JAMB UTME - Mathematics - 1991

Factorize m3 - 2m2 - m + 2

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m3 - 2m2 - m + 2
Let f(m) = m3 - 2m2 - m2 + 2
= f(1)
= 1 - 2 - 2 + 2 = 0
∴ m - 1 is factor $\frac{m^3-2m^2-m^2+2}{m-1}$
= m2 - m - 2
= (m - 1)m2 - m - 2
= (m - 1)(m + 1)(m - 2)

At what value of x is the function y = x${}^2$ - 2x -3 minimum?

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Find the total area of the surface of a solid cylinder whose base radius is 4cm and height is 5cm

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PMN and PQR are two secants of the circle MQTRN and PT is a tangent. If PM = 5cm, PN = 12cm and PQ = 4.8cm, calculate the respective lengths of PR and PT in centimeters

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In this problem, we are given a circle MQTRN and two secants PMN and PQR intersecting at point T and a tangent PT. First, we can use the theorem that states that the product of the lengths of the two segments of each secant is equal. So we have: PM * PN = PT * PR Substituting the given values, we get: 5cm * 12cm = PT * PR 60cm² = PT * PR Next, we need to find the respective lengths of PR and PT. To do this, 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 triangle PRT, PT is the hypotenuse, so we have: PT² = PR² + RT² We can also use the fact that RT is perpendicular to PT, which means that triangle PRT is a right triangle. To find RT, we can use the theorem that states that the product of the segments of a secant and its external part is equal. So we have: PT * TP = QT * TR Substituting the given values, we get: PT * PT = 4.8cm * TR PT² = 4.8cm * TR TR = PT² / 4.8cm Now we can substitute RT in the Pythagorean theorem and simplify: PT² = PR² + (PT² / 4.8cm)² PT⁴ / (4.8cm)² = PR² PR = sqrt(PT⁴ / (4.8cm)²) = PT² / 4.8cm Finally, we can substitute the value of PR in the first equation we obtained and solve for PT: 60cm² = PT * (PT² / 4.8cm) PT³ = 288cm³ PT = cube root of 288cm³ ≈ 6.67cm Now we can substitute the value of PT in the equation we obtained for PR: PR = PT² / 4.8cm ≈ 7.7cm Therefore, the respective lengths of PR and PT are approximately 7.7cm and 6.67cm, respectively. So the answer is 7.7, 12.5.

Simplify $\frac{\sqrt{1+x}+\sqrt{x}}{\sqrt{1+x}-\sqrt{x}}$

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Rationalize $\frac{2\sqrt{3}+3\sqrt{2}}{3\sqrt{2}-2\sqrt{3}}$

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$\frac{2\sqrt{3}+3\sqrt{2}}{3\sqrt{2}-2\sqrt{3}}$
= $\frac{2\sqrt{3}+3\sqrt{2}}{3\sqrt{2}-2\sqrt{3}}$ x $\frac{3\sqrt{2}+2\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}$
$\frac{4\left(3\right)+9\left(2\right)+2\left(6\right)\sqrt{6}}{9\left(2\right)-4\left(3\right)}$
$\frac{12+18+12\sqrt{6}}{1‘8-12}$
= $\frac{30+12\sqrt{6}}{6}$
= 5 + 2$\sqrt{6}$

If g(x) = x2 + 3x + 4, find g(x + 1) - g(x)

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The function g(x) is defined as g(x) = x^2 + 3x + 4. We are asked to find g(x + 1) - g(x). To find g(x + 1), we simply substitute (x + 1) in place of x in the function g(x): g(x + 1) = (x + 1)^2 + 3(x + 1) + 4 = x^2 + 2x + 1 + 3x + 3 + 4 = x^2 + 5x + 8 To find g(x), we substitute x in place of x in the function g(x): g(x) = x^2 + 3x + 4 Now we can find g(x + 1) - g(x) by subtracting g(x) from g(x + 1): g(x + 1) - g(x) = (x^2 + 5x + 8) - (x^2 + 3x + 4) = x^2 + 5x + 8 - x^2 - 3x - 4 = 2x + 4 Therefore, the correct option is 2(x + 2), which is equivalent to 2x + 4.

If 3 gallons of spirit containing 20% water are added to 5 gallons of another spirit containing 15% water, what percentage of the mixture is water?

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% of water in the mixture
= $\frac{\text{Total Amount of water}}{\text{Total quantity of spirit}}$ x $\frac{100}{1}$
$\frac{3\left(\frac{20}{100}\right)+5\left(\frac{15}{100}\right)}{3+5}$ x $\frac{100}{1}$
= $\frac{\frac{6}{10}+\frac{75}{100}}{8}$ x $\frac{100}{1}$
= $\frac{0.6+0.75}{8}$ x $\frac{100}{1}$

= $\frac{1.35}{8}$ x $\frac{100}{1}$
= $\frac{33.75}{2}$
= 16.875
= 16$\frac{7}{8}$

What is the product of $\frac{27}{5^1}$(3)-3 and $\frac{(1{)}^{-1}}{5}$?

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$\frac{27}{5^1}$(3)-3 x $\frac{(1{)}^{-1}}{5}$ = $\frac{27}{5}$ x $\frac{1}{3^3}$ x $\frac{1}{\frac{1}{5}}$
= $\frac{27}{5}$ x $\frac{1}{27}$ x $\frac{5}{1}$
= 1

In the diagram above, |PQ| = |QR|, |PS| = |RS|, ∠PSR = 30o and ∠PQR = 80o. Find ∠SPQ.

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oin PR
QRP = QPR
= 180 - 80 = 100/20 = 50o
SRP = SPR
= 180 - 30 = 150/2 = 75o
∴ SPQ = SPR - QPR
= 75 - 50 = 25o

From a point Z, 60 m north of X, a man walks 60√3m eastwards to another point Y. Find the bearing of Y from X.

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To find the bearing of Y from X, we need to determine the angle between the direction of due north and the direction of XY. First, we can draw a diagram to represent the problem. Let X be the origin, and let Y be the point that is 60√3 m east and some distance north of X. Let Z be the point that is 60 m due north of X, and let W be the point that is due west of Y and due south of Z. [Diagram not included as plain text] We can see that △XYZ is a right-angled triangle, with XY as the hypotenuse, XZ as the adjacent side, and YZ as the opposite side. We know that XZ has a length of 60 m, and YZ has a length of 60√3 m. Therefore, we can use the tangent ratio to find the angle between XZ and XY: tan θ = YZ/XZ tan θ = (60√3)/60 tan θ = √3/1 θ = tan⁻¹(√3) θ = 60° Therefore, the bearing of Y from X is 060°. Option (C) is the correct answer.

Obi borrows ₦10.00 at 2% per month simple interest and repays ₦8.00 after 4 months, how much does he still owe?

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Multiply (x2 - 3x + 1) by (x - a)

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To multiply the expression (x^2 - 3x + 1) by (x - a), we need to use the distributive property of multiplication over addition. We start by multiplying each term in the first expression (x^2 - 3x + 1) by (x - a) and then add the resulting terms. (x^2 - 3x + 1) * (x - a) = x^3 - ax^2 - 3x^2 + 3ax + x - a = x^3 - (a+3)x^2 + (3a+1)x - a Therefore, the correct option is: - x^3 - (a+3)x^2 + (3a+1)x - a

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To find the angles subtended at the center of the pie chart under education and food, we need to first calculate the percentage of the chart that each category occupies. The family's income is divided into five categories: electricity, food, transport, education, and extended family. The percentages of income spent on each category are given as follows: - Electricity: 3% - Food: 58% - Transport: 20% - Education: 11% - Extended family: 7% To calculate the percentage of the chart that each category occupies, we can multiply each percentage by 360 (the total number of degrees in a circle) and divide by 100. This gives us: - Electricity: (3/100) x 360 = 10.8 degrees - Food: (58/100) x 360 = 208.8 degrees - Transport: (20/100) x 360 = 72 degrees - Education: (11/100) x 360 = 39.6 degrees - Extended family: (7/100) x 360 = 25.2 degrees So, the angles subtended at the center of the pie chart under education and food are 39.6 degrees and 208.8 degrees, respectively. However, the options given in the question are in terms of rounded values, so we need to find the closest options to our calculated values. The closest option to 39.6 degrees is 39.6 degrees itself. The closest option to 208.8 degrees is 212.4 degrees. Therefore, the answer is (D) 39.6 degrees and 212.4 degrees.

Evaluate x2(x2 - 1)-$\frac{1}{2}$ - (x2 - 1)$\frac{1}{2}$

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x2(x2 - 1)-$\frac{1}{2}$ - (x2 - 1)$\frac{1}{2}$ = $\frac{x^2}{(x^2?1{)}^{\frac{1}{2}}}$ - $\frac{(x^2?1{)}^{\frac{1}{2}}}{1}$
= $\frac{x^2?(x^2?1)}{(x^2?1{)}^{\frac{1}{2}}}$
= $\frac{x^2?x^2+1}{(x^2?1{)}^{\frac{1}{2}}}$
= (x2 - 1)-$\frac{1}{2}$

Musa borrows N10.00 at 2% per month simple interest and repays N8.00 after 4 months. How much does he still owe?

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I = PRT100 $\frac{PRT}{100}$

= 10×2×4100 $\frac{10\times 2\times 4}{100}$

= 45 $\frac{4}{5}$

= 0.8

Total amount = N10.80

He pays N8.00

Remainder = 10.80 - 8.00

= N2.80

Simplify 3$\frac{1}{3}$ - 1$\frac{1}{4}$ x $\frac{2}{3}$ + 1$\frac{2}{5}$

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3 - $\frac{1}{3}$ - ($\frac{5}{4}$ x $\frac{2}{3}$) + 1$\frac{2}{5}$
= $\frac{10}{3}$ - $\frac{5}{6}$ + $\frac{7}{5}$
= $\frac{100-25+42}{30}$
= $\frac{117}{30}$
= 3.9
$\approx$ 4

Fifty boxes each of 50 bolts were inspected for the number which were defective. The following was the result:

$\begin{array}{ccccccc}\text{No. defective per box}& 4& 5& 6& 7& 8& 9\\ \text{No. of boxes}& 2& 7& 17& 10& 8& 6\end{array}$

The mean and the median of the distribution are respectively

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To find the mean of the distribution, we need to calculate the sum of all the defective bolts in the boxes and divide it by the total number of boxes: Mean = (4x2 + 5x7 + 6x17 + 7x10 + 8x8 + 9x6) / 50 = 335 / 50 = 6.7 To find the median of the distribution, we need to arrange the data in order of increasing magnitude and find the middle value. In this case, we have an even number of observations (50), so the median will be the average of the two middle values, which are the 25th and 26th observations. From the table, we can see that the 25th and 26th observations are both 6, so the median is: Median = (6 + 6) / 2 = 6 Therefore, the mean and the median of the distribution are 6.7 and 6, respectively.

Find correct to 3 decimal places ($\frac{1}{0.05}$ + $\frac{1}{5.005}$) - (0.05 x 2.05)

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($\frac{1}{0.05}$ + $\frac{1}{5.005}$) - (0.05 x 2.05)
($\frac{1}{0.05}$ x 5.005) - (0.05 x 2.05)

$\frac{500.5}{0.05}$ - 1025
100.1 - 0.1025
= 99.9975
99.998 to 3 sig. fiq.

If the exterior angles of a pentagon are xo, (x + 5)o, (x + 10), (x + 15)o and (x + 20)o, find x

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Evaluate $\frac{x{y}^2-x^{2}y}{x^2-x{y}^1}$, when x = -2 and y = 2

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$\frac{x{y}^2-x^{2}y}{x^2-x{y}^1}$
= $\frac{(-2)(3{)}^2-(-2{)}^2\left(3\right)}{(-2{)}^2-(-2\left)\right(3)}$
= -3

What is the nth term of the progression 27, 9, 3,......?

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The given progression is decreasing by a factor of 3 in each term, starting from 27. Therefore, the nth term can be calculated by multiplying 27 by 1/3 raised to the power of (n-1). This is because in the nth term, the factor of 3 has been applied (n-1) times to the first term 27. Therefore, the answer is: 27 x (1/3)^(n-1)

If 2257 is the result of subtracting 4577 from 7056 in base n, find n

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$\begin{array}{c}7056\\ \text{- 4577}\\ 2257\end{array}$
By trial and error method
Let the base to 8
i.e. Let n = 8 and it is easily verified that the subtraction holds.
The subtraction does not hold when other values of n are tried
n = 8

Find the two values of y which satisfy the simultaneous equation 3x + y = 8, x2 + xy = 6

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We can solve this system of equations by substitution or elimination method. Here, we will use substitution method. First, we solve the first equation for y in terms of x: 3x + y = 8 y = 8 - 3x Now we substitute this expression for y into the second equation and solve for x: x^2 + xy = 6 x^2 + x(8 - 3x) = 6 x^2 + 8x - 3x^2 = 6 -2x^2 + 8x - 6 = 0 x^2 - 4x + 3 = 0 (x - 3)(x - 1) = 0 x = 3 or x = 1 Now we substitute each of these values of x into the expression we found for y earlier: y = 8 - 3x When x = 3, y = -1 When x = 1, y = 5 Therefore, the two values of y that satisfy the system of equations are -1 and 5. So, the correct answer is: - -1 and 5

Factorize 1 - (a - b)2

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1 - (a - b)2 = [1 + (a - b)][1 - a + b]
= (1 + a - b)(1 - a + b)

Given that 1/2 log10 P = 1, find the value Of P

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1/2log10P
log10P1/2 = 1
P1/2 = 10
p = 102

Solve the equation (x - 2) (x - 3) = 12

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(x - 2) (x - 3) = 12
x2 - 3x - 2x + 6 = 12
x2 - 5x - 6 = 0
(x - 1)(x - 6) = 0
x = -1 or 6

If cos x = $\sqrt{\frac{a}{b}}$ find coses x

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cosx = $\sqrt{\frac{a}{b}}$
y2 + $\sqrt{(a{)}^2}$ = $\sqrt{(b{)}^2}$ by pythagoras
y2 = b - a
∴ y = b - a
cosec x = $\frac{1}{sinx}$ = $\frac{1}{y}$
$\frac{b}{y}$ = $\frac{\sqrt{b}}{\sqrt{b-a}}$
= $\sqrt{\frac{b}{b-a}}$

A crate of soft drinks contains 10 bottles of Coca-cola, 8 of Fanta and 6 of sprite. If one bottle is selected at random, what is the probability that it is NOT a Cocacola bottle?

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Coca-cola = 10 bottles, Fanta = 8 bottles
Sprite = 6 bottles, Total = 24
P(cola-cola) = $\frac{10}{24}$
P(not coca-cola) = 1 - $\frac{10}{24}$
$\frac{24?10}{24}$ = $\frac{14}{24}$
= $\frac{7}{24}$

Simplify cos2 x (sec2x + sec2 x tan2x)

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To simplify cos2 x (sec2x + sec2 x tan2x), we can use the trigonometric identity: 1 + tan2 x = sec2 x. First, we can simplify sec2 x + sec2 x tan2 x: sec2 x + sec2 x tan2 x = sec2 x (1 + tan2 x) Next, we can substitute sec2 x (1 + tan2 x) into the original equation: cos2 x (sec2 x + sec2 x tan2 x) = cos2 x (sec2 x (1 + tan2 x)) Using the identity 1 + tan2 x = sec2 x, we can simplify: cos2 x (sec2 x (1 + tan2 x)) = cos2 x (sec2 x)(sec2 x) Finally, using the identity sec2 x = 1/cos2 x, we can simplify: cos2 x (sec2 x)(sec2 x) = cos2 x (1/cos2 x)(1/cos2 x) = 1/cos2 x = sec2 x Therefore, the answer is 1+tan2xsec2x.

Express $62+3$ as decimal correct to 3 significant figures.

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Simplify $\frac{3}{x}+\frac{15}{2y}+\frac{6}{xy}$

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Which of the following is a factor of rs + tr - pt - ps?

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Find the sum of the 20 terms in an arithmetic progression whose first term is 7 and last term 117?

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An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. Let's call this constant difference "d". We know that the first term of the sequence is 7 and the last term is 117. Let's call the number of terms in the sequence "n". Using this information, we can write two equations: 1. The nth term of an arithmetic progression formula: an = a1 + (n - 1)d, where "an" is the nth term of the sequence, "a1" is the first term of the sequence, "n" is the number of terms in the sequence, and "d" is the constant difference between any two consecutive terms in the sequence. We can use this equation to find the value of "d": 117 = 7 + (n - 1)d 110 = (n - 1)d d = 110/(n-1) 2. The sum of an arithmetic progression formula: Sn = n/2 * (a1 + an), where "Sn" is the sum of the first "n" terms of the sequence. We can use this equation to find the sum of the first 20 terms of the sequence: Sn = 20/2 * (7 + (7 + 19d)) Sn = 10 * (14 + 2090/(n-1)) Now we can substitute the value of "d" from equation (1) into equation (2) and simplify: Sn = 10 * (14 + 2090/(n-1)) Sn = 10 * (14 + 2090/(20-1)) (since there are 20 terms in the sequence) Sn = 10 * (14 + 110) Sn = 1240 Therefore, the sum of the 20 terms in the arithmetic progression is 1240

Simplify $\frac{1}{1+\sqrt{5}}$ - $\frac{1}{1?\sqrt{5}}$

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$\frac{1}{1+\sqrt{5}}$ - $\frac{1}{1-\sqrt{5}}$
= $\frac{3-\sqrt{5}-3-\sqrt{5}}{(3+\sqrt{5})(3-\sqrt{5}}$
= $\frac{-2\sqrt{5}}{9-5}$
= $\frac{-2\sqrt{5}}{4}$
= - $\frac{1}{2}\sqrt{5}$

Simplify 2log $\frac{2}{5}$ - log$\frac{72}{125}$ + log 9

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2log $\frac{2}{5}$ - log$\frac{72}{125}$ + log 9
[$\frac{2}{5}$)2 x 9] = log $\frac{4}{25}$ x $\frac{9}{1}$ x $\frac{125}{72}$
= log $\frac{72}{125}$
= log $\frac{5}{2}$
= log $\frac{10}{4}$
= log 10 - log 4
= log10 10 - log10 22
= 1 - 2 log2

Find the volume of the figure.

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The area of a square is 144 sq cm. Find the length of its diagonal

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The area of a square is given as 144 sq cm. To find the length of its diagonal, we need to use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In the case of a square, we know that all sides are equal in length, so we can use this fact to simplify the equation. Let's call the length of one side of the square "s". Then, we know that the area of the square is s^2 = 144 sq cm. Solving for "s", we get s = sqrt(144) = 12 cm. Now, let's draw a diagonal line across the square, splitting it into two right-angled triangles. We can label the hypotenuse of each triangle as "d" (which is the length of the diagonal we want to find), and the other two sides as "s". Using the Pythagorean theorem, we get: d^2 = s^2 + s^2 d^2 = 2s^2 d^2 = 2(12^2) d^2 = 288 d = sqrt(288) = 12sqrt(2) So the length of the diagonal is 12sqrt(2) cm.

One angle of a rhombus is 60o. The shorter of the two diagonals is 8cm long. Find the length of the longer one.

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In a rhombus, all sides are congruent, and opposite angles are equal. If one angle of the rhombus is 60 degrees, then the opposite angle is also 60 degrees. Let the shorter diagonal of the rhombus be divided into two equal halves. Let the half of the shorter diagonal be `a`. Using the properties of a 30-60-90 degree triangle, we know that the length of the longer diagonal of the rhombus is `2a√3`. Since the shorter diagonal is 8cm, `a = 4cm`. Therefore, the length of the longer diagonal is: `2a√3 = 2(4cm)√3 = 8√3 cm` Therefore, the length of the longer diagonal of the rhombus is `8√3 cm`. So the correct answer is: 8√3.

If w varies inversely as $\frac{ur}{u+r}$ and is equal to 8 when u = 2 and r = 6, find a relationship between u, v, w.

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W $\alpha$ $\frac{\frac{1}{uv}}{u+v}$
∴ w = $\frac{\frac{k}{uv}}{u+v}$
= $\frac{k(u+v)}{uv}$
w = $\frac{k(u+v)}{uv}$
w = 8, u = 2 and v = 6
8 = $\frac{k(2+6)}{2\left(6\right)}$
= $\frac{k\left(8\right)}{12}$
k = 12

In this figure, PQ = PR = PS and SRT = 68${}^{\circ }$. Find QPS.

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Since PQRS is quadrilateral 2y + 2x = QPS = 360${}^{\circ }$

i.e. 2(y + x) + QPS = 360${}^{\circ }$

QPS = 360${}^{\circ }$ - 2(y + x)

But x + y + 68${}^{\circ }$ = 180${}^{\circ }$

x + y = 180${}^{\circ }$ - 68${}^{\circ }$ = 180${}^{\circ }$

x + y = 180${}^{\circ }$ - 68${}^{\circ }$

= 112${}^{\circ }$

QPS = 360${}^{\circ }$ - 2(112${}^{\circ }$)

360${}^{\circ }$ - 224${}^{\circ }$ = 136${}^{\circ }$

Factory P produces 20,000 bags of cement per day while factory Q produces 15,000 bags per day. If P reduces production by 5% and Q increases production by 5%, determine the effective loss in the number of bags produced per day by the two factories

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Factory P produces 20,000 bags of cement per day and reduces production by 5%, which means it produces: 20,000 - 5% of 20,000 = 20,000 - 1,000 = 19,000 bags per day. Factory Q produces 15,000 bags of cement per day and increases production by 5%, which means it produces: 15,000 + 5% of 15,000 = 15,000 + 750 = 15,750 bags per day. The total number of bags produced per day by both factories before the changes were: 20,000 + 15,000 = 35,000 bags per day. The total number of bags produced per day by both factories after the changes are: 19,000 + 15,750 = 34,750 bags per day. The effective loss in the number of bags produced per day by the two factories is the difference between the original number of bags produced and the new number of bags produced: 35,000 - 34,750 = 250 bags per day. Therefore, the effective loss in the number of bags produced per day by the two factories is 250

A flagstaff stands on the top of a vertical tower. A man standing 60 m away from the tower observes that the angles of elevation of the top and bottom of the flagstaff are 64o and 62o respectively. Find the length of the flagstaff.

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$\frac{BC}{60}$ = $\frac{tan62}{1}$
BC = 60 tan 62
$\frac{AC}{60}$ = $\frac{tan62}{1}$
AC = 60 tan 64
AB = AC - BC
= 60(tan 64o - tan 62o)

Fifty boxes each of 50 bolts were inspected for the number which were defective. The following was the result:

$\begin{array}{ccccccc}\text{No. defective per box}& 4& 5& 6& 7& 8& 9\\ \text{No. of boxes}& 2& 7& 17& 10& 8& 6\end{array}$

Find the percentage of boxes containing at least 5 defective bolts each.

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To find the percentage of boxes containing at least 5 defective bolts each, we need to first add up the number of boxes that have 5, 6, 7, 8, or 9 defective bolts. These are the boxes that contain at least 5 defective bolts. From the table, we can see that there are: - 7 boxes with 5 defective bolts - 17 boxes with 6 defective bolts - 10 boxes with 7 defective bolts - 8 boxes with 8 defective bolts - 6 boxes with 9 defective bolts Adding these up, we get a total of 7+17+10+8+6 = 48 boxes. Therefore, out of the 50 boxes that were inspected, 48 of them contain at least 5 defective bolts each. To find the percentage, we can use the formula: percentage = (part/whole) x 100 In this case, the "part" is the number of boxes containing at least 5 defective bolts (48), and the "whole" is the total number of boxes inspected (50). Substituting in the values, we get: percentage = (48/50) x 100 = 96% Therefore, the answer is 96%, option (A).

A car travels from calabar to Enugu, a distance of p km with an average speed of u km per hour and continues to benin, a distance of q km, with an average speed of w km per hour. Find its average speed from Calabar to Benin

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Average speed = $\frac{\text{total Distance}}{\text{Total Time}}$
from Calabar to Enugu in time t1, hence
t1 = $\frac{p}{u}$ also from Enugu to Benin
t2 = $\frac{q}{w}$
Average speed = $\frac{p+q}{t_1+t_2}$
= $\frac{p+q}{\frac{p}{u}+\frac{q}{w}}$
= p + q × $\frac{uw}{pw+qu}$
= $\frac{uw(p+q)}{pw+qu}$

Find the positive number n such that thrice its square is equal to 12 times the number.

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The problem can be translated into an equation using algebra. Let's start by translating the words into math symbols. Let n be the positive number we are looking for. The phrase "thrice its square" means 3 times n^2 or 3n^2. The phrase "12 times the number" means 12 times n or 12n. So we can translate the sentence "thrice its square is equal to 12 times the number" into the equation: 3n^2 = 12n Now we can solve for n. First, we can simplify the equation by dividing both sides by 3: n^2 = 4n Next, we can rearrange the equation by subtracting 4n from both sides: n^2 - 4n = 0 Now we can factor out an n: n(n - 4) = 0 So the solutions to this equation are n = 0 and n - 4 = 0, which gives n = 4. However, we are looking for a positive value of n, so we can discard the solution n = 0. Therefore, the positive number n such that thrice its square is equal to 12 times the number is n = 4. Therefore, the answer is 4.

In the figure, PQRS is a square of sides 8cm. What is the area of $△$UVW?

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Find the gradient of the line passing through the points (-2, 0) and (0, -4).

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To find the gradient of a line passing through two given points, we need to use the formula: Gradient = (change in y-coordinate) / (change in x-coordinate) Let's take the two given points: (-2, 0) and (0, -4) The change in y-coordinate is -4 - 0 = -4 (final y-coordinate - initial y-coordinate) The change in x-coordinate is 0 - (-2) = 2 (final x-coordinate - initial x-coordinate) Now, we can substitute the values in the formula: Gradient = -4 / 2 = -2 Therefore, the gradient of the line passing through the points (-2, 0) and (0, -4) is -2. Note: The gradient (also known as slope) of a line represents how steep the line is. A positive gradient means the line is sloping upwards, while a negative gradient means the line is sloping downwards. The larger the magnitude of the gradient, the steeper the line.