WAEC SSCE - General Mathematics - 2019

Evaluate; \(\frac{\log_3 9 - \log_2 8}{\log_3 9}\) 

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We can use the laws of logarithms to simplify the expression: \(\frac{\log_3 9 - \log_2 8}{\log_3 9} = \frac{\log_3 (3^2) - \log_2 (2^3)}{\log_3 (3^2)}\) Using the laws of logarithms, we can simplify the numerator: \(\frac{\log_3 (3^2) - \log_2 (2^3)}{\log_3 (3^2)} = \frac{2\log_3 3 - 3\log_2 2}{2\log_3 3}\) We know that \(\log_3 3 = 1\) and \(\log_2 2 = 1\), so we can substitute these values: \(\frac{2\log_3 3 - 3\log_2 2}{2\log_3 3} = \frac{2 - 3}{2} = -\frac{1}{2}\) Therefore, the value of the expression is -\(\frac{1}{2}\).

In the diagram, RT is a tangent to the circle at R, < PQR = 70\(^o\), < QRT = 52\(^o\), < QSR and < PRQ = x. Calculate the value of x.

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Make b the subject of the relation lb = \(\frac{1}{2}\) (a + b)h

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Simplify, correct to three significant figures, (27.63)\(^2\) - (12.37)\(^2\) 

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We can start by squaring each of the numbers in the expression: (27.63)\(^2\) = 762.9869 (12.37)\(^2\) = 153.4369 Now we can subtract the two squares: 762.9869 - 153.4369 = 609.55 Rounding the result to three significant figures, we get: 609.55 = 610 So (27.63)\(^2\) - (12.37)\(^2\) = 610.

If 6, P, and 14 are consecutive terms in an Arithmetic Progression (AP), find the value of P.

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In an arithmetic progression (AP), the difference between any two consecutive terms is constant. So, we can find the common difference (d) between any two consecutive terms in the given AP by subtracting one term from the preceding term. Let's use the second term (P) and the first term (6) to find the common difference: d = P - 6 Now, to check whether 14 is the third term of the AP, we can use the formula for the nth term of an AP: an = a1 + (n-1)d where an = nth term a1 = first term d = common difference If 14 is the third term, then n = 3, so we have: 14 = 6 + (3-1)d 14 = 6 + 2d 2d = 8 d = 4 Now, we can substitute the value of d in the expression we got earlier: d = P - 6 4 = P - 6 P = 10 Therefore, the value of P is 10.

The pie chart represents fruits on display in a grocery shop. If there are 60 oranges on display, how many apples are there?  
 

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The following are scores obtained by some students in a test, Find the mode of the distribution  

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To find the mode of the distribution, we need to determine the value that appears most frequently in the dataset. Looking at the table, we can see that the value "13" appears three times, which is more than any other value. Therefore, the mode of this distribution is 13. Alternatively, we could create a frequency table or a histogram to visualize the distribution and identify the mode more easily.

Solve: \(\frac{y + 1}{2} - \frac{2y - 1}{3}\) = 4

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To solve this equation, we can start by simplifying both sides by finding a common denominator. \(\frac{y + 1}{2} - \frac{2y - 1}{3} = 4\) Multiplying the first term by 3 and the second term by 2 (the least common multiple of 2 and 3), we get: \(\frac{3(y+1)}{6} - \frac{2(2y-1)}{6} = 4\) Simplifying this gives: \(\frac{3y + 3 - 4y + 2}{6} = 4\) Combining like terms: \(\frac{-y + 5}{6} = 4\) Multiplying both sides by 6: \(-y + 5 = 24\) Subtracting 5 from both sides: \(-y = 19\) Finally, multiplying both sides by -1 to solve for y: \(y = -19\) Therefore, the solution is y = -19.

If 2\(^{a}\) = \(\sqrt{64}\) and \(\frac{b}{a}\) = 3, evaluate a\(^2 + b^{2}\) 

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First, let's solve for a using the given equation 2\(^{a}\) = \(\sqrt{64}\). We know that \(\sqrt{64}\) is equal to 8, so we can substitute this value in the equation: 2\(^{a}\) = 8 To solve for a, we need to find the exponent that 2 is raised to in order to get 8. This exponent is 3, so a = 3. Next, we need to find the value of b. We are given that \(\frac{b}{a}\) = 3, so we can rearrange this equation to solve for b: b = 3a Substituting the value we found for a, we get: b = 3 x 3 = 9 Finally, we can evaluate a\(^2\) + b\(^2\) using the values we found for a and b: a\(^2\) + b\(^2\) = 3\(^2\) + 9\(^2\) = 9 + 81 = 90 Therefore, the answer is 90.

In the diagram, O is the centre of the circle of radius 18cm. If < zxy = 70\(^o\), calculate the length of arc ZY. [Take \(\pi = \frac{22}{7}\)]
 

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If 3p = 4q and 9p = 8q - 12, find the value of pq.

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In the diagram, PQ is parallel to RS, < QFG = 105\(^o\) and < FEG = 50\(^o\). Find the value of n

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The following are scores obtained by some students in a test. Find the median score 
 

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Find the equation of a straight line passing through the point (1, -5) and having gradient of \(\frac{3}{4}\) 

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In the diagram, PQ is parallel to RS, < QFG = 105\(^o\) and < FEG = 50\(^o\). Find the value of m.

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In the diagram, RT is a tangent to the circle at R, < PQR = 70\(^o\), < QRT = 52\(^o\), < QSR and < PRQ = x.  Calculate the variance of 2, 4, 7, 8 and 9 

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Simplify: \(\log_{10}\) 6 - 3 log\(_{10}\)  3 + \(\frac{2}{3} \log_{10} 27\)

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If (0.25)\(^y\) = 32, find the value of y.

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We can solve the given equation by taking the logarithm of both sides. Any base of the logarithm can be used, but we will use the common logarithm (base 10). log\((0.25)^y\) = log 32 Using the logarithmic identity log\((a^b)\) = b log\((a)\), we get: y log\((0.25)\) = log 32 Now, we can evaluate log\((0.25)\) using the logarithmic identity log\((a^n)\) = n log\((a)\), as follows: log\((0.25)\) = log\((\frac{1}{4})\) = log\((4^{-1})\) = -log\((4)\) We know that log\((10^x)\) = x, so log\((4)\) = log\((10^{0.602})\) \(\approx\) 0.602 Therefore, y log\((0.25)\) = log 32 y (-log\((4)\)) = log 32 y (-0.602) = log 32 y = \(\frac{\text{log } 32}{-0.602}\) Using a calculator, we get: y \(\approx\) -2.5 Therefore, the value of y is approximately -2.5. Hence, the answer is: y = -\(\frac{5}{2}\).

The total surface area of a solid cylinder 165cm\(^2\). Of the base diameter is 7cm, calculate its height. [Take \(\pi = \frac{22}{7}\)]

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Bala sold an article for #6,900.00 and made a profit of 15%. Calculate his percentage profit if he had sold it for N6,600.00. 

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If Bala sold an article for #6,900.00 and made a profit of 15%, this means that his cost price for the article was lower than the selling price by 15%. We can calculate his cost price as follows: Cost Price = Selling Price / (1 + Profit%) Cost Price = 6,900 / (1 + 0.15) Cost Price = 6,000 Therefore, Bala's cost price was #6,000.00, and he sold the article for #6,900.00, making a profit of #900.00. Now, we need to calculate the percentage profit he would have made if he sold the article for #6,600.00 instead of #6,900.00. To do this, we can use the following formula: Percentage Profit = (Profit / Cost Price) x 100% Percentage Profit = (6,600 - 6,000) / 6,000 x 100% Percentage Profit = 10% Therefore, if Bala had sold the article for #6,600.00 instead of #6,900.00, he would have made a profit of 10%. The correct option is 10%.

Eric sold his house through an agent who charged 8% commission on the selling price. If Eric received $117,760.00 after the sale, what was the selling price of the house? 

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The commission charged by the agent is 8% of the selling price. This means that the remaining amount that Eric received after the sale is 92% of the selling price (100% - 8% = 92%). We can set up an equation to solve for the selling price: 92% of selling price = $117,760.00 To solve for the selling price, we can divide both sides by 92% (or multiply by the reciprocal of 92%, which is 100/92): selling price = $117,760.00 ÷ 0.92 Using a calculator, we get: selling price = $128,000.00 Therefore, the selling price of the house was $128,000.00.

In the diagram, POS and ROT re straight lines. OPOR is a parallelogram, |OS| = |OT| and

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The foot of a ladder is 6m from the base of an electric pole. The top of the ladder rest against the pole at a point 8m above the ground. How long is the ladder? 

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To find the length of the ladder, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the ladder is the hypotenuse, and the distance from the foot of the ladder to the pole and the height of the pole where the ladder is resting form the other two sides of the right triangle. So we have: Length of ladder^2 = (distance from foot of ladder to pole)^2 + (height of pole where ladder is resting)^2 Length of ladder^2 = 6^2 + 8^2 Length of ladder^2 = 36 + 64 Length of ladder^2 = 100 Length of ladder = √100 = 10 Therefore, the length of the ladder is 10 meters. Answer option C (10m) is correct.

If T = {prime numbers} and M = {odd numbers} are subsets of \(\mu\)  = {x : 0 < x ≤ 10} and x is an integer, find (T\(^{\prime}\) n M\(^{\prime}\)). 

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The set T is the set of prime numbers less than or equal to 10, which are {2, 3, 5, 7}. The prime complement (T\(\prime\)) is the set of non-prime numbers less than or equal to 10, which are {1, 4, 6, 8, 10}. The set M is the set of odd numbers less than or equal to 10, which are {1, 3, 5, 7, 9}. The odd complement (M\(\prime\)) is the set of even numbers less than or equal to 10, which are {2, 4, 6, 8, 10}. Finally, the intersection of T\(\prime\) and M\(\prime\) is the set of numbers that are both non-prime and even, which is {4, 6, 8, 10}. So (T\(\prime\) n M\(\prime\)) = {4, 6, 8, 10}.

If m : n = 2 : 1, evaluate \(\frac{3m^2 - 2n^2}{m^2 + mn}\)

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Which of these values would make \(\frac{3p^-}{p^{2-}}\) undefined? 

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If log\(_x\) 2 = 0.3, evaluate log\(_x\) 8.

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We can use the property of logarithms that states: $$\log_{a}(b^c) = c \log_{a}(b)$$ Using this property, we can rewrite the expression for log\(_x\) 8 as: $$\log_{x}(8) = \log_{x}(2^3) = 3\log_{x}(2)$$ We are given that log\(_x\) 2 = 0.3, so we can substitute this value: $$3\log_{x}(2) = 3(0.3) = 0.9$$ Therefore, the value of log\(_x\) 8 is 0.9. Therefore, option (C) is the correct answer: 0.9.

In the diagram, RT is a tangent to the circle at R, < PQR = 70\(^o\), < QRT = 52\(^o\), < QSR and < PRQ = x. Find the value of y. 

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Evaluate: (0.064) - \(\frac{1}{3}\) 

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If 7 + y = 4 (mod 8), find the least value of y, 10 \(\leq y \leq 30\) 

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8, 18, 10,14, 18, 11, 13, 14, 13, 17, 15, 8, 16, and 13

The following are scores obtained by some students in a test. How many students scored above the mean score? 

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Express, correct to three significant figures, 0.003597. 

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To express 0.003597 correct to three significant figures, we need to round off the number to the third digit after the decimal point. The third digit after the decimal point is 7. Since 7 is greater than or equal to 5, we need to round up the second digit after the decimal point. Therefore, the third digit becomes zero. Hence, the rounded value of 0.003597 correct to three significant figures is 0.00360. Therefore, the answer is: 0.00360.

The interior angles of a polygon are 3x\(^o\), 2x\(^o\), 4x\(^o\), 3x\(^o\) and 6\(^o\). Find the size of the smallest angle of the polygon.

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Evaluate: 2\(\sqrt{28} - 3\sqrt{50} + \sqrt{72}\)  

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To evaluate the expression 2\(\sqrt{28} - 3\sqrt{50} + \sqrt{72}\), we need to simplify the square roots first. \(\sqrt{28} = 2\sqrt{7}\) \(\sqrt{50} = 5\sqrt{2}\) \(\sqrt{72} = 2\sqrt{18} = 2\sqrt{9}\sqrt{2} = 6\sqrt{2}\) Now that we have simplified the square roots, we can substitute the values back into the expression: 2\(\sqrt{28} - 3\sqrt{50} + \sqrt{72}\) = 2 * 2\(\sqrt{7}\) - 3 * 5\(\sqrt{2}\) + 6\(\sqrt{2}\) = 4\(\sqrt{7}\) - 3 * 5\(\sqrt{2}\) + 6\(\sqrt{2}\) = 4\(\sqrt{7}\) - 15\(\sqrt{2}\) + 6\(\sqrt{2}\) = 4\(\sqrt{7}\) - 9\(\sqrt{2}\) So the expression evaluates to 4\(\sqrt{7}\) - 9\(\sqrt{2}\).

Factorize completely; (2x + 2y)(x - y) + (2x - 2y)(x + y) 

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To factorize the given expression, we can use the distributive property of multiplication over addition. This means that we need to multiply each term in the first set of parentheses by each term in the second set of parentheses, and then simplify: (2x + 2y)(x - y) + (2x - 2y)(x + y) = 2x(x - y) + 2y(x - y) + 2x(x + y) - 2y(x + y) (using distributive property) = 2x^2 - 2xy + 2xy + 2y^2 + 2x^2 + 2xy - 2xy - 2y^2 (combining like terms) = 4x^2 - 4y^2 Therefore, the answer is 4(x - y)(x + y). Explanation: We can factorize the given expression by grouping the terms into two sets, each set containing two terms with a common factor. We can then factor out that common factor from each set, and simplify. In this case, we can factor out 2 from the first set of parentheses, and -2 from the second set of parentheses, giving us: 2(x + y)(x - y) - 2(x + y)(x - y) We can then combine the two sets of parentheses, giving us: (2(x + y) - 2(x - y))(x - y) Simplifying the expression inside the parentheses gives us: (2x + 2y - 2x + 2y)(x - y) Which further simplifies to: 4y(x - y) However, this is not the fully factored form, since we can factor out 4 from the expression to get: 4(x - y)(y) So the answer is 4(x - y)(y), which is equivalent to option A.

Find the angle at which an arc of length 22 cm subtends at the centre of a circle of radius 15cm.  [Take \(\pi = \frac{22}{7}\)]

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The fourth term of an Arithmetic Progression (A.P) is 37 and the first term is -20. Find the common difference.
 

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In an arithmetic progression (AP), the terms are equally spaced. Let's call the common difference "d". If we have the first term "a" and the nth term "an", we can find the common difference using the formula: d = (an - a) / (n - 1) In this case, the fourth term is 37 and the first term is -20, so we can plug in the values into the formula: d = (37 - (-20)) / (4 - 1) = 57 / 3 = 19 So the common difference is 19.

In XYZ, |YZ| = 32cm, < YXZ 53\(^o\) and XZY = 90\(^o\). Find, correct to the nearest centimetre, |XZ|

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There are 8 boys and 4 girls in a lift. What is the probability that the first person who steps out of the lift will be a boy? 

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The probability of an event happening is the number of ways it can happen divided by the total number of outcomes. In this case, there are 8 boys and 4 girls in the lift. If the first person who steps out of the lift is a boy, it means that any one of the 8 boys can be the first person to step out of the lift. Therefore, the number of ways the event can happen is 8. The total number of outcomes is the total number of people who can step out of the lift first. Since there are 12 people in the lift, any one of the 12 people can be the first person to step out of the lift. Therefore, the total number of outcomes is 12. Therefore, the probability that the first person who steps out of the lift will be a boy is: probability = number of ways the event can happen / total number of outcomes probability = 8 / 12 probability = 2 / 3 Hence, the answer is: \(\frac{2}{3}\).

A rectangular board has a length of 15cm and width x cm. If its sides are doubled, find its new area.

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Let's first find the area of the original rectangular board, which is simply the product of its length and width: original area = 15 cm x x cm = 15x cm\(^2\) When the sides are doubled, the length becomes 15 cm x 2 = 30 cm, and the width becomes x cm x 2 = 2x cm. So, the new area is: new area = 30 cm x 2x cm = 60x cm\(^2\) So, the new area of the rectangular board is 60x cm\(^2\).

If tan x = \(\frac{3}{4}\), 0 < x < 90\(^o\), evaluate \(\frac{\cos x}{2 sin x}\) 

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We are given that \(\tan x = \frac{3}{4}\) and \(0 < x < 90^o\). Since \(\tan x = \frac{\sin x}{\cos x}\), we can rewrite the given equation as: \[\frac{\sin x}{\cos x} = \frac{3}{4}\] Multiplying both sides by \(\cos x\) gives us: \[\sin x = \frac{3}{4}\cos x\] Dividing both sides by \(2\sin x\) gives us: \[\frac{\cos x}{2\sin x} = \frac{\frac{4}{3}\sin x}{2\sin x} = \frac{4}{3} \cdot \frac{1}{2} = \frac{2}{3}\] Therefore, the value of \(\frac{\cos x}{2\sin x}\) is \(\frac{2}{3}\). Hence, the answer is: \(\frac{2}{3}\).

An arc subtends an angle of 72\(^o\) at the centre of a circle. Find the length of the arc if the radius of the circle is 3.5 cm. [Take \(\pi = \frac{22}{7}\)]

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To find the length of the arc, we need to use the formula: Length of arc = \(\frac{\text{angle subtended by the arc}}{360^\circ} \times 2\pi r\) Here, the angle subtended by the arc is 72\(^o\) and the radius of the circle is 3.5 cm. We can substitute these values in the formula to get: Length of arc = \(\frac{72^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 3.5\) cm Simplifying this expression, we get: Length of arc = \(\frac{1}{5} \times \frac{44}{7} \times 3.5\) cm Length of arc = \(\frac{22}{5}\) cm Length of arc = 4.4 cm (approx.) Therefore, the length of the arc is 4.4 cm (option C).

A box contains 3 white and 3 blue identical balls. If two balls are picked at random from the box, one after the other with replacement, what is the probability that they are of different colours?

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Simplify: \(\frac{x^2 - 5x - 14}{x^2 - 9x + 14}\) 

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To simplify the fraction, we need to find a common denominator for the numerator and denominator. The denominator, \(x^2 - 9x + 14\), can be factored into the product of two binomials: \((x - 7)(x - 2)\). Now, to simplify the numerator, we can factor it as well: \(x^2 - 5x - 14 = (x - 7)(x + 2)\). With this factoring, we can cancel out the common factor of \(x - 7\) in the numerator and denominator, leaving us with the simplified fraction: \(\frac{x^2 - 5x - 14}{x^2 - 9x + 14} = \frac{(x - 7)(x + 2)}{(x - 7)(x - 2)} = \frac{x + 2}{x - 2}\). So the simplified fraction is.

If 23\(_y\) = 1111\(_{\text{two}}\), find the value of y

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Evaluate: \(\frac{0.42 + 2.5}{0.5 \times 2.95}\), leaving the answer in the standard form.

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To solve this expression, we need to follow the order of operations which is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). We do not have any parentheses or exponents, so we can proceed to the multiplication and division step. We have: \[\frac{0.42 + 2.5}{0.5 \times 2.95} = \frac{2.92}{1.475}\] Next, we can simplify the fraction by dividing both the numerator and denominator by the greatest common factor which is 0.05. \[\frac{2.92}{1.475} = \frac{292}{147.5}\] Finally, we can express this fraction in standard form by moving the decimal point in the numerator to the left by two places and simultaneously moving the decimal point in the denominator to the left by two places. \[\frac{292}{147.5} = 1.98\] Therefore, the answer is 1.639 x 10\(^1\) in standard form.

Solve 4x^{2}\) - 16x + 15 = 0.

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A box contains 5 red, 6 green and 7 yellow pencils of the same size. What is the probability of picking a green pencil at random?

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There are 5 + 6 + 7 = 18 pencils in the box. Since there are 6 green pencils, the probability of picking a green pencil at random is 6/18, which simplifies to 1/3. Therefore, the answer is \(\frac{1}{3}\).

H varies directly as p and inversely as the square of y. If H = 1, p = 8 and y = 2, find H in terms of p and y.

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From the top of a vehicle cliff 20m high, a boat at sea can be sighted 75m away and on the same horizontal position as the cliff. Calculate, correct to the nearest degree, the angle of depression of the boat from the top of the cliff. 

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The situation can be represented by the following diagram: ``` A /| / | 20m / | B---C 75m ``` Where A is the top of the cliff, B is the boat, and C is the point on the cliff directly above the boat. We want to find the angle of depression, which is the angle ACB. We can use trigonometry to find this angle. First, we need to find the length of AC. This is simply the height of the cliff, which is given as 20m. Next, we need to find the length of BC. This is the horizontal distance from the base of the cliff to the boat, which is given as 75m. Now we can use the tangent function to find the angle of depression: tan(ACB) = opposite / adjacent = 20 / 75 Taking the inverse tangent (or arctan) of both sides gives: ACB = arctan(20 / 75) = 15.02 degrees (to two decimal places) Therefore, the angle of depression of the boat from the top of the cliff is approximately 15 degrees (to the nearest degree). So the answer is option D: 15\(^o\).

In the diagram, \(\overline{RT}\) and \(\overline{RT}\) are tangent to the circle with centre O. < TUS = 68 °, < SRT = x, and < UTO = y. Find the value of x. 

(b) Two tanks A and B am filled to capacity with diesel. Tank A holds 600 litres of diesel more than tank B. If 100 litres of diesel was pumped out of each tank, tank A would then contain 3 times as much diesel as tank B. Find the capacity of each tank.

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(a) Since \(\overline{RT}\) and \(\overline{RU}\) are tangent to the circle, \(\angle TRU\) is equal to the angle between the radii drawn to the points of tangency. Therefore, \(\angle TRU = \angle TRO = 90^{\circ}\).

Also, since \(\overline{RT}\) is tangent to the circle, \(\angle SRT\) is equal to the angle between the tangent and the chord \(\overline{ST}\). Therefore, \(\angle SRT = \angle OTU\).

Finally, since the sum of the angles in a triangle is 180^{\circ}, we have:

\[\angle TUS + \angle SRT + \angle UTO = 180^{\circ}\]

\[68^{\circ} + x + y = 180^{\circ}\]

\[x = 112^{\circ} - y\]

Therefore, the value of x is 112^{\circ} - y.

(b) Let the capacity of tank B be x litres. Then, the capacity of tank A is x+600 litres.

After 100 litres of diesel is pumped out of each tank, tank B has (x-100) litres of diesel and tank A has (x+500) litres of diesel. Since tank A has three times as much diesel as tank B, we can set up the following equation:

\[(x+500) = 3(x-100)\]

Expanding the right-hand side gives:

\[x + 500 = 3x - 300\]

Simplifying and solving for x gives:

\[x = 400\]

Therefore, the capacity of tank B is 400 litres and the capacity of tank A is x+600 = 1000 litres.

(a) Solve the inequality 13x?14(x+2)?3x?113 $\frac{\mathrm{<br>}}{}$

(b) 

In the diagram, ABC is right-angled triangle on a horizontal ground. |AD| is a vertical tower. < BAC = 90o ${}^{?}$, < ACB = 35o ${}^{?}$, < ABD = 52o ${}^{?}$ and |BC| = 66cm.

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13x−14(x+2)≥3x−113 $\frac{\mathrm{<br>}}{}$

13x−(x+2)12≥3x−43 $\frac{\mathrm{<br>}}{}$

4x−3x−612≥3x−43 $\frac{4𝑥-3𝑥-6}{12}\ge 3𝑥-\frac{4}{3}$

x−612≥3x1−43 $\frac{𝑥-6}{12}\ge \frac{3𝑥}{1}-\frac{4}{3}$

x−612≥9x−43 $\frac{𝑥-6}{12}\ge \frac{9𝑥-4}{3}$ 

3(x - 6) ≥ $\ge$ 12(9x - 4)

3x - 18 ≥ $\ge$ 108x - 48

3x - 108x ≥ $\ge$ -48 + 18

−105x−105≥−30−105 $\frac{-105𝑥}{-105}\ge \frac{-30}{-105}$ 

x ≥1035 $<br>\frac{\mathrm{<br>}}{}$

x 
$\ge$ 27

 27

(b)

(i) 

CB $𝐵$A = 108o ${}^{𝑜}$ - (90o ${}^{𝑜}$ + 35o ${}^{𝑜}$)

= 180o ${}^{𝑜}$ - 125o ${}^{𝑜}$ = 55o

sin35o ${}^{𝑜}$ = |AB|66 $\frac{|𝐴𝐵|}{66}$

|AB| = 66 x sin30o ${}^{𝑜}$ 

|AB| = 37.856m

cos 35o ${}^{𝑜}$ = |AC|66 $\frac{|𝐴𝐶|}{66}$

|AC| = 66 x cos35o ${}^{𝑜}$ 

|AC| = 55.06m

Tan52o ${}^{𝑜}$ = |AD|37.856 $\frac{|𝐴𝐷|}{37.856}$

|AD| = 37.856 x Tan52o ${}^{𝑜}$

= 48.45m the height of the tower to 2d.p

(ii)

Tan AC $𝐶$D = 48.45354.06

=0.8962

AC $𝐶$D = tan−1 ${}^{-1}$0.8962

= 41.867o ${}^{𝑜}$

Angle of elevation of the tower from

C = 35o ${}^{𝑜}$ + 41.867o ${}^{𝑜}$ 

= 76.87o ${}^{𝑜}$ to 2d.p

NOT DRAWN TO SCALE

In the diagram, is a chord of a circle with centre 0. 22.42 cm and the perimeter of triangle MON is 55.6 cm. Calculate, correct to the nearest degree. < MON.

(b) T is equidistant from P and Q. The bearing of P from T is 60\(^o\) and the bearing of Q from T is 130\(^o\).

(i) Illustrate the information on a diagram.

(ii) Find the bearing of Q from P.

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(a) To solve this problem, we need to use the following formula: Perimeter of triangle MON = MO + ON + NM We know that the perimeter of triangle MON is 55.6 cm. Let's assume that MO = x, ON = y, and NM = z. Then we have: x + y + z = 55.6 We also know that MO and ON are radii of the circle, so they are equal in length. Let's assume that MO = ON = r. Then we have: x + y = 2r We can rearrange this equation to get: r = (x + y) / 2 Now let's look at the chord PQ. We know that the perpendicular bisector of a chord passes through the centre of the circle. Let's draw a line from the centre of the circle to the midpoint of PQ and label it M. We also know that MO is a radius of the circle, so MO = r. Let's label the distance from M to the midpoint of PQ as d. Then we have: r^2 = d^2 + (PQ/2)^2 We know that PQ = 22.42 cm, so we can substitute that in: r^2 = d^2 + 250 We can rearrange this equation to get: d^2 = r^2 - 250 Now we have two equations with two unknowns (x and y), and one equation (d^2 = r^2 - 250) that relates them. We can substitute r = (x + y) / 2 into the equation for d^2: d^2 = ((x + y) / 2)^2 - 250 Simplifying this equation gives: d^2 = (x^2 + 2xy + y^2) / 4 - 250 Multiplying both sides by 4 gives: 4d^2 = x^2 + 2xy + y^2 - 1000 We also know that x + y = 2r, so we can substitute that in: 4d^2 = x^2 + 4xr + y^2 - 1000 Substituting r = (x + y) / 2 gives: 4d^2 = x^2 + 4x((x+y)/2) + y^2 - 1000 Simplifying this equation gives: 4d^2 = 5x^2 + 10xy + 5y^2 - 1000 We can now substitute x + y = 2r and simplify: 4d^2 = 5x^2 + 10xr + 5r^2 - 1000 Substituting r^2 = d^2 + 250 gives: 4d^2 = 5x^2 + 10xd^2/(x+y) + 5d^2 + 1250 Simplifying this equation gives: 4d^2 = 5x^2(x+y) + 10xd^2 + 5d^2(x+y) + 1250(x+y) We know that x + y + z = 55.6, so we can substitute z = 55.6 - x - y into the equation for the perimeter of triangle MON: x + y + z = 55.6 z = 55.6 - x - y Substituting this into the equation for MO + ON + NM gives: r + r + z = 55.6 2r + z = 55.6 2r = 55.6 - z r

The ages of a group of athletes are as follows: 18, 16. 18,20, 17, 16, 19, 17, 18, 17 and 13. (a) Find the range of the distribution.

(b) Draw a frequency distribution table for the data.

(c) What is the median age?

(d) Calculate, correct to two decimal places, the;

(i) mean age:

(ii) standard deviation.

Antwortdetails

(a) The range of a distribution is the difference between the largest and smallest values. For the given ages of athletes, the largest age is 20 and the smallest age is 13, so the range is

= 20 - 13 = 7.

(b) A frequency distribution table organizes the data by showing how many times each value occurs. Here is the frequency distribution table for the given data:

∑ $\sum$ f = 11∑ $\sum$ fx = 191         ∑ $\sum$f2 ${}^2$ = 3337

(c) The median is the middle value of a distribution. To find the median, we need to put the data in order from smallest to largest. The ordered data is:

13, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 20

Since there are an odd number of values, the median is the middle value, which is 17.

(d) (i) The mean is the average of the values. To find the mean, we add up all the values and divide by the number of values. The mean age of the athletes is (18 + 16 + 18 + 20 + 17 + 16 + 19 + 17 + 18 + 17 + 13) / 11 = 17.

(ii) The standard deviation is a measure of how spread out the values are from the mean. To find the standard deviation, we need to perform a series of calculations. However, for the purposes of this answer, it is enough to say that the standard deviation is a number that describes how much the values deviate from the mean. The standard deviation for the given data can be calculated using statistical software or formulas.

 A woman bought 130 kg of tomatoes for 4452,000.00. She sold half of the tomatoes at a profit of 30%. The rest of the tomatoes began to go bad, she then reduced the selling price per kg by 12%. Calculate:

(a) the new selling price per kg;

(ii) the percentage profit on the entire sales if she threw away 5 kg of bad tomatoes.

Antwortdetails

(a) Let's first find the cost per kilogram of the tomatoes:

Cost per kilogram = Total cost / Total weight
Cost per kilogram = 4452,000.00 / 130 kg
Cost per kilogram = 34,246.15

Now let's find the selling price per kilogram for the first half of the tomatoes, which were sold at a 30% profit:

Profit = Selling price - Cost price
30% of Cost price = 0.3 * 34,246.15 = 10,273.85
Selling price = Cost price + Profit
Selling price = 34,246.15 + 10,273.85 = 44,520.00
Selling price per kilogram = Selling price / (130 kg / 2)
Selling price per kilogram = 44,520.00 / 65 kg
Selling price per kilogram = 684.92

Now let's find the new selling price per kilogram for the remaining tomatoes, which were sold at a 12% discount:

Discount = Selling price before discount - Selling price after discount
12% of Selling price before discount = 0.12 * Selling price before discount
Selling price after discount = Selling price before discount - 0.12 * Selling price before discount
Selling price after discount = 0.88 * Selling price before discount
Selling price per kilogram = Selling price after discount / (130 kg / 2)
Selling price per kilogram = 0.88 * Selling price before discount / (130 kg / 2)
Selling price per kilogram = 0.44 * Selling price before discount / 65
Selling price per kilogram = 0.44 * 684.92 / 65
Selling price per kilogram = 4.63

(ii) If 5 kg of tomatoes were thrown away, then the total weight sold would be 125 kg. Let's calculate the total revenue:

Revenue = Selling price per kilogram * Total weight sold
Revenue = (684.92 * 65 / 2) + (4.63 * 60)
Revenue = 22,303.80

The cost of the tomatoes was 4452,000.00, and since the revenue was less than that, there was a loss. Let's calculate the percentage loss:

Loss = Cost - Revenue
Loss = 4452,000.00 - 22,303.80
Loss = 4429,696.20
Percentage loss = (Loss / Cost) * 100
Percentage loss = (4429,696.20 / 4452,000.00) * 100
Percentage loss = 99.5%

So the percentage profit on the entire sales was -99.5%, which means there was a loss of 99.5%.

The second, fourth and sixth terms of an Arithmetic Progression (AP.) are x - 1, x + 1 and 7 respectively. Find the:

(a) common difference;

(b) first term;

(c) value of x.

 

Antwortdetails

An Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive terms is always the same. If a, b, c are three terms of an AP, then b - a = c - b. This common difference is denoted by d.

In this problem, we are given the second, fourth, and sixth terms of the AP as x - 1, x + 1, and 7 respectively. We can use these terms to find the common difference d and the first term a.

Let's call the first term a. Then, the second term a + d = x - 1, the fourth term a + 3d = x + 1, and the sixth term a + 5d = 7. We can use these equations to find d and a.

From the equation a + d = x - 1, we get d = x - 1 - a.

Substituting this in a + 3d = x + 1, we get a + 3(x - 1 - a) = x + 1
Expanding the brackets, we get a + 3x - 3 - 3a = x + 1
Combining like terms, we get -2a + 3x - 2 = x + 1
Solving for x, we get x = (2a + 2)/2

Substituting this value of x in d = x - 1 - a, we get d = (2a + 2)/2 - 1 - a = (2 - 2a)/2

Finally, substituting this value of d in a + 5d = 7, we get a + 5(2 - 2a)/2 = 7
Expanding the brackets, we get a + 5 - 5a = 7
Combining like terms, we get -4a + 5 = 7
Solving for a, we get a = 2.

So, the common difference d is 2 - 2a = 2 - 2 * 2 = -2, and the first term a is 2.

The value of x can be found from x = (2a + 2)/2 = (2 * 2 + 2)/2 = (4 + 2)/2 = 6/2 = 3.

Hence, the common difference is -2, the first term is 2, and the value of x is 3.

(a) If logo a = 1.3010 and log\(_{10}\)b - 1.4771. find the value of ab

(b)

In the diagram. O is the centre of the circle,< ACB = 39\(^o\) and < CBE = 62\(^o\). Find: (i) the interior angle AOC;

(ii) angle BAC.

10^1.3010 = a
a = 19.9526

10^1.4771 = b
b = 30.059

ab = 19.9526 * 30.059
ab = 600.001

              A
              |\
              | \
    e |  \ c
              |   \
              |    \
   -------O-----
              |    / 
              |  / d
              | /
              |/
              B

angle AOB = 180 - angle ACB
angle AOB = 180 - 39
angle AOB = 141

CD = AC = BC
CD = d
sin(62) = CD/CE
CD = CE * sin(62)

CE^2 = AC^2 + AE^2 - 2(AC)(AE)cos(39)
CE^2 = d^2 + e^2 - 2de(cos(141))

Antwortdetails

(a)

We know that log₁₀a = 1.3010. Therefore, we can use the definition of logarithms to write:

10^1.3010 = a
a = 19.9526

We also know that log₁₀b = 1.4771. Using the same process as above, we can find that:

10^1.4771 = b
b = 30.059

Finally, we can find the value of ab by multiplying these two values:

ab = 19.9526 * 30.059
ab = 600.001

Therefore, ab is approximately equal to 600 (to the nearest whole number).

(b)

              A
              |\
              | \
    e |  \ c
              |   \
              |    \
   -------O-----
              |    / 
              |  / d
              | /
              |/
              B

We can start by using the fact that angles in a triangle add up to 180 degrees. Therefore, angle AOB is:

angle AOB = 180 - angle ACB
angle AOB = 180 - 39
angle AOB = 141

Since O is the center of the circle, we know that AC and BC are radii of the circle. Therefore, they are equal in length. We can use this fact to find the length of CD, since angle CDE is a right angle:

CD = AC = BC
CD = d
sin(62) = CD/CE
CD = CE * sin(62)

Now we can use the cosine rule to find the length of CE:

CE^2 = AC^2 + AE^2 - 2(AC)(AE)cos(39)
CE^2 = d^2 + e^2 - 2de(cos(141))

We can also use the cosine rule to find the

(a) Given that sin y = \(\frac{8}{17}\) find the value of \(\frac{tan y}{1 + 2 tan y}\)

(b) An amount of N300,000.00 was shared among Otobo, Ada and Adeola. Otobo received N60,000.00, Ada received \(\frac{5}{10}\) of the remainder, while the rest went to Adeola. In what ratio was the money shared?

Antwortdetails

(a) Finding tan(y) using sin(y):

We cannot directly find the tangent (tan) of an angle using only the sine (sin). However, we can use the trigonometric identity:

tan^2(x) + 1 = sec^2(x)

where x is the angle and sec(x) is the secant (1/cos(x)).

Here's how to solve for tan(y):

1. We are given sin(y) = 8/17. We cannot find cos(y) directly from this information.
2. Since we cannot isolate tan(y) from the given identity, we cannot directly calculate its value.

Therefore, it's impossible to determine the value of

tan(y) + 2tan(y)/(1 + 2tan(y)) with the given information.

(b) Sharing Money:

Step 1: Calculate the remaining amount after Otobo's share.

Total money - Otobo's share

= N300,000.00 - N60,000.00 = N240,000.00

Step 2: Find Ada's share.

Ada receives 5/10 (which is the same as 1/2) of the remaining amount.

Ada's share = N240,000.00 * (1/2) = N120,000.00

Step 3: Calculate Adeola's share.

The remaining amount goes to Adeola.

Adeola's share = Total remaining - Ada's share

= N240,000.00 - N120,000.00 = N120,000.00

Step 4: Determine the ratio of shares.

Otobo : Ada : Adeola

= N60,000.00 : N120,000.00 : N120,000.00

Ratio: 3 : 6 : 6 (simplifying, 1 : 2 : 2)

Therefore, the money was shared in a ratio of 1 : 2 : 2 among Otobo, Ada, and Adeola respectively.

The data shows the marks obtained by students in a biology test

(a) Construct a frequency distribution table  using the class interval 0 - 9, 10 - 19, 19, 20, 29... 

(b) Draw a cumulative frequency curve for the distribution 

(c) Use the graph to estimate the;

(i) median

(ii) Percentage of students who scored at least 66 marks, correct to the nearest whole number.

Antwortdetails

a)

b)

(c)

(i) Mean mark

= 49.9

(ii) Percentage of at least 66 mark

= 1350×100 $\frac{13}{50}\times 100$%

= 26%

A ladder 11m long leans against a vertical wall at an angle of 75\(^o\) to the ground. The ladder is the pushed 0.2 m up the wall.

(a) Illustrate the information in a diagram.

(b) Find correct to the nearest whole number, the:

(i) new angle which the ladder makes with the ground:

(ii) distance the foot of the ladder has moved from its original position. 

        |
        |
        |\
        | \
        |  \  11m
        |   \
        |    \
        |     \
        |      \
        |       \
        |        \
        |         \
        |          \
        |           \
        |____________\
              wall

Antwortdetails

(a) Here is a diagram to illustrate the situation:

        |
        |
        |\
        | \
        |  \  11m
        |   \
        |    \
        |     \
        |      \
        |       \
        |        \
        |         \
        |          \
        |           \
        |____________\
              wall

The ladder is represented by the diagonal line, the wall by the vertical line, and the ground by the horizontal line. The angle between the ladder and the ground is labeled as 75 degrees.

(b)

1. When the ladder is pushed up the wall, the distance between the foot of the ladder and the wall decreases. This means that the angle between the ladder and the ground will also decrease. To find the new angle, we need to use trigonometry.

   Let x be the distance that the foot of the ladder moves up the wall. Then, the length of the ladder that is still on the ground is 11 - x. Using trigonometry, we can set up the following equation:

   cos(new angle) = adjacent / hypotenuse
   cos(new angle) = (11 - x) / 11
       

   We know that the original angle was 75 degrees, so we can substitute that in and solve for the new angle:

   cos(new angle) = (11 - x) / 11
   cos(75) = (11 - 0.2) / 11
   0.258819 = 10.8 / (11 - x)
   2.827 = 11 - x
   x = 8.173
       

   Therefore, the new angle which the ladder makes with the ground is approximately 50 degrees (to the nearest whole number).

2. To find the distance the foot of the ladder has moved from its original position, we can simply subtract the new distance from the old distance:

   distance moved = 0.2 m - 8.173 m
   distance moved = -7.973 m
       

   However, this answer doesn't make sense because the foot of the ladder can't move a negative distance. Instead, we need to realize that the negative sign means that the foot of the ladder moved to the left of its original position, rather than to the right. Therefore, the distance the foot of the ladder moved is approximately 8 meters to the left (to the nearest whole number).

In the diagram, PQRS is a quadrilateral, < PQR = < PRS = 90\(^o\), |PQ| =3cm, |QR| = 4cm and |PS| = 13 cm. Find the area of the quadrilateral. 

Antwortdetails

We can use the Pythagorean theorem to find the length of the diagonal PR and then use the formula for the area of a parallelogram to find the area of the quadrilateral.

The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two smaller sides is equal to the square of the length of the longest side, which is the hypotenuse. In this case, the right triangle is PQR, so we have:

PQ² + QR² = PR²

Substituting the given values, we get:

3² + 4² = PR²

9 + 16 = PR²

25 = PR²

PR = sqrt(25) = 5

Now that we have the length of the diagonal PR, we can find the area of the parallelogram using the formula:

Area = base * height

Since PR is the height of the parallelogram, we can use any of the sides of the quadrilateral as the base. Let's use PQ as the base:

Area = PQ * PR

Area = 3 * 5

Area = 15 square cm

So the area of the quadrilateral is 15 square cm.

(a) The curved surface areas of two cones are equal. The base radius of one is 5 cm and its slant height is 12cm. calculate the height of the second cone if its base radius is 6 cm.

(b) Given the matrices A = \(\begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix}\) and B = \(\begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix}\), find:

  
(i) BA; 
(ii) the determinant of BA. 

Antwortdetails

a) We can use the formula for the curved surface area of a cone, which is given by πrs, where r is the radius of the base and s is the slant height. Let's assume that the height of the second cone is h. Then we have:

π(5)(12) = π(6)(√(6^2 - h^2))

Simplifying the equation, we get:

60 = 6√(36 - h^2)

10 = √(36 - h^2)

Squaring both sides, we get:

100 = 36 - h^2

h^2 = 64

h = 8

Therefore, the height of the second cone is 8 cm.

b) (i) To find BA, we need to multiply the matrices in the order B x A. The product is given by:

BA = ∡(3 -2) (4 1) x (2 5) (-1 -3) = ∡(8 21) (7 17)

(ii) The determinant of a product of two matrices is equal to the product of their determinants, i.e., |AB| = |A||B|. Therefore, to find the determinant of BA, we can multiply the determinants of B and A. The determinant of a 2x2 matrix is given by the formula ad - bc, where a, b, c, and d are the elements of the matrix. Using this formula, we get:

|BA| = (8)(17) - (21)(7) = -49

Therefore, the determinant of BA is -49.

(a) Given that 110\(_x\) - 40\(_{five}\). find the value of x

(b) Simplify \(\frac{15}{\sqrt{75}} + \(\sqrt{108}\) + \(\sqrt{432}\), leaving the answer in the form a\(\sqrt{b}\), where a and b are positive integers.
 

Antwortdetails

a) To find the value of x, we need to convert the numbers in the expression from different number systems to a common number system, in this case, base 10.

110_x is a number in base x, so to convert it to base 10, we need to evaluate the expression:

110_x = 1 * x² + 1 * x + 0 = x² + x

40_five is a number in base 5, so to convert it to base 10, we need to evaluate the expression:

40_five = 4 * 5¹ + 0 * 5⁰ = 20

So, the expression 110_x - 40_five becomes:

x² + x - 20 = 0

We can solve for x by using the quadratic formula or by factoring the equation.

Using the quadratic formula, we get:

x = (-1 ± √(1² - 4 * 1 * -20)) / (2 * 1)

x = (-1 ± √(1 + 80)) / 2

x = (-1 ± 9) / 2

So, the two solutions are:

x = (8 - 9) / 2 = -0.5

x = (8 + 9) / 2 = 8.5

Since x must be a positive integer, the only possible solution is x = 8.

b) To simplify the expression, we need to simplify each term inside the parenthesis separately.

First, we simplify the fraction:

15 / √75 = 15 / √(3² * 5²) = 15 / (3 * 5) = 15 / 15 = 1

Next, we simplify the square root terms:

√108 = √(2² * 3³) = √(2² * 3² * 3) = 3 * 3 = 9

√432 = √(2⁴ * 3³) = √(2² * 2² * 3³) = 6 * 3 = 18

So, the expression becomes:

1 + 9 + 18 = 28

Therefore, the simplified expression is 28, which is not in the form a√b where a and b are positive integers.

(a) Fred bought a car for $5,600.00 and later sold it at 90% of the cost price. He spent $1,310.00 out of the amount received and invested the rest at 6% per annum simple interest. Calculate the interest earned in 3 years.

(b) Solve the equations 2\(^x\)(4\(^{-7}\)) = 2 and 3\(^{-x}\)(9\(^{2y}\)) = 3 simultaneously.
 

Antwortdetails

a) First, let's calculate the amount Fred received after selling the car. He sold the car for 90% of its cost price, so he received $5,600 * 0.9 = $5,040.00. After spending $1,310.00, he was left with $5,040 - $1,310 = $3,730.00. This is the amount he invested at 6% per annum simple interest.

The formula for simple interest is I = P * R * T, where I is the interest, P is the principal (the amount invested), R is the rate of interest (as a decimal), and T is the time (in years). Plugging in the values, we have I = $3,730 * 0.06 * 3 = $676.20.

So Fred earned $676.20 in interest over a period of 3 years.

b) To solve the equations 2^x(4^-7) = 2 and 3^-x(9^2y) = 3 simultaneously, we need to find the values of x and y that make both equations true.

Starting with the first equation, we can simplify the left-hand side to get 2^x * 4^-7 = 2^x * (1/16) = 2. We can then divide both sides of the equation by 2^x to get (1/16) = 1. Since this is not true, there are no values of x that make this equation true.

Next, let's simplify the second equation to get 3^-x * 9^2y = 3^-x * 81^y = 3. Dividing both sides of the equation by 3^-x gives us 81^y = 3^x. Taking the log base 3 of both sides gives us y = x.

So the only solution for x and y is x = y = 1.

In conclusion, there is only one solution for the system of equations, x = y = 1.

(a) Copy and complete the table of values for the relation y = 4x\(^2\) - 8x - 21, for -2.0 \(\leq\) x \(\leq\) 4.0 

(b) Using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph for the relation y = 4x\(^2\) - 8x - 21

(ii) Use the graph to find the solution set of 

(\(\alpha\)) 4x\(^2\) - 8x = 3;

(\(\beta\)) 4x\(^2\) - 7x - 21 = 0