WAEC SSCE - General Mathematics - 2013

If sin x = \(\frac{5}{13}\) and 0^o \(\leq\) x \(\leq\) 90^o, find the value of (cos x - tan x)

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We know that sin x = \(\frac{5}{13}\) and 0^o \(\leq\) x \(\leq\) 90^o. First, we can find the value of cos x using the identity: sin² x + cos² x = 1 sin² x + cos² x = 1 \(\frac{25}{169}\) + cos² x = 1 cos² x = \(\frac{144}{169}\) cos x = \(\pm\)\(\frac{12}{13}\) Since 0^o \(\leq\) x \(\leq\) 90^o, we know that cos x is positive. Therefore, cos x = \(\frac{12}{13}\). Next, we can find the value of tan x using the identity: tan x = \(\frac{sin x}{cos x}\) tan x = \(\frac{sin x}{cos x}\) = \(\frac{\frac{5}{13}}{\frac{12}{13}}\) = \(\frac{5}{12}\) Finally, we can find the value of (cos x - tan x) as: cos x - tan x = \(\frac{12}{13}\) - \(\frac{5}{12}\) = \(\frac{79}{156}\) Therefore, the answer is (cos x - tan x) = \(\frac{79}{156}\).

In the diagram, ST//PQ reflex angle SRQ = 198^o and < RQp = 72^o. Find the value of y

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In the diagrams, |XZ| = |MN|, |ZY| = |MO| and |XY| = |NO|. Which of the following statements is true?

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The pie chart shows the distribution of 600 mathematics textbooks for Arts, Business, Science and Technical Classes. How many textbooks are for the technical class?

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If y varies directly s the square root of (x + 1) and y = 6 when x = 3, find x when y = 9

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We are given that y varies directly with the square root of (x+1), which can be written as y=k√(x+1), where k is the constant of variation. To find k, we use the given values of x and y: y=k√(x+1) 6=k√(3+1) 6=k√4 6=2k k=3 So the equation for y in terms of x is y=3√(x+1). To find x when y=9, we substitute these values into the equation and solve for x: 9=3√(x+1) 3=√(x+1) 9=x+1 x=8 Therefore, x = 8 when y = 9.

If 9^{(2 - x)} = 3, find x

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In the diagram, PRST is a square. If |PQ| = 24cm. |QR| = 10cm and < PQR = 90^o, find the perimeter of the polygon PQRST.

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A cube and cuboid have the same base area. The volume of the cube is 64cm\(^3\) while that of the cuboid is 80cm\(^3\). Find the height of the cuboid

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Let the base area of the cube be x, then the length of one side of the cube is \(\sqrt[3]{64}\) = 4 cm. Since the base area of the cube and cuboid are equal, the base of the cuboid must also have an area of x. The volume of the cuboid is given as 80cm\(^3\) which can be expressed as: 80 = x × h, where h is the height of the cuboid We know that the length of the base of the cuboid is equal to the length of the side of the cube. Therefore, the dimensions of the cuboid are 4 cm by 4 cm by h cm. Using the formula for the volume of a cuboid, we get: Volume of cuboid = length × width × height = 4 × 4 × h = 16h Substituting 80 for the volume of the cuboid, we get: 16h = 80 Solving for h, we get: h = 5cm Therefore, the height of the cuboid is 5cm. Hence, the correct answer is 5cm.

If 2 log x (3\(\frac{3}{8}\)) = 6, find the value of x

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In the diagram, PQ is a straight line. Calculate the value of the angle labelled 2y

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An open cone with base radius 28cm and perpendicular height 96cm was stretched to form sector of a circle. calculate the arc of the sector (Take \(\pi = \frac{22}{7}\))

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If p = (y : 2y \(\geq\) 6) and Q = (y : y -3 \(\geq\) 4), where y is an integer, find p\(\cap\)Q

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The set p represents all values of y such that 2y is greater than or equal to 6. Simplifying the inequality 2y \(\geq\) 6 gives y \(\geq\) 3. Therefore, p can be written as p = {y: y \(\geq\) 3}. Similarly, the set Q represents all values of y such that y - 3 is greater than or equal to 4. Simplifying the inequality y - 3 \(\geq\) 4 gives y \(\geq\) 7. Therefore, Q can be written as Q = {y: y \(\geq\) 7}. The intersection of p and Q, denoted by p\(\cap\)Q, is the set of all values of y that are in both p and Q. Since p contains all values of y greater than or equal to 3 and Q contains all values of y greater than or equal to 7, the intersection of p and Q is {y: y \(\geq\) 7}. Therefore, p\(\cap\)Q = {7, 8, 9, 10, ...}. The correct option is (c) {3, 4, 5, 6, 7}.

If \(\frac{1}{2}\)x + 2y = 3 and \(\frac{3}{2}\)x and \(\frac{3}{2}\)x - 2y = 1, find (x + y)

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Given the equations: \begin{align*} \frac{1}{2}x + 2y &= 3 \\ \frac{3}{2}x - 2y &= 1 \\ \end{align*} We can solve for x and y using simultaneous equations: First, multiply the first equation by 2: \begin{align*} x + 4y &= 6 \\ \frac{3}{2}x - 2y &= 1 \\ \end{align*} Then, multiply the second equation by 2: \begin{align*} x + 4y &= 6 \\ 3x - 4y &= 2 \\ \end{align*} Add the equations together: \begin{align*} 4x &= 8 \\ x &= 2 \\ \end{align*} Substitute x = 2 back into the first equation: \begin{align*} \frac{1}{2}(2) + 2y &= 3 \\ 1 + 2y &= 3 \\ 2y &= 2 \\ y &= 1 \\ \end{align*} Finally, substitute x = 2 and y = 1 into (x + y): \begin{align*} x + y &= 2 + 1 \\ &= 3 \\ \end{align*} Therefore, (x + y) = 3, and the answer is.

In the diagram, PQRS is a rhombus and < PSQ = 35^o. Calculate the size of < PRO

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Using the histogram, estimate the mode of distribution

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In a histogram, the mode is the value with the highest frequency, or the tallest bar. Looking at the given histogram, it appears that the tallest bar is the one corresponding to the interval between 52 and 54, which means that the mode lies somewhere in that interval. Since the interval width is 2, we can estimate the mode to be around the middle of the interval, which is (52 + 54)/2 = 53. Therefore, the estimated mode of the distribution is 53.5. So, the answer is (c) 53.5.

Using the histogram, what is the median class?

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To find the median class in a histogram, we need to identify the class interval that contains the median value. The median is the middle value in a dataset, so we need to find the midpoint of the data. To find the median class: 1. Add up the frequencies in the histogram starting from the left-hand side until the total is greater than or equal to the total number of data points divided by 2. 2. The median class is the class interval that contains the midpoint of the data. In this histogram, the total number of data points is 80. The midpoint is (80 + 1)/2 = 40.5. Starting from the left-hand side, we can add up the frequencies: 8 + 19 + 24 = 51. The median falls within the interval 40.5 - 50.5, which is the third interval. Therefore, the median class is 40.5 - 50.5.

Using the venn diagram, find n(x \(\cap\) y¹)

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The intersection of two sets x and y1 is represented by the overlapping region of the two circles. From the diagram, we can see that the number of elements in the intersection is 2. Therefore, n(x \(\cap\) y1) = 2.

in the diagram, the height of a flagpole |TF| and the length of its shadow |FL| re in the ratio 6:8. Using k as a constant of proportionality, find the shortest distance between T and L

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A chord is 2cm from the centre of a circle. If the radius of the circle is 5cm, find the length of the chord

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To solve this problem, we can use the Pythagorean theorem to find the length of the chord. We know that the radius of the circle is 5cm and the distance from the centre of the circle to the chord is 2cm. We can draw a perpendicular line from the centre of the circle to the chord, which will divide the chord into two equal parts. This perpendicular line will also bisect the chord and form a right triangle with the radius of the circle and half of the chord. The hypotenuse of this triangle is the radius of the circle, which is 5cm, and one leg is half of the chord, which we can call x. The other leg is the distance from the centre of the circle to the chord, which is 2cm. Using the Pythagorean theorem, we can solve for x: 5^2 = x^2 + 2^2 25 = x^2 + 4 x^2 = 21 x = √21 Therefore, the length of the chord is twice the value of x, which is 2√21cm. Hence, the correct answer is option A.

The bearing of Y from X is 060^o and the bearing of Z from Y = 060^o. Find the bearing of X from Z

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The slant height of a cone is 5cm and the radius of its base is 3cm. Find, correct to the nearest whole number, the volume of the cone. ( Take \(\pi = \frac{22}{7}\))

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The distance between two towns is 50km. It is represented on a map by 5cm. Find the scale used

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In the diagram, O is the centre of the circle. OM||XZ and < ZOM = 25^o

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An interior angle of a regular polygon is 5 times each exterior angle. How many sides has the polygon?

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In a regular polygon, each exterior angle is equal to 360 degrees divided by the number of sides. Let the number of sides be represented by n. Therefore, the measure of each exterior angle is 360/n. Since the interior angle and exterior angle are supplementary, we can write: Interior angle + Exterior angle = 180 degrees Let x be the measure of each interior angle. We are given that each interior angle is 5 times each exterior angle, so: x = 5(360/n - x) Simplifying and solving for x, we get: x = 150 degrees The sum of the interior angles of an n-sided polygon is (n - 2) × 180 degrees. Since each interior angle in our polygon is 150 degrees, we can write: n × 150 = (n - 2) × 180 Simplifying and solving for n, we get: n = 12 Therefore, the polygon has 12 sides. So, the correct answer is 12.

The graph of the relation y = x2 + 2x + k passes through the point (2, 0). Find the values of k

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We are given the equation y = x2 + 2x + k and we know that it passes through the point (2,0). We can substitute x = 2 and y = 0 into the equation to find k. Substituting, we get: 0 = 22 + 2(2) + k 0 = 4 + 4 + k 0 = 8 + k k = -8 Therefore, the value of k is -8, which is the fourth option.

Find the values of k in the equation 6k² = 5k + 6

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What is the locus of the point X which moves relative to two fixed points P and M on a plane such that < PXM = 30^o

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Simplify: \(\frac{x^2 - y^2}{(x + y)^2} + \frac{(x - y)^2}{(3x + 3y)}\)

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If Un = n(n² + 1), evaluate U5 - U4

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Express \(\frac{2}{x + 3} - \frac{1}{x - 2}\) as a simple fraction

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To add fractions, we need to have a common denominator. In this case, the common denominator is \((x+3)(x-2)\). Therefore, we need to convert each fraction to have this denominator. \[\frac{2}{x+3} - \frac{1}{x-2} = \frac{2(x-2)}{(x+3)(x-2)} - \frac{(x+3)}{(x+3)(x-2)}\] Simplifying the above expression, we have: \[\frac{2(x-2) - (x+3)}{(x+3)(x-2)} = \frac{2x - 4 - x - 3}{(x+3)(x-2)} = \frac{x-7}{(x+3)(x-2)}\] Therefore, \(\frac{2}{x + 3} - \frac{1}{x - 2} = \boxed{\frac{x-7}{(x+3)(x-2)}}\). The correct option is (a).

If \(\sqrt{50} - K\sqrt{8} = \frac{2}{\sqrt{2}}\), find K

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We can start by simplifying the left-hand side of the equation using the laws of square roots: \begin{align*} \sqrt{50} - K\sqrt{8} &= \sqrt{25\cdot 2} - K\sqrt{4\cdot 2} \\ &= 5\sqrt{2} - 2K\sqrt{2} \\ &= \sqrt{2}(5 - 2K). \end{align*} Now, we can rewrite the given equation as: \begin{align*} \sqrt{2}(5 - 2K) &= \frac{2}{\sqrt{2}} \\ 5 - 2K &= \frac{2}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \\ 5 - 2K &= \frac{2}{2} \\ 5 - 2K &= 1 \\ -2K &= -4 \\ K &= 2. \end{align*} Therefore, the value of K that satisfies the equation is 2.

Solve the inequality: \(\frac{2x - 5}{2} < (2 - x)\)

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A pyramid has a rectangular base with dimensions 12m by 8m. If its height is 14m, calculate the volume

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The formula for the volume of a pyramid is given by V = 1/3 * B * h, where B is the area of the base and h is the height of the pyramid. In this case, the base of the pyramid is a rectangle with dimensions 12m by 8m, so its area is A = 12m * 8m = 96m². The height of the pyramid is given as 14m. Substituting these values into the formula, we have V = 1/3 * 96m² * 14m = 448m³. Therefore, the volume of the pyramid is 448m³.

Simplify \(\frac{1\frac{7}{8} \times 2\frac{2}{5}}{6\frac{3}{4} \div \frac{3}{4}}\)

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Given that (x + 2)(x² - 3x + 2) + 2(x + 2)(x - 1) = (x + 2) M, find M

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We can begin by factoring out (x + 2) from both terms on the left side of the equation: (x + 2)(x² - 3x + 2) + 2(x + 2)(x - 1) = (x + 2) M (x + 2)[(x² - 3x + 2) + 2(x - 1)] = (x + 2) M Simplifying the expression in the brackets, we get: (x + 2)(x² - x) = (x + 2) M Now, we can cancel out (x + 2) from both sides of the equation: x² - x = M Therefore, the value of M is simply x² - x.

Which of the following is not a probability of Mary scoring 85% in a mathematics test?

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The probability of an event happening can never be greater than 1. Therefore, 1.01 cannot be a probability of Mary scoring 85% in a mathematics test.

In the diagram, PQRST is a regular polygon with sides QR and TS produced to meet at V. Find the size of < RVS

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Multiply 2.7 x 10^-4 by 6.3 x 10⁶ and leave your answers in standard form

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To multiply two numbers in scientific notation, we simply multiply their coefficients and add their exponents. (2.7 x 10^-4) x (6.3 x 10⁶) = (2.7 x 6.3) x 10^-4+6 = 17.01 x 10² We can express 17.01 x 10² in standard form by moving the decimal point to the left two places, which gives us: 1.701 x 10³ Therefore, the correct answer is (c) 1.701 x 10³.

Find the value of m in the diagram

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Given that P = x² + 4x - 2, Q = 2x - 1 and Q - p = 2, find x

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If x = 64 and y = 27, evaluate: \(\frac{x^{\frac{1}{2}} - y^{\frac{1}{3}}}{y - x^{\frac{2}{3}}}\)

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We can start by substituting the given values: \begin{align*} \frac{x^{\frac{1}{2}} - y^{\frac{1}{3}}}{y - x^{\frac{2}{3}}} &= \frac{64^{\frac{1}{2}} - 27^{\frac{1}{3}}}{27 - 64^{\frac{2}{3}}} \\ &= \frac{8 - 3}{27 - 16} \\ &= \frac{5}{11} \end{align*} Therefore, the answer is $\frac{5}{11}$.

When a number is subtracted from 2, the result equals 4 less than one-fifth of the number. Find the number

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Let's call the number we are looking for "x". According to the problem, when we subtract x from 2, the result is equal to 4 less than one-fifth of the number. In mathematical terms, this can be written as: 2 - x = (1/5)x - 4 To solve for x, we can start by simplifying the equation by adding x to both sides: 2 = (6/5)x - 4 Next, we can add 4 to both sides: 6 = (6/5)x Finally, we can multiply both sides by 5/6 to isolate x: x = 5 Therefore, the answer is 5.

The pie chart shows the distribution of 600 mathematics textbooks for Arts, Business, Science and Technical Classes. What percentage of the total number of textbooks belongs to science?

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Given that p^{\(\frac{1}{3}\)} = \(\frac{3\sqrt{q}}{r}\), make q the subject of the equation

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We want to solve for q, so we need to isolate it on one side of the equation. First, we can isolate the cube root of p by cubing both sides of the equation: p^(1/3) = (3√q)/r (p^(1/3))³ = (3√q)³/r³ p = (27q)/r³ Next, we can isolate q by multiplying both sides by r³/27: (p/27)r³ = q Therefore, the solution is q = (p/27)r³, which is equivalent to q = pr^(1/3).

An object is 6m away from the base of a mast. The angle of depression of the object from the top pf the mast is 50^o, Find, correct to 2 decimal places, the height of the mast

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In what number base is the addition 465 + 24 + 225 = 1050?

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To solve this problem, we need to find out in what base the addition statement is true. Let's assume the base is "b". Then, in the units column, we have: - 5 + 4 = 10, so we write down 0 and carry-over 1. - In the "b" column, we have: 6 + 2 + 2 + 1 = 11, so we write down 1 and carry-over 1. - In the "b^2" column, we have: 4 + 2 + 5 + 1 = 12, so we write down 2 and carry-over 1. - In the "b^3" column, we have: 4 + 2 + 2 = 8. Putting these digits together, we get the number 8201 in base "b". Now, we need to check if this number is equal to 1050 in base 10. 8201 in base "b" means: 8 x b^3 + 2 x b^2 + 0 x b + 1 x 1 = 1050 Rearranging the terms, we get: 8 b^3 + 2 b^2 + 1 = 1050 Subtracting 1 from both sides: 8 b^3 + 2 b^2 = 1049 Since b is a positive integer, we can see that b must be greater than 5. Trying b = 6, we get: 8 x 6^3 + 2 x 6^2 = 1048 This is not equal to 1049, so we need to try a larger base. Trying b = 7, we get: 8 x 7^3 + 2 x 7^2 = 1049 This is equal to 1049, so the base is 7. Therefore, the answer is seven.

A sales boy gave a change of N68 instead of N72. Calculate his percentage error

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The sales boy gave a change of N68 instead of N72, which means he gave N4 less than he should have. To find the percentage error, we use the formula: Percentage Error = (Error / True Value) × 100% In this case, the True Value is N72 and the Error is N4. Substituting these values into the formula, we get: Percentage Error = (4 / 72) × 100% Percentage Error = 0.055555... × 100% Percentage Error = 5.5555...% Rounding off to the nearest whole number, we get: Percentage Error ≈ 6% Therefore, the sales boy's percentage error is approximately 6%. The closest option to this answer is 5\(\frac{5}{9}\)%, so the answer would be: - 5\(\frac{5}{9}\)%

Four oranges sell for Nx and three mangoes sell for Ny. Olu bought 24 oranges and 12 mangoes. How much did he pay in terms of x and y?

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Four oranges sell for Nx, so one orange costs \(\frac{N}{4}x\). Three mangoes sell for Ny, so one mango costs \(\frac{N}{3}y\). Olu bought 24 oranges and 12 mangoes, so he paid: \[24 \cdot \frac{N}{4}x + 12 \cdot \frac{N}{3}y = 6Nx + 4Ny\] Therefore, Olu paid N(6x + 4y) in terms of x and y. The answer is option (B).

(a) Copy and complete the table of values for the relation \(y = 3x^{2} - 5x - 7\).

(b) Using scales of 2 cm to 1 unit on the x- axis and 2 cm to 5 units on the y- axis, draw the graph of \(y = 3x^{2} - 5x - 7, -3 \leq x \leq 4\).

(c) From the graph : (i) find the roots of the equation \(3x^{2} - 5x - 7 = 0\) ; (ii) estimate the minimum value of y ; (iii) calculate the gradient of the curve at the point x = 2.

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A boy 1.2m tall, stands 6m away from the foot of a vertical lamp pole 4.2m long. If the lamp is at the tip of the pole,

(a) represent this information in a diagram ;

(b) calculate the (i) length of the shadow of the boy cast by the lamp ; (ii) angle of elevation of the lamp from the boy, correct to the nearest degree.

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The frequency distribution table shows the marks obtained by 100 students in a Mathematics test.

(a) Draw the cumulative curve for the distribution.

(b) Use the graph to find the : (i) 60th percentile ; (ii) probability that a student passed the test if the pass mark was fixed at 35%.

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The area of a circle is \(154cm^{2}\). It is divided into three sectors such that two of the sectors are equal in size and the third sector is three times the size of the other two put together. Calculate the perimeter of the third sector. [Take \(\pi = \frac{22}{7}\)].

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(a) Two positive whole numbers p and q are such that p is greater than q and their sum is equal to three times their difference;

(i) Express p in terms of q ; (ii) Hence, evaluate \(\frac{p^{2} + q^{2}}{pq}\).

(b) A man sold 100 articles at 25 for N66.00 and made a gain of 32%. Calculate his gain or loss percent if he sold them at 20 for N50.00.

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(a) If (3 - x), 6, (7 - 5x) are consecutive terms of a geometric progression (GP) with constant ratio r > 0, find the :

(i) values of x ; (ii) constant ratio.

(b) In the diagram, |AB| = 3 cm, |BC| = 4 cm, |CD| = 6 cm and |DA| = 7 cm. Calculate <ADC, correct to the nearest degree.

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(a) Simplify, without using tables or calculator : \(\frac{\frac{3}{4}(3\frac{3}{8} + 1\frac{5}{8})}{2\frac{1}{8} - 1\frac{1}{2}}\).

(b) Given that \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\), evaluate, correct to 2 significant figures and without using tables or calculator, \(\log_{10} 1.125\). 

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When one end of a ladder, LM, is placed against a vertical wall at a point 5 metres above the ground, the ladder makes an angle of 37° with the horizontal ground.

(a) Represent this information in a diagram ;

(b) Calculate, correct to 3 significant figures, the length of the ladder ;

(c) If the foot of the ladder is pushed towards the wall by 2 metres, calculate,correct to the nearest degree, the angle which the ladder nows makes with the ground.

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(a) The present ages of a father and his son are in the ratio 10 : 3. If the son is 15 years old now, in how many years will the ratio of their ages be 2 : 1?

(b) The arithmetic mean of x, y and z is 6 while that of x, y, z, l, u, v and w is 9. Calculate the arithmetic mean of l, u, v and w.

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A segment of a circle is cut off from a rectangular board as shown in the diagram. If the radius of the circle is \(1\frac{1}{2}\) times the length of the chord; calculate, correct to 2 decimal places, the perimeter of the remaining portion. [Take \(\pi = \frac{22}{7}\)]

(b) Evaluate without using calculators or tables, \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6})\).

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(a) Using ruler and a pair of compasses only, construct : (i) a trapezium WXYZ such that |WX| = 10.2 cm, |XY| = 5.6 cm, |YZ| = 5.8 cm, < WXY = 60° and WX is parallel to YZ  (ii) a perpendicular from Z to meet \(\overline{WX}\) at N.

(b) Measure : (i) |WZ| ; (ii) |ZN| .

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An aeroplane flies due North from a town T on the equator at a speed of 950km per hour for 4 hours to another town P. It then flies eastwards to town Q on longitude 65°E. If the longitude of T is 15°E,

(a) represent this information in a diagram ;

(b) calculate the : (i) latitude of P, correct to the nearest degree ; (ii) distance between P and Q, correct to four significant figures. [Take \(\pi = \frac{22}{7}\); Radius of the earth = 6400km].

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(a) Solve : \(7x + 4 < \frac{1}{2}(4x + 3)\).

(b) Salem, Sunday and Shaka shared a sum of N1,100.00. For every N2.00 that Salem gets, Sunday gets 50 kobo and for every N4.00 Sunday gets, Shaka gets N2.00. Find Shaka's share.

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