WAEC SSCE - General Mathematics - 2008

In the diagram, PQ//RS, QU//PT and < PSR = 42^o. Find angle x.

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PQR is a sector of a circle centre O, radius 4cm. If PQR = 30^o, find, correct to 3 significant figures, the area of sector PQR. [Take \(\pi = \frac{22}{7}\)]

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To find the area of the sector PQR, we need to know the angle at the center of the circle that the sector subtends. Since the radius of the circle is 4 cm and PQR = 30^o, the circumference of the circle is 2πr = 8π cm. The fraction of the circle that the sector PQR subtends is 30/360, or 1/12. Therefore, the area of the sector PQR is 1/12 of the total area of the circle: Area of sector PQR = (1/12) × πr² = (1/12) × π × (4 cm)² ≈ 3.347 cm² Rounded to 3 significant figures, the area of sector PQR is approximately 3.35 cm², which corresponds to the first option: 4.19 cm².

Given the sets A = [2, 4, 6, 8] and B = [2, 3, 5, 9]. If a number is selected at random from set B, what is the probability that the number is prime?

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In the diagram, < QPR = 60^o

< PQR = 50^o

< QRS = 2x^o

< SRP = 3x^o

< UQP = y^o and RS//TU

calculate y

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Simplify \(\sqrt{50} + \frac{10}{\sqrt{2}}\)

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Every staff in an office owns either a Mercedes and/or a Toyota car. 20 own Mercedes, 15 own Toyota and 5 own both. How many staff are there in the office?

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To find the total number of staff in the office, we need to add the number of staff who own a Mercedes and the number of staff who own a Toyota, but we must subtract the number of staff who own both cars to avoid counting them twice. Using the given information, we know that there are 20 staff who own a Mercedes, 15 staff who own a Toyota, and 5 staff who own both. Therefore, the number of staff who own either a Mercedes or a Toyota is: 20 + 15 - 5 = 30 Therefore, there are 30 staff in the office who own either a Mercedes or a Toyota.

If the perimeter of \(\bigtriangleup\)PQR in thr diagram is 24cm, what is the area of \(\bigtriangleup\)PRS?

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Simplify \(\frac{\log \sqrt{8}}{\log 4 - \log 2}\)

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We can start by simplifying the numerator first: \begin{align*} \log \sqrt{8} &= \log \left(\sqrt{4}\cdot\sqrt{2}\right) \\ &= \log 4 + \log \sqrt{2} \\ &= 2 + \frac{1}{2} \log 2 \end{align*} Similarly, we can simplify the denominator: \begin{align*} \log 4 - \log 2 &= \log\frac{4}{2} \\ &= \log 2 \end{align*} Now we can substitute these simplified expressions back into the original expression: \begin{align*} \frac{\log \sqrt{8}}{\log 4 - \log 2} &= \frac{2 + \frac{1}{2} \log 2}{\log 2} \\ &= \frac{2}{\log 2} + \frac{1}{2} \\ &= \boxed{\frac{3}{2}} \end{align*} Therefore, the correct option is (c) \(\frac{3}{2}\).

A train travels 60km in M minutes. If its average speed is 400km per hour, find the value of M

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The average speed of the train is given as 400km/hr. This means that in one hour, the train travels 400km. We can convert this to km/minute by dividing by 60 (the number of minutes in an hour). Therefore, the speed of the train in km/minute is: 400km/hr ÷ 60 = 6.67km/min (rounded to two decimal places) We are given that the train travels 60km in M minutes. We can use the formula: speed = distance/time to find the value of M. Rearranging the formula gives: time = distance/speed Substituting the given values, we get: M = 60km ÷ 6.67km/min M = 9 minutes (rounded to the nearest minute) Therefore, the value of M is 9.

Find the mean deviation of these numbers 10, 12, 14, 15, 17, 19.

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To find the mean deviation of a set of numbers, we first find the mean (average) of the numbers. The mean of the numbers 10, 12, 14, 15, 17, and 19 is: (mean) = (10 + 12 + 14 + 15 + 17 + 19) / 6 = 87 / 6 = 14.5 Next, we find the deviation of each number from the mean by subtracting the mean from each number: 10 - 14.5 = -4.5 12 - 14.5 = -2.5 14 - 14.5 = -0.5 15 - 14.5 = 0.5 17 - 14.5 = 2.5 19 - 14.5 = 4.5 We take the absolute value of each deviation to ensure that they are all positive: |-4.5| = 4.5 |-2.5| = 2.5 |-0.5| = 0.5 |0.5| = 0.5 |2.5| = 2.5 |4.5| = 4.5 Then we find the mean of the absolute deviations: (mean deviation) = (4.5 + 2.5 + 0.5 + 0.5 + 2.5 + 4.5) / 6 = 2.5 Therefore, the mean deviation of the numbers is 2.5. Therefore, the correct option is (a) 2.5.

Given the sets A = [2, 4, 6, 8] and B = [2, 3, 5, 9]. If a number is picked at random from each of the two sets, what is the probability that their difference is 6 or 7?

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P naira invested for 4 years invested for 4 years at r% simple interest per annum yields 0.36 p naira interest. Find the value of r

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We can use the formula for simple interest, which is: I = P * r * t Where I is the interest earned, P is the principal (initial amount invested), r is the annual interest rate, and t is the time in years. In this problem, we know that the interest earned is 0.36P, the principal is P, and the time is 4 years. Substituting these values into the formula, we get: 0.36P = P * r * 4 Simplifying the equation, we can divide both sides by P * 4: 0.36 = r * 4 r = 0.36 / 4 r = 0.09 or 9% Therefore, the answer is option (C) 9.

If the volume of a cube is 343cm³, find the length of its side

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To find the length of a side of a cube, we need to take the cube root of its volume. In this case, the volume is given as 343cm³. Taking the cube root of 343, we get: ∛343 = 7 Therefore, the length of the side of the cube is 7cm. So the correct answer is 7cm.

In the figure shown, PQs is a straight line. What is the value of < PRQ?

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If p = \(\frac{3}{5} \sqrt{\frac{q}{r}}\), express q in terms of p and r

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Given that p = \(\frac{3}{5} \sqrt{\frac{q}{r}}\), we need to express q in terms of p and r. Let's first isolate the term containing q on one side of the equation: \begin{align*} p &= \frac{3}{5} \sqrt{\frac{q}{r}} \\ \frac{5}{3}p &= \sqrt{\frac{q}{r}} \\ \left(\frac{5}{3}p\right)^2 &= \frac{q}{r} \\ \frac{25}{9} p^2r &= q \\ \end{align*} Therefore, q is equal to \(\frac{25}{9} p^2r\). So the correct option is: \(\frac{25}{9} p^2r\)

Evaluate \(\frac{(3.2)^2 - (4.8)^2}{3.2 + 4.8}\)

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If x \(\alpha\) (45 + \(\frac{1}{2}y\)), which of the following is true>?

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In the diagram, /PQ/ = /QR/ and /PR/ = /RS/ = /SP/, calculate the side of < QRS

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If x^o is obtuse, which of the following is true?

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If tan x = 1, evaluate sin x + cos x, leaving your answer in the surd form

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If \(\tan x = 1\), then we can find the values of sin x and cos x using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(\tan x = \frac{\sin x}{\cos x} = 1\), we can say that \(\sin x = \cos x\). So we have, \[\sin^2 x + \cos^2 x = 2\cos^2 x = 1 \implies \cos x = \pm\frac{1}{\sqrt{2}}\] Using \(\sin x = \cos x\), we can say that \(\sin x = \pm\frac{1}{\sqrt{2}}\) Now, \[\sin x + \cos x = \pm\frac{1}{\sqrt{2}} \pm \frac{1}{\sqrt{2}} = \pm\sqrt{2}\] Since \(\tan x = 1\), x must be in the first quadrant, where both \(\sin x\) and \(\cos x\) are positive. Therefore, we have \[\sin x + \cos x = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}\] Hence, the answer is \(\sqrt{2}\). Therefore, the option that represents this answer is: $\sqrt{2}$.

If p = \(\frac{1}{2}\) and \(\frac{1}{p - 1} = \frac{2}{p + x}\), find the value of x

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Given that p = \(\frac{1}{2}\) and \(\frac{1}{p - 1} = \frac{2}{p + x}\), we want to find the value of x. We can start by substituting the value of p in the second equation: \begin{aligned} \frac{1}{p - 1} &= \frac{2}{p + x} \\ \\ \frac{1}{\frac{1}{2} - 1} &= \frac{2}{\frac{1}{2} + x} && \text{(Substitute } p = \frac{1}{2} \text{)}\\ \\ \frac{1}{-\frac{1}{2}} &= \frac{2}{\frac{1}{2} + x} && \text{(Simplify)}\\ \\ -2 &= \frac{2}{\frac{1}{2} + x} && \text{(Simplify)}\\ \\ -2\left(\frac{1}{2} + x\right) &= 2 && \text{(Cross-multiply)}\\ \\ -1 - 2x &= 1 && \text{(Simplify)}\\ \\ -2x &= 2 \\ \\ x &= -1 \end{aligned} Therefore, the value of x is -1.5, which corresponds to option B: -1\(\frac{1}{2}\).

In the diagram, PX is a tangent to the circle and RST is an equilateral triangle. Calculate < PTS

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In the diagram, < QPR = 90^o. If q² = 25 - r². Find the value of p

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If cos (x + 25)^o = sin 45^o, find the value of x

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We know that cos (90^o - A) = sin A. So, cos (x + 25^o) = cos (90^o - 45^o) = cos 45^o. Since cos A = cos B implies A = ±B + 360^on, we have: x + 25^o = ±45^o + 360^on x = -25^o ± 45^o + 360^on x = 20^o + 360^on or x = 70^o + 360^on Since we are given the range of x, we take x = 20^o. Therefore, the value of x is 20^o.

Simplify \(\frac{\frac{1}{x} + \frac{1}{y}}{x + y}\)

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To simplify the expression, we can start by getting a common denominator for the two fractions in the numerator. The common denominator is xy. So, we can write: \begin{align*} \frac{\frac{1}{x} + \frac{1}{y}}{x + y} &= \frac{\frac{y}{xy} + \frac{x}{xy}}{x + y} \\ &= \frac{\frac{x + y}{xy}}{x + y} \\ &= \frac{1}{xy} \end{align*} Therefore, the simplified expression is \(\frac{1}{xy}\).

An arc of a circle, radius 14cm, is 18.33cm long. Calculate to the nearest degree, the angle which the arc subtends at the centre of the circle. [T = \(\frac{22}{7}\)]

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The formula for the length of an arc is L = rθ where L is the length of the arc, r is the radius of the circle, and θ is the angle subtended by the arc at the centre of the circle in radians. To find the angle in degrees, we need to convert radians to degrees by multiplying by 180/π. In this problem, the radius of the circle is 14cm and the length of the arc is 18.33cm. Plugging these values into the formula, we get: 18.33 = 14θ Solving for θ, we get: θ = 18.33/14 θ = 1.3093 radians To convert to degrees, we multiply by 180/π: θ = 1.3093 × 180/π θ ≈ 75 degrees Therefore, the angle which the arc subtends at the centre of the circle is approximately 75 degrees. Answer: 75^o

If p-2g + 1 = g + 3p and p - 2 = 0, find g

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Substitute p = 2 in the equation p - 2g + 1 = g + 3p to get: 2 - 2g + 1 = g + 3(2) -2g = 3 - 2 -2g = 1 g = -1/2 Therefore, the answer is -1.

Solve the inequality 1 - 2x < - \(\frac{1}{3}\)

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The given inequality is: 1 - 2x < -\(\frac{1}{3}\) To solve this inequality, we need to isolate the variable, which in this case is x. First, we can simplify the inequality by adding \(\frac{1}{3}\) to both sides: 1 - 2x + \(\frac{1}{3}\) < 0 Next, we can combine like terms: \(\frac{4}{3}\) - 2x < 0 Now, we can isolate x by subtracting \(\frac{4}{3}\) from both sides: -2x < -\(\frac{4}{3}\) Finally, we can solve for x by dividing both sides by -2, remembering to flip the inequality because we are dividing by a negative number: x > \(\frac{2}{3}\) Therefore, the solution to the inequality is x > \(\frac{2}{3}\).

If 4y is 9 greater than the sum of y and 3x, by how much is greater than x?

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Let's start by translating the sentence into an equation: 4y = 9 + y + 3x We can simplify this equation by subtracting y from both sides: 3y = 9 + 3x Now, let's isolate x by subtracting 9 from both sides and then dividing by 3: x = (3y - 9)/3 = y - 3 So we see that x is 3 less than y. Therefore, the answer is (A) 3.

A trader bought 100 tubers at 5 for N350.00. She sold them in sets of 4 for N290.00. Find her gain percent.

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In the diagram, find the size of the angle marked a°

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Simplify 3\(\sqrt{27x^3y^9}\)

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In the diagram, PQRS is a rhombus. /PR/ = 10cm and /QS/ = 24cm. Calculate the perimeter of the rhombus

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What is the length of an edge of a cube whose total surface area is X cm² and whose total surface area is \(\frac{X}{2}\)cm³?

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In the figure /PX/ = /XQ/, PQ//YZ and XV//QR. What is the ratio of the area of XYZQ to te area of \(\bigtriangleup\)YZR?

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Given that x = 2 and y = -\(\frac{1}{4}\), evaluate \(\frac{x^2y - 2xy}{5}\)

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In the diagram, O is the centre of the circle and PQRS is a cyclic quadrilateral. Find the value of x.

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Factorize 5y2 + 2ay - 3a2

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To factorize 5y^2 + 2ay - 3a^2, we need to find two numbers that multiply to give -15a^2 and add up to 2a. These numbers are 5a and -3a. We can then rewrite the expression as: 5y^2 + 5ay - 3ay - 3a^2 Factorising by grouping: (5y^2 + 5ay) - (3ay + 3a^2) Factorising out the common factor in each group: 5y(y + a) - 3a(y + a) Factorising out the common factor (y + a): (5y - 3a)(y + a) Therefore, the answer is (5y - 3a)(y + a).

Simplify: \(\frac{2x^2 - 5x - 12}{4x^2 - 9}\)

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To simplify the given expression, we need to factor the numerator and the denominator and then simplify the fraction by canceling out any common factors. The numerator can be factored as follows: \begin{align*} 2x^2 - 5x - 12 &= 2x^2 - 8x + 3x - 12 \\ &= 2x(x - 4) + 3(x - 4) \\ &= (2x + 3)(x - 4) \end{align*} The denominator can be factored using the difference of squares formula as: \begin{align*} 4x^2 - 9 &= (2x)^2 - 3^2 \\ &= (2x - 3)(2x + 3) \end{align*} Now we can simplify the fraction as follows: \begin{align*} \frac{2x^2 - 5x - 12}{4x^2 - 9} &= \frac{(2x + 3)(x - 4)}{(2x - 3)(2x + 3)} \\ &= \frac{x - 4}{2x - 3} \end{align*} Therefore, the simplified form of the expression is \(\frac{x - 4}{2x - 3}\). So, option (B) is the correct answer.

Given the sets A = [2, 4, 6, 8] and B = [2, 3, 5, 9]. If a number is picked at random from each of the two sets, what is the probability that their difference is odd?

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If 2ⁿ = 128, find the value of (2^{n - 1})(5^{n - 1})

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If x% of 240 equals 12, find x

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We can start solving the problem by setting up an equation based on the information given. The problem states that "x% of 240 equals 12". In other words, we can translate this into an equation: x/100 * 240 = 12 To solve for x, we can isolate it by dividing both sides of the equation by 240/100, which simplifies to 2.4: x/100 * 240 = 12 x = 12 / 2.4 x = 5 Therefore, the value of x that satisfies the given equation is 5.

Find the quadratic equation whose roots are c and -c

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If the roots of the quadratic equation are c and -c, then the equation can be expressed in the factored form as: (x - c)(x + c) = 0 Expanding the above expression: x^2 - c^2 = 0 Therefore, the quadratic equation whose roots are c and -c is x^2 - c^2 = 0. Hence, the correct option is: x² - c² = 0.

The angles of a quadrilateral are (x + 10)^o, 2y^o, 90^o and (100 - y)^o, Find y in terms of x

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The sum of the angles of a quadrilateral is 360^o. Therefore, (x + 10)^o + 2y^o + 90^o + (100 - y)^o = 360^o. Simplifying the equation, we have: x + 10 + 2y + 90 + 100 - y = 360 x + 200 + y = 360 y = 160 - x Therefore, y is equal to 160 subtracted by x. Hence, the correct option is y = 160 - x.

XY is a chord of circle centre O and radius 7cm. The chord XY which is 8cm long subtends an angle of 120^o at the centre of the circle. Calculate the perimeter of the minor segment. [Take \(\pi = \frac{22}{7}\)]

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The table below shows how a man spends his income in a month.

(a) Represent the information on a pie chart.

(b) What percentage of his income is spent on transportation?

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(a) 

In the diagram, AB // CD and BC // FE. \(\stackrel\frown{CDE} = 75°\) and \(\stackrel\frown{DEF} = 26°\). Find the angles marked x and y.

(b) 

The diagram shows a circle ABCD with centre O and radius 7 cm. The reflex angle AOC = 190° and < DAO = 35°. Find :

(i) < ABC ; (ii) < ADC.

(c) Using the diagram in (b) above, calculate, correct to 3 significant figures, the length of : (i) arc ABC ; (ii) the chord AD. [Take \(\pi = 3.142\)].

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(a) If 3, x, y, 18 are the terms of an Arithmetic Progression (A.P), find the values of x and y.

(b)(i) The sum of the second and third terms of a grometric progression is six times the fourth term. Find the two possible values of the common ratio.

(ii) If the second term is 8 and the common ratio is positive, find the first six terms.

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(a) If \(2^{x + y} = 16\) and \(4^{x - y} = \frac{1}{32}\), find the value of x and y.

(b) P, Q and R are related in such a way that \(P \propto \frac{Q^{2}}{R}\). When P = 36, Q = 3 and R = 4. Calculate Q when P = 200 and R = 2.

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The ages, in years, of 50 teachers in a school are given below :

21 37 49 27 49 42 26 33 46 40 50 29 23 24 29 31 36 22 27 38 30 26 42 39 34 23 21 32 41 46 46 31 33 29 28 43 47 40 34 44 26 38 34 49 45 27 25 33 39 40

(a) Form a frequency distribution table of the data using the intervals : 21 - 25, 26 - 30, 31 - 35 etc.

(b) Draw the histogram of the distribution 

(c) Use your histogram to estimate the mode

(d) Calculate the mean age. 

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(a) A pentagon is such that one of its exterior sides is 60°. Two others are (90 - m)° each while the remaining angles are (30 + 2m)° each. Find the value of m.

(b) 

In the diagram, PQR is a straight line, \(\overline{QR} = \sqrt{3} cm\) and \(\overline{SQ} = 2 cm\). Calculate, correct to one decimal place, < PQS.

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(a) Copy and complete the table of values for \(y = 3\sin x + 2\cos x\) for \(0° \leq x \leq 360°\).

(b) Using a scale of 2 cm to 60° on x- axis and 2 cm to 1 unit on the y- axis, draw the graph of \(y = 3 \sin x + 2 \cos x\) for \(0° \leq x \leq 360°\).

(c) Use your graph to solve the equation : \(3 \sin x + 2 \cos x = 1.5\).

(d) Find the range of values of x for which \(3\sin x + 2\cos x < -1\).

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(a) Without using calculator or tables, find the value of \(\log 3.6\) given that \(\log 2 = 0.3010, \log 3 = 0.4771\) and \(\log 5 = 0.6990\).

(b) If all numbers in the equation \(\frac{y}{y + 101} = \frac{11}{10010}\) are in base two, solve for y.

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(a) The triangle ABC has sides AB = 17m, BC = 12m and AC = 10m. Calculate the :

(i) largest angle of the triangle ; (ii) area of the triangle.

(b) From a point T on a horizontal ground, the angle of elevation of the top R of a tower RS, 38m high is 63°. Calculate, correct to the nearest metre, the distance between T and S.

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(a) Solve, correct to two decimal places, the equation \(4x^{2} = 11x + 21\).

(b) A man invests £1500 for two years at compound interest. After one year, his money amounts to £1560. Find the :

(i) rate of interest ; (ii) interest for the second year.

(c) A car costs N300,000.00. It depreciates by 25% in the first year and 20% in the second year. Find its value after 2 years.

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(a) Solve the inequality : \(\frac{2}{5}(x - 2) - \frac{1}{6}(x + 5) \leq 0\).

(b) Given that P = \(\frac{x^{2} - y^{2}}{x^{2} + xy}\),

(i) express P in its simplest form ; (ii) find the value of P if x = -4 and y = -6.

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(a) Using ruler and a pair of compasses only, construct :

(i) a quadrilateral PQRS such that /PQ/ = 7 cm, < QPS = 60°, /PS/ = 6.5 cm, < PQR = 135° and /QS/ = /QR/ ;

(ii) locus, \(l_{1}\) of points equidistant from P and Q ;

(iii) locus, \(l_{2}\) of points equidistant from P and S.

(b)(i) Label the point T where \(l_{1}\) and \(l_{2}\) intersect. (ii) With center T and radius /TP/, construct a circle \(l_{3}\).

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(a) A rectangular field is l metres long and b metres wide. Its perimeter is 280 metres. If the length is two and a half times the breadth, find the values of l and b.

(b) The base of a pyramid is a 4.5 metres rectangle. The height of the pyramid is 4 metres. Calculate its volume.

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