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**Question 1**
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If m = 4, n = 9 and r = 16., evaluate \(\frac{m}{n}\) - 1\(\frac{7}{9}\) + \(\frac{n}{r}\)

**Question 2**
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Describe the shaded portion in the diagram

**Answer Details**

The shaded portion in the diagram represents the set of elements that belong to the complement of P and the intersection of Q and R. In other words, it represents the set of elements that are not in P, but are in both Q and R. Therefore, the correct answer is P' \(\cap\) Q \(\cap\) R'.

**Question 4**
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Calculate the mean deviation of 5, 3, 0, 7, 2, 1

**Answer Details**

To find the mean deviation, we need to first calculate the mean of the given set of numbers: Mean = (5 + 3 + 0 + 7 + 2 + 1) / 6 = 18 / 6 = 3 Next, we need to find the absolute deviation of each number from the mean: |5 - 3| = 2 |3 - 3| = 0 |0 - 3| = 3 |7 - 3| = 4 |2 - 3| = 1 |1 - 3| = 2 To calculate the mean deviation, we add up these absolute deviations and divide by the number of numbers: Mean deviation = (2 + 0 + 3 + 4 + 1 + 2) / 6 = 12 / 6 = 2 Therefore, the answer is option (B) 2.0.

**Question 5**
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The angle of elevation of an aircraft from a point K on the horizontal ground 30\(\alpha\). If the aircraft is 800m above the ground, how far is it from K?

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**Question 6**
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Adding 42 to a given positive number gives the same result as squaring the number. Find the number

**Answer Details**

Let's call the unknown number "x". According to the problem, adding 42 to x gives the same result as squaring x. We can write this as an equation: x + 42 = x^2 To solve for x, we can rearrange this equation to get it in standard quadratic form: x^2 - x - 42 = 0 Now we can factor this quadratic equation: (x - 7)(x + 6) = 0 This gives us two possible solutions: x = 7 and x = -6. However, we were given that x is a positive number, so we can discard the negative solution. Therefore, the answer is x = 7. To check our answer, we can substitute x = 7 into the original equation: 7 + 42 = 49 7^2 = 49 So adding 42 to 7 does indeed give the same result as squaring 7.

**Question 8**
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In the diagram, O is the centre. If PQ//RS and ∠ONS = 140°, find the size of ∠POM.

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**Question 9**
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A trader bought an engine for $15,000.00 outside Nigeria. If the exchange rate is $0.075 to N1.00, how much did the engine cost in Naira?

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**Question 10**
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The volume of a cone of height 3cm is 38\(\frac{1}{2}\)cm^{3}. Find the radius of its base. [Take \(\pi = \frac{22}{7}\)]

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**Question 11**
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\(\begin{array}{c|c}

Scores & 0 - 4 & 5 - 9 & 10 - 14\\

\hline Frequency & 2 & 1 & 2\end{array}\)

The table shows the distribution of the scores of some students in a test. Calculate the mean scores.

**Question 12**
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The rate of consumption of petrol by a vehicle varies directly as the square of the distance covered. If 4 litres of petrol is consumed on a distance of 15km. how far would the vehicle go on 9 litres of petrol?

**Answer Details**

Since the rate of consumption of petrol varies directly as the square of the distance covered, let's assume that the rate is k\(\frac{\text{litres}}{\text{km}^2}\). Then we can write the relationship as: rate of consumption = k(distance covered)\(^2\) We are given that when the distance covered is 15 km, the rate of consumption is 4 litres. Substituting these values in the above equation, we get: 4 = k(15)\(^2\) k = \(\frac{4}{15^2}\) Now, we need to find how far the vehicle can go on 9 litres of petrol. Let's assume that the vehicle can travel a distance of x km on 9 litres of petrol. Using the same equation as before, we get: 9 = \(\frac{4}{15^2}\)(x)\(^2\) Simplifying this equation, we get: x\(^2\) = \(\frac{9}{\frac{4}{15^2}}\) x\(^2\) = 506.25 x = 22.5 Therefore, the vehicle can travel 22.5 km on 9 litres of petrol. So the answer is 22\(\frac{1}{2}\)km.

**Question 13**
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Given that logx 64 = 3, evaluate x log\(_2\)8

**Question 14**
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The population of students in a school is 810. If this is represented on a pie chart, calculate the sectoral angle for a class of 7 students

**Answer Details**

To find the sectoral angle for a class of 7 students, we first need to calculate what percentage of the total population they represent: Percentage of class of 7 students = (Number of students in the class / Total population) x 100% Percentage of class of 7 students = (7 / 810) x 100% Percentage of class of 7 students = 0.864% (rounded to three decimal places) Next, we need to convert this percentage to an angle of the pie chart. We know that a full circle is 360 degrees, and the percentage of the class of 7 students represents a certain portion of this circle. We can use the following formula to calculate the sectoral angle: Sectoral angle = Percentage of class of 7 students x 360 degrees Plugging in the percentage we calculated earlier, we get: Sectoral angle = 0.864% x 360 degrees Sectoral angle ≈ 3.11 degrees (rounded to two decimal places) Therefore, the sectoral angle for a class of 7 students in the pie chart is approximately 3.11 degrees. So, the correct option is (a) 32^{o}.

**Question 15**
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A farmer uses \(\frac{2}{5}\) of his land to grow cassava, \(\frac{1}{3}\) of the remaining for yam and the rest for maize. Find the part of the land used for maize

**Answer Details**

Let's assume the farmer has 15 parts of land. The fraction of land used for cassava is \(\frac{2}{5}\) of 15, which is 6 parts. The remaining land is 15 - 6 = 9 parts. The fraction of the remaining land used for yam is \(\frac{1}{3}\) of 9, which is 3 parts. Therefore, the part of the land used for maize is the remaining land after cassava and yam are planted, which is 9 - 3 = 6 parts. So the fraction of the land used for maize is \(\frac{6}{15}\), which simplifies to \(\frac{2}{5}\). Therefore, the answer is option (B) \(\frac{2}{5}\).

**Question 17**
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Two angles of a pentagon are in the ratio 2:3. The others are 60^{o} each. Calculate the smaller of the two angles

**Answer Details**

Let's call the two angles that are in the ratio 2:3 as "2x" and "3x". The sum of the interior angles of a pentagon is given by the formula (5-2) x 180 degrees = 540 degrees. Since the other three angles are all 60 degrees each, we can find the sum of those angles by multiplying 60 by 3, which gives us 180 degrees. Now we can set up an equation to solve for the unknown angle, which is 2x + 3x. 2x + 3x + 180 = 540 Simplifying this equation, we get: 5x + 180 = 540 Subtracting 180 from both sides, we get: 5x = 360 Dividing both sides by 5, we get: x = 72 So the smaller angle, which is 2x, is equal to 2 times 72, which is 144 degrees. Therefore, the correct answer is 144 degrees.

**Question 18**
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PQRT is square. If x is the midpoint of PQ, Calculate correct to the nearest degree, LPXS

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**Question 19**
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The scores of twenty students in a test are as follows: 44, 47, 48, 49, 50, 51, 52, 53, 53, 54, 58, 59, 60, 61, 63, 65, 67, 70, 73, 75. Find the third quartile.

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**Question 20**
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In the diagram, VW//YZ, |WX| = 6cm, |XY| = 16cm, |YZ| = 20cm and |ZX| = 12cm. Calculate |VX|

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**Question 21**
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Which of the following is used to determine the mode of a grouped data?

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**Question 22**
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In the diagram, PTR is a tangent to the centre O. If angles TON = 108°, Calculate the size of angle PTN

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**Question 23**
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Simplify: \(\frac{3x - y}{xy} - \frac{2x + 3y}{2xy} + \frac{1}{2}\)

**Answer Details**

To simplify the expression, we first need to find a common denominator for the fractions. The lowest common multiple of the denominators xy and 2xy is 2xy. \(\frac{3x - y}{xy} - \frac{2x + 3y}{2xy} + \frac{1}{2} = \frac{3x\cdot2 - y\cdot2}{xy\cdot2} - \frac{2x\cdot1 + 3y\cdot1}{2xy\cdot1} + \frac{1\cdot xy}{2\cdot xy}\) Simplifying further, we get: \(\frac{6x - 2y}{2xy} - \frac{2x + 3y}{2xy} + \frac{xy}{2xy}\) Combining like terms, we get: \(\frac{4x - 5y + xy}{2xy}\) Therefore, the answer is option D: \(\frac{4x - 5y + xy}{2xy}\).

**Question 24**
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The sum 11011_{2}, 1111_{2} and 10_{m}10_{n}0_{2}. Find the value of m and n.

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**Question 25**
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If {X: 2 d- x d- 19; X integer} and 7 + x = 4 (mod 9), find the highest value of x

**Answer Details**

To solve this problem, we need to use the given conditions to find the possible values of x and then determine the highest value among them. From the first condition, we know that x is an integer between 2 and 19 (inclusive) and can be represented as {X: 2 d- x d- 19; X integer}. This means that x can take on any of the following values: 2, 3, 4, ..., 18, 19. The second condition tells us that 7 + x is congruent to 4 modulo 9, which can be written as: 7 + x ≡ 4 (mod 9) To solve for x, we can subtract 7 from both sides of the congruence: x ≡ -3 (mod 9) Since we want x to be between 2 and 19, we can add or subtract multiples of 9 to -3 until we get a value within the range of possible values for x. Doing so, we obtain: x ≡ -3 (mod 9) x ≡ 15 (mod 9) Since 15 is the largest value of x that satisfies both conditions, our answer is 15.

**Question 26**
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In the diagram, O is the centre, \(\bar{RT}\) is a diameter, < PQT = 33\(^o\) and < TOS = 76\(^o\). Using the diagram, find the size of angle PRS.

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**Question 27**
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The probability that kebba, Ebou and Omar will hit a target are \(\frac{2}{3}\), \(\frac{3}{4}\) and \(\frac{4}{5}\) respectively. Find the probability that only Kebba will hit the target.

**Answer Details**

The probability that only Kebba will hit the target means that Kebba hits the target and the other two miss the target. We can find this probability by multiplying the probability of Kebba hitting the target with the probability of Ebou missing the target and the probability of Omar missing the target. Probability that only Kebba hits the target = Probability of Kebba hitting the target × Probability of Ebou missing the target × Probability of Omar missing the target = \(\frac{2}{3}\) × \(\frac{1}{4}\) × \(\frac{1}{5}\) = \(\frac{2}{3\times4\times5}\) = \(\frac{1}{30}\) Therefore, the probability that only Kebba will hit the target is \(\frac{1}{30}\).

**Question 28**
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A letter is selected from the letters of the English alphabet. What is the probability that the letter selected is from the word MATHEMATICS?

**Answer Details**

The word MATHEMATICS contains the letters A, C, E, H, I, M, S, and T. Since there are 26 letters in the English alphabet, the probability of selecting any letter from the alphabet is 1 out of 26. Out of the letters in the word MATHEMATICS, there are 8 letters, so the probability of selecting a letter from the word MATHEMATICS is 8 out of 26. Therefore, the probability of selecting a letter from the word MATHEMATICS is: $$\frac{8}{26}=\frac{4}{13}$$ Therefore, the correct answer is option (C).

**Question 29**
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Tom will be 25 years old in n years' time. If he is 5 years younger than Bade's present age.

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**Question 30**
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In the diagram, \(\bar{OX}\) bisects < YXZ and \(\bar{OZ}\) bisects < YZX. If < XYZ = 68^{o}, calculate the value of < XOZ

**Question 31**
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If 2^{n} = y, Find 2\(^{(2 + \frac{n}{3})}\)

**Answer Details**

We know that 2\(^n\) = y. Therefore, 2\(^{(2 + \frac{n}{3})}\) can be written as 2\(^2\) × 2\(^{\frac{n}{3}}\) = 4 × 2\(^{\frac{n}{3}}\). Substituting 2\(^n\) = y in the above equation, we get: 2\(^{(2 + \frac{n}{3})}\) = 4y\(^\frac{1}{3}\) Therefore, the answer is (a) 4y\(^\frac{1}{3}\).

**Question 32**
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If \(\frac{27^x \times 3^{1 - x}}{9^{2x}} = 1\), find the value of x.

**Question 33**
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Make K the subject of the relation T = \(\sqrt{\frac{TK - H}{K - H}}\)

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**Question 35**
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In the given diagram, \(\bar{QT}\) and \(\bar{PR}\) are straight lines, < ROS = (3n - 20), < SOT = n, < POL = m and < QOL is a right angle. Find the value of n.

**Answer Details**

Since < QOL is a right angle, we have: < QOT + < SOT = 90^{o} Since < QOT = 180^{o} - < POL, we have: < QOT = 180^{o} - m Substituting into the first equation, we get: 180^{o} - m + n = 90^{o} n - m = 90^{o} - 180^{o} = -90^{o} Also, from the diagram, we have: < QOT + < ROS + < SOT = 180^{o} Substituting < QOT = 180^{o} - < POL and simplifying, we get: 180^{o} - m + (3n - 20) + n = 180^{o} 4n - m = 20 We now have a system of two equations: n - m = -90^{o} 4n - m = 20 Solving simultaneously, we get: n = 35^{o}

**Question 36**
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In the diagram, the shaded part is carpet laid in a room with dimensions 3.5m by 2.2m leaving a margin of 0.5m round it. Find area of the margin

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**Question 37**
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The area of a rhombus is 110 cm\(^2\). If the diagonals are 20 cm and (2x + 1) cm long, find the value of x.

**Question 38**
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If \(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{3}}\) is simplified as m + n\(\sqrt{6}\), find the value of (m + n)

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**Question 39**
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\(\begin{array}{c|c}

x & 0 & 1\frac{1}{4} & 2 & 4\\

\hline

y & 3 & 5\frac{1}{2} & &

\end{array}\)

The table given shows some values for a linear graph. Find the gradient of the line

**Answer Details**

To find the gradient of a line, we need to calculate the change in y divided by the change in x between two points on the line. Let's choose two points on the line from the given table. We can select the points (0,3) and (2,y), where y is the value of y for x = 2. The change in y is y - 3, and the change in x is 2 - 0 = 2. Therefore, the gradient of the line is: gradient = change in y / change in x = (y - 3) / 2 We don't know the value of y yet, but we can use the other point on the line to find it. Using the points (1.25, 5.5) and (2, y), we get: gradient = (y - 5.5) / (2 - 1.25) = (y - 5.5) / 0.75 Setting the two expressions for the gradient equal to each other, we get: (y - 3) / 2 = (y - 5.5) / 0.75 Solving for y gives: y = 7 Therefore, the value of y for x = 2 is 7, and the gradient of the line is: gradient = (y - 3) / 2 = (7 - 3) / 2 = 2 So the answer is 2, and the gradient of the line is 2. In simple terms, the gradient of a line tells us how steep the line is. If the gradient is positive, the line is sloping upwards from left to right. If the gradient is negative, the line is sloping downwards from left to right. The larger the absolute value of the gradient, the steeper the line.

**Question 40**
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Ada draws the graph of y = x^{2} - x - 2 and y = 2x - 1 on the same axes. Which of these equations is she solving?

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**Question 41**
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In a circle radius rcm, a chord 16\(\sqrt{3}cm\) long is 10cmfrom the centre of the circle. Find, correct to the nearest cm, the value of r

**Answer Details**

In a circle, a radius is a line segment that connects the center of the circle to any point on the circle. A chord is a line segment that connects two points on a circle. Given that a chord of length 16\(\sqrt{3}cm\) is 10cm from the center of the circle, we can use the Pythagorean theorem to find the length of the radius. First, we draw a line from the center of the circle perpendicular to the chord, creating a right triangle with the radius, the perpendicular line, and half the chord length (which is \(8\sqrt{3}\) cm). Using the Pythagorean theorem: \begin{align*} r^2 &= (8\sqrt{3})^2 + 10^2 \\ r^2 &= 192 + 100 \\ r^2 &= 292 \\ r &\approx \sqrt{292} \\ r &\approx 17 \text{ cm} \end{align*} Rounding to the nearest whole number, we get the answer of 17cm. Therefore, the correct option is (b) 17cm.

**Question 42**
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A trader bought 100 oranges at 5 for N40.00 and 20 for N120.00. Find the profit or loss percent

**Question 43**
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Find the equation whose roots are \(\frac{3}{4}\) and -4

**Answer Details**

If a quadratic equation has roots **p** and **q**, then it can be expressed in the factored form as:

(x - p)(x - q) = 0

Expanding this equation, we get:

x^{2}- (p + q)x + pq = 0

In this case, the roots are given as \(\frac{3}{4}\) and -4. Thus, the equation can be expressed as:

(x - \(\frac{3}{4}\))(x + 4) = 0

Expanding this equation, we get:

x^{2}+ (4 - \(\frac{3}{4}\))x - 3 = 0

Multiplying throughout by 4 to eliminate fractions, we get:

4x^{2}+ 13x - 12 = 0

Therefore, the equation whose roots are \(\frac{3}{4}\) and -4 is **4x ^{2} + 13x - 12 = 0**.

Hence, the answer is **4x ^{2} + 13x - 12 = 0**.

**Question 44**
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