|QP|=\frac{500\times\sin 110^\circ}{\sin 40^\circ}=\frac{500\times0.9397}{0.6428}\approx\mathbf{731\text{ m}}.\]

(ii) Height of the mountain \(NQ\). In right-angled triangle \(QNP\) (right angle at \(N\)), with \(\angle QPN=30^\circ\):

\[NQ=|QP|\sin 30^\circ=730.95\times0.5\approx\mathbf{365\text{ m}}.\]

(Check via triangle \(QRN\): \(QR=\dfrac{500\sin30^\circ}{\sin40^\circ}=388.9\text{ m}\), and \(NQ=QR\sin70^\circ=388.9\times0.9397\approx365\text{ m}\).)

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(a) Construction: Draw \(AB = 7\text{ cm}\). At \(B\), construct an angle of \(75^\circ\) by constructing \(60^\circ\) and \(90^\circ\), then bisecting the angle between them. On the \(75^\circ\) ray, mark \(C\) such that \(BC = 5\text{ cm}\). Join \(AC\).

\(|AC| = 7.5\text{ cm}\).

(b) Construction: Through \(C\), construct a line parallel to \(AB\). Construct the perpendicular bisector of \(AC\). Let it meet the parallel line at \(D\). Join \(AD\).

\(\angle BAD = 80^\circ\).

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(a) \(\dfrac{5y - x}{8y + 3x} = \dfrac15\). Cross-multiplying:

\[5(5y - x) = 8y + 3x \;\Rightarrow\; 25y - 5x = 8y + 3x\]

\[17y = 8x \;\Rightarrow\; \frac{x}{y} = \frac{17}{8} = 2.13 \text{ (2 d.p.)}\]

(b) Since \(3\) is a root of \(x^2 + bx - 15 = 0\):

\[3^2 + 3b - 15 = 0 \;\Rightarrow\; 9 + 3b - 15 = 0 \;\Rightarrow\; 3b = 6 \;\Rightarrow\; b = 2\]

The equation becomes \(x^2 + 2x - 15 = 0 = (x + 5)(x - 3)\). Hence the other root is \(x = -5\).

(Check by product of roots: \(3 \times (-5) = -15\), matching the constant term.)

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From the diagram the solid is a closed cylinder capped on top by a hemisphere. The hemisphere has height \(7\text{ cm}\); since a hemisphere's height equals its radius, the radius is \(r = 7\text{ cm}\), which is also the radius of the cylinder. The cylinder height is \(h = 10\text{ cm}\). Take \(\pi = \dfrac{22}{7}\).

(a) Total surface area.

The exposed surfaces are: the curved surface of the cylinder, the flat circular base of the cylinder, and the curved surface of the hemisphere. (The flat top of the cylinder is exactly covered by the flat face of the hemisphere, so it is not counted.)

Curved surface of cylinder:

\[2\pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440 \text{ cm}^2\]

Base circle of cylinder:

\[\pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{ cm}^2\]

Curved surface of hemisphere:

\[2\pi r^2 = 2 \times 154 = 308 \text{ cm}^2\]

Total surface area:

\[440 + 154 + 308 = 902 \text{ cm}^2\]

Total surface area \(= 902\text{ cm}^2\).

(b) Volume.

Volume of cylinder:

\[\pi r^2 h = \frac{22}{7} \times 49 \times 10 = 22 \times 7 \times 10 = 1540 \text{ cm}^3\]

Volume of hemisphere:

\[\frac{2}{3}\pi r^3 = \frac{2}{3} \times \frac{22}{7} \times 7^3 = \frac{2}{3} \times \frac{22}{7} \times 343 = \frac{2}{3} \times 22 \times 49 = \frac{2156}{3} \approx 718.67 \text{ cm}^3\]

Total volume:

\[1540 + 718.67 = 2258.67 \text{ cm}^3\]

Total volume \(\approx 2258.67\text{ cm}^3\) (i.e. \(2258\tfrac{2}{3}\text{ cm}^3\)).

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None

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(a) Let \(\log_{10}x = u\) and \(\log_{10}y = v\).

\[u + v = 4 \quad\text{...(1)}\qquad u + 2v = 3 \quad\text{...(2)}\]

Subtract (1) from (2): \(v = -1\). Then \(u = 4 - (-1) = 5\).

\[\log_{10}x = 5 \Rightarrow x = 10^5 = 100000, \qquad \log_{10}y = -1 \Rightarrow y = 10^{-1} = 0.1\]

(b) \(t\) varies directly as \(V\) and inversely as \(P\): \(t = \dfrac{kV}{P}\).

With \(P = 5,\, V = 20,\, t = 10\): \(10 = \dfrac{k(20)}{5} = 4k \Rightarrow k = 2.5\).

(i) \(t = \dfrac{2.5V}{P}\) (equivalently \(t = \dfrac{5V}{2P}\)).

(ii) \(V = 50,\, P = 2\): \(t = \dfrac{2.5(50)}{2} = 62.5\) minutes.

(iii) \(V = 40,\, t = 20\): \(20 = \dfrac{2.5(40)}{P} \Rightarrow 20 = \dfrac{100}{P} \Rightarrow P = 5\) pumps.