JAMB UTME - Mathematics - 1992

Question 1 Report

$p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}$

$q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}$ Where $p^T$ is the transpose p calculate $/p^T/$ when x = 0, y = 1 and z = 2

48

24

-24

-48

Answer Details

=

| \begin{matrix} 0 & 3 & 0 \\ 2 & 1 & 3 \\ 4 & 2 & 2 \end{matrix} |

PT =

| \begin{matrix} 0 & 2 & 4 \\ 2 & 1 & 3 \\ 0 & 3 & 2 \end{matrix} |

/pT/ =

| \begin{matrix} 0 & 2 & 4 \\ 3 & 1 & 3 \\ 0 & 3 & 2 \end{matrix} |

= 0[2 - 6] - 2[6 - 0] + 4[9 - 0]
= 0 - 12 + 36 = 24

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Matrices And Determinants

Number Bases

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Question 2 Report

If $x = 3 - \sqrt{3}$, find $x^2 + \frac{36}{x^2}$

9

18

24

27

Answer Details

x = 3 -

\sqrt{3}

x2 = (3 -

\sqrt{3}

)2
= 9 + 3 - 6

\sqrt{34}

= 12 - 6

\sqrt{3}

= 6(2 -

\sqrt{3}

)
∴ x2 +

\frac{36}{x^{2}}

= 6(2 -

\sqrt{3}

) +

\frac{36}{6 (2 - \sqrt{3})}

6(2 -

\sqrt{3}

) +

\frac{6}{2 - \sqrt{3}}

= 6(-

\sqrt{3}

) +

\frac{6 (2 + \sqrt{3})}{(2 - \sqrt{3}) (2 + \sqrt{3})}

= 6(2 -

\sqrt{3}

) +

\frac{6 (2 + \sqrt{3})}{4 - 3}

6(2 -

\sqrt{3}

) + 6(2 +

\sqrt{3}

) = 12 + 12
= 24

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Statistics

Quadratic Equations

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Question 3 Report

$x$	2	4	6	8
$f$	4	$y$	6	5

If the mean of the above frequency distribution is 5.2, find y

2, 1

1, 2

1, 5

5, 2

Answer Details

Mean

\bar{x}

=

\frac{\sum f x}{\sum f}

=

\frac{5.2}{1}

=

\frac{8 + 4 y + 36 + 40}{4 + y + 6 + 5}

=

\frac{5.2}{1}

=

\frac{84 + 4 y}{15 + y}

= 5.2(15 + y)
= 84 + 4y
= 5.2 x 15 + 5.2y
= 84 + 4y
= 78 + 5.2y
= 84 = 4y
= 5.2y - 4y
= 84 - 78
1.2y = 6
y =

\frac{6}{1.2}

=

\frac{60}{12}

= 5

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Probability

Linear Inequalities

Statistics

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Question 4 Report

If x = (all prime factors of 44) and y = (all prime factors of 60), the elements of X ∪ Y and X ∩ Y respectively are

(2, 4, 3, 5, 11) and (4)

(4, 3, 5, 11) and (3, 4)

(2, 5, 11) and (2)

(2, 3, 5, 11) and (2)

Answer Details

The prime factors of 44 are 2, 2, 11. The prime factors of 60 are 2, 2, 3, 5. To find X ∪ Y (the union of X and Y), we need to list all the distinct prime factors from both 44 and 60. These are 2, 3, 5, 11. Therefore, X ∪ Y is (2, 3, 5, 11). To find X ∩ Y (the intersection of X and Y), we need to list all the prime factors that are common to both 44 and 60. These are 2. Therefore, X ∩ Y is (2). Therefore, the answer is (2, 3, 5, 11) and (2), which corresponds to option D.

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Sets

Algebraic Fractions

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Question 5 Report

A cliff on the bank of a river is 300 meter high. if the angle of depression of a point on the opposite side of the river is $60^\circ$, find the width of the river.

100

75√3 m

100√3m

200√3m

Answer Details

The problem involves finding the width of a river given the height of a cliff and the angle of depression of a point on the opposite side of the river. The angle of depression is the angle formed between the horizontal line and the line of sight from the point on the opposite side of the river to the top of the cliff. We can use trigonometry to solve the problem. Let x be the width of the river. Then we have a right triangle with the height of the cliff as the opposite side, x as the adjacent side, and the angle of depression as 60 degrees. Using the tangent function, we have: tan(60) = opposite/adjacent sqrt(3) = 300/x x = 300/sqrt(3) x = 100sqrt(3) Therefore, the width of the river is 100sqrt(3) meters. So, the correct option is: - 100 - 75√3 m - 100√3m - 200√3m (100√3m) is the correct answer.

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Trigonometry

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Question 6 Report

Ice forms on a refrigerator ice box at rate of (4 - 0.61)g per minute after 1 minute. If initially, there were 2g of ice, find the mass of ice formed in 5 minutes.

19.5

17.0

14.5

12.5

Introductory Calculus

Probability

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Question 7 Report

Solve the equation $y^2 - 11y + 24 = 0$

8,3

4,9

6,4

9,-8

Answer Details

To solve the equation: y^2 - 11y + 24 = 0 We can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by: x = (-b ± sqrt(b^2 - 4ac)) / (2a) In this case, we have a = 1, b = -11, and c = 24. Substituting these values into the quadratic formula, we get: y = (-(-11) ± sqrt((-11)^2 - 4(1)(24))) / (2(1)) Simplifying this expression, we get: y = (11 ± sqrt(121 - 96)) / 2 y = (11 ± sqrt(25)) / 2 y = (11 ± 5) / 2 So the solutions to the equation are: y = 8 or y = 3 Therefore, the equation y^2 - 11y + 24 = 0 has two solutions: y = 8 and y = 3.

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Question 8 Report

Given that $ \theta $ is an acute angle and $ \sin \theta = \frac{m}{n} $, find $ \cos \theta $

\frac{\sqrt{n^{2} - m^{2}}}{m}

\frac{\sqrt{(n + m) (n - m)}}{n}

\frac{m}{\sqrt{n^{2} - m^{2}}}

\sqrt{\frac{n}{n^{2} - m^{2}}}

Sine, Cosine And Tangent Of An Angle

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Question 9 Report

Evaluate $ \frac{(x-2)(x^2+3x-2)}{x^2-4} $

7

2

3

4

Algebraic Expressions

Quadratic Equations

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Question 10 Report

The radius of a circle is given as 5cm subject to an error of 0.1cm. What is the percentage error in the area of the circle?

\frac{1}{25}

\frac{1}{4}

4

25

Answer Details

Area =

π r^{2} = π \times 5 \times 5 = 25 π

Area error =

π \times {0.1}^{2} = 0.01 π

∴

% error =

\frac{0.01 π}{25 π} \times 10

= \frac{0.01}{25} \times 100 = \frac{1}{25}

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Representation Of Data

Percentages

Probability

Fractions, Decimals, Approximations And Percentage

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Question 11 Report

Obtain a maximum value of the function $f(x)=x^3-12x+11$.

-5

-2

2

27

Answer Details

To obtain the maximum value of the given function f(x) = x^3 - 12x + 11, we need to find its critical points. The critical points of a function are the points where the derivative of the function is either zero or undefined. So, let's find the derivative of the given function: f'(x) = 3x^2 - 12 Now, we'll set f'(x) = 0 and solve for x: 3x^2 - 12 = 0 x^2 - 4 = 0 (x - 2)(x + 2) = 0 So, the critical points of the function are x = 2 and x = -2. To determine whether these are maximum or minimum points, we'll take the second derivative of the function: f''(x) = 6x Now, we'll substitute the critical points into the second derivative: f''(2) = 12 f''(-2) = -12 Since f''(2) is positive, the critical point x = 2 is a minimum point. Since f''(-2) is negative, the critical point x = -2 is a maximum point. Therefore, the maximum value of the given function is obtained when x = -2. Now, we'll substitute x = -2 into the original function: f(-2) = (-2)^3 - 12(-2) + 11 f(-2) = -8 + 24 + 11 f(-2) = 27 So, the maximum value of the given function is 27, and the correct option is (D).

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Question 12 Report

If the scores of 3 students in a test are 5, 6 and 7, find the standard deviation of their scores

\frac{2}{3}

\frac{2}{3} \sqrt{3}

\sqrt{\frac{2}{3}}

\sqrt{\frac{3}{2}}

Answer Details

To find the standard deviation of the scores, we need to first calculate the mean (average) of the scores: mean = (5 + 6 + 7) / 3 = 6 Next, we calculate the deviation of each score from the mean: deviation of 5 = 5 - 6 = -1 deviation of 6 = 6 - 6 = 0 deviation of 7 = 7 - 6 = 1 To calculate the standard deviation, we take the square root of the average of the squared deviations: standard deviation = √[(1^2 + 0^2 + (-1)^2) / 3] = √(2/3) ≈ 0.82 Therefore, the answer is option (D), √(2/3).

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Measures Of Dispersion

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Question 13 Report

If the binary operation $ \ast $ is defined by $ m \ast n = mn + m + n $ for any real number m and n, find the identity of the elements under this operation

e = 1

e = -1

e = -2

e = 0

Answer Details

m

*

n = mn + m + n
m

*

e = me + m + e, e

*

m = e
∴ me + m + e, m(e + 1)e - e = 0
e + 1 = 0
∴ e = -1

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Binary Operations

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Question 14 Report

If in the diagram above, FG is parallel to KM, find the value of x

75o

95o

105o

125o

Angles And Intercepts On Parallel Lines

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Question 15 Report

What is the value of x satisfying the equation $ \frac{4^{2x}}{4^{3x}} = 2 $?

-2

-

\frac{1}{2}

\frac{1}{2}

2

Answer Details

\frac{4^{2 x}}{4^{3 x}}

= 2
42x - 3x = 2
4-x = 2
(22)-x
= 21
Equating coefficients: -2x = 1
x = -

\frac{1}{2}

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Question 16 Report

The locus of a point which is equidistant from two given fixed points is the

perpendicular bisector of the straight line joining them

parallel line to the straight line joining them

transverse to the straight line joining them

angle bisector of 90o which the straight line joining them makes with the horizontal

Answer Details

The locus of a point which is equidistant from two given fixed points is the perpendicular bisector of the straight line joining them. This means that any point on the perpendicular bisector of the straight line joining the two fixed points is equidistant from those points. To understand this concept, imagine two points on a plane and draw a straight line joining them. Now draw a perpendicular line bisecting the straight line joining them. Any point on this perpendicular line is equidistant from the two fixed points, since it is the same distance away from each of them. This is because the perpendicular line creates right angles with the straight line joining the two fixed points, and all points on a perpendicular line are equidistant from the two endpoints of the straight line it intersects.

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Loci

Circles

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Question 17 Report

The angle between latitudes 30oS and 13oN is

17o

33o

43o

53o

Answer Details

The distance between two latitudes is always the same, and this distance is equal to 1/360th of the circumference of the earth. The circumference of the earth is approximately 40,000 km, so 1/360th of the circumference is approximately 111 km. To find the angle between latitudes 30°S and 13°N, we need to add the distance between these latitudes to the angle at the center of the earth. The distance between 30°S and the equator (0°) is 30°, and the distance between the equator and 13°N is 13°, for a total distance of 43°. Therefore, the angle between latitudes 30°S and 13°N is 43°. So, the answer is

43o

.

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Angles

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Question 18 Report

Factorize 9p2 - q2 + 6qr - 9r2

(3p - 3q + r)(3p - q - 3r)

(6p - 3q - 3r)(3p - q - 4r)

(3p - q + 3r)(3p - q - 3r)

(3q - p + 3r)(3q - p + 3r)

Algebraic Expressions

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Question 19 Report

If the angles of quadrilateral are (P + 10)o(2P - 30)o(3P + 20)o and 4po, find p

63

40

36

28

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Question 20 Report

x is a point due east of point Y on a coast Z is another point on the coast but 63m due south of y. If the distance ZX is 12Km. Calculate the bearing of Z from X

240o

210o

150o

60o

Bearings

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Question 21 Report

A man invested a sum of ₦280.00 partly at 5% and partly at 4%, if the total interest is ₦12.80 per annum, find the amount invested at 5%.

₦14.00

₦120.00

₦140.00

₦160.00

Answer Details

Let x be the amount invested at 5% and y be the amount invested at 4%. Then we have the following system of equations: x + y = 280 --- (1) (since the man invested a total of ₦280.00) 0.05x + 0.04y = 12.80 --- (2) (since the total interest earned is ₦12.80) To solve for x, we can use the first equation to get y = 280 - x, and substitute into the second equation to get: 0.05x + 0.04(280 - x) = 12.80 Simplifying and solving for x, we get: 0.05x + 11.20 - 0.04x = 12.80 0.01x = 1.60 x = 160 Therefore, the amount invested at 5% is ₦160.00.

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Modular Arithmetic

Ratio, Proportions And Rates

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Question 22 Report

Simplify $\frac{(1.25 \times 10^{-4}) \times (2.0 \times 10^{-1})}{(6.25 \times 10^{5})}$

4.0 x 10-3

5.0 x 10-2

2.0 x 10-1

5.0 x 10-3

Answer Details

\frac{(1.25 \times 10^{- 4}) \times (2.0 \times 10^{- 1})}{(6.25 \times 10^{5})}

=

\frac{1.25 \times 2}{6.25}

x 104 - 1 - 5

\frac{2.50}{6.25}

x 10-2 =

\frac{250}{625}

x 10-2
0.4 x 10-2 = 4.0 x 10-3

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Number Bases

Sets

Logarithms

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Question 23 Report

Five people are to be arranged in a row for a group photograph. How many arrangements are the if a married couple in the group insist on sitting next to each other?

48

24

20

10

Permutation And Combination

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Question 24 Report

$p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}$

$q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}$

pq is equivalent to

ppT

pp-1

qp

pp

Answer Details

=

| \begin{matrix} 0 & 3 & 0 \\ 2 & 1 & 3 \\ 4 & 2 & 2 \end{matrix} |

Q =

| \begin{matrix} 0 & 2 & 4 \\ 3 & 1 & 2 \\ 0 & 3 & 2 \end{matrix} |

= pT
pq = ppT

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Matrices And Determinants

Quadratic Equations

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Question 25 Report

For an arithmetical sequence, the first term is 2 and the common difference is 3. Find the sum of the first 11 terms

157

187

197

200

Answer Details

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. In this case, the first term is 2 and the common difference is 3, so the sequence is: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32 To find the sum of the first 11 terms of this sequence, we can use the formula for the sum of an arithmetic series: Sn = n/2 * (a1 + an) where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term. In this case, n = 11, a1 = 2, and an = 32 (the 11th term can be found by adding the common difference 3, 10 times to the first term 2). So, substituting the values in the formula: Sn = 11/2 * (2 + 32) Sn = 11/2 * 34 Sn = 187 Therefore, the sum of the first 11 terms is 187. So the correct answer is.

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Sequence And Series

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Question 26 Report

Determine the distance on the earth's surface between two town P (latoN, Long 20oN) and Q(Lat 60oN, Long 25oW) (Radius of the earth = 6400km)

\frac{800 π}{9}

km

\frac{800 \sqrt{3 π}}{9}

km

800

π

km

800

\sqrt{3 π}

km

Coordinate Geometry

Angles And Intercepts On Parallel Lines

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Question 27 Report

The probability of an event P is $ \frac{3}{4} $ while that of another event Q is $ \frac{1}{6} $. If the probability of both P and Q is $ \frac{1}{2} $. What is the probability of either P or Q.

\frac{1}{96}

\frac{1}{8}

\frac{5}{6}

\frac{11}{12}

Probability

Surds (radicals)

Probability

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Question 28 Report

If two dice are thrown together, what is the probability of obtaining at least a score of 10?

\frac{1}{6}

\frac{1}{12}

\frac{5}{6}

\frac{11}{12}

Answer Details

The total sample space when two dice are thrown together is 6 x 6 = 36

\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1. & 1.1 & 1.2 & 1.3 & 1.4 & 1.5 & 1.6 \\ 2 & 2.1 & 2.2 & 2.3 & 2.4 & 2.5 & 2.6 \\ 3 & 3.1 & 3.2 & 3.3 & 3.4 & 3.5 & 3.6 \\ 4 & 4.1 & 4.2 & 4.3 & 4.4 & 4.5 & 4.6 \\ 5 & 5.1 & 5.2 & 5.3 & 5.4 & 5.5 & 5.6 \\ 6 & 6.1 & 6.2 & 6.3 & 6.4 & 6.5 & 6.6 \end{matrix}

At least 10 means 10 and above
P(at least 10) =

\frac{6}{36}

=

\frac{1}{6}

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Probability

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Question 29 Report

ind then equation line through (5, 7) parallel to the line 7x + 5y = 12

5x + 7y = 120

7x + 5y = 70

x + y = 7

15x + 17y = 90

Answer Details

Equation (5, 7) parallel to the line 7x + 5y = 12
5Y = -7x + 12
y =

\frac{- 7 x}{5}

+

\frac{12}{5}

Gradient =

\frac{- 7}{5}

∴ Required equation =

\frac{y - 7}{x - 5}

=

\frac{- 7}{5}

i.e. 5y - 35 = -7x + 35
5y + 7x = 70

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Question 30 Report

Resolve $ \frac{3}{x^2+x-2} $ into partial fractions

\frac{1}{x - 1} + \frac{1}{x + 2}

$\frac{1}{x + 1} + \frac{1}{x - 2}$

\frac{1}{x + 1} - \frac{1}{x - 2}

\frac{1}{x - 2} + \frac{1}{x + 2}

Answer Details

\frac{3}{x^{2} + x - 2}

=

\frac{3}{(x - 1) (x + 1)}

\frac{A}{x - 1}

+

\frac{B}{x + 2}

A(x + 2) + B(x - 1) = 3
when x = 1, 3A = 3

\to

a = 1
when x = -2, -3B = 3

\to

B = -1
3 = 1 - 1
=

\frac{1}{x - 1} + \frac{1}{x + 2}

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Algebraic Fractions

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Question 31 Report

Make t the subject of formula $S = ut + \frac{1}{2}at^2$

\frac{1}{a} - u + \sqrt{u^{2} - 2 a s}

\frac{1}{a} u \pm (u^{2} - 2 a s)

\frac{1}{a} u \pm \sqrt{2 a s}

\frac{1}{a} - u + \sqrt{2 a s}

Answer Details

Given S = ut +

\frac{1}{2} a t^{2}

S = ut +

\frac{1}{2} a t^{2}

∴ 2S = 2ut + at2
= at2 + 2ut - 2s = 0
t =

\frac{- 2 u \pm 4 u^{2} + 2 a s}{2 a}

= -2u

π

\frac{\sqrt{u^{2} 4 u^{2} + 2 a s}}{2 a}

= $\frac{1}{a}$ (-u + $\sqrt{U^{2} - 2 a s}$ )

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Change Of Subject Of A Formula/Relation

Statistics

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Question 32 Report

Find n if 34n = 100112

5

6

7

8

Answer Details

To find n if 34n = 100112, convert both sides to base 10
= 3n + 4 = (1 x 24) + (0 x23) + (0 x 22) + (1 x 21) + 1 x 2o
= 3n + 4 = 16 + 0 + 0 + 2 + 1
3n + 4 = 19
3n = 15
n = 5

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Number Bases

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Question 33 Report

a student blows a balloon and its volume increases at a rate of $ \pi(20 - t^2)\text{ cm}^3\text{ s}^{-1} $ after t seconds. If the initial volume is 0 cm³, find the volume of the balloon after 2 seconds

37.00 $π$

37.33 $π$

40.00 $π$

42.67 $π$

Answer Details

The volume of the balloon increases at a rate of π(20-t^2) cm^3S^-1 after t seconds. If we want to find the volume of the balloon after 2 seconds, we need to integrate the given rate with respect to time from 0 to 2, since we want to know the change in volume from the initial volume of 0 cm^3 after 2 seconds. Integrating the given rate with respect to time, we get: ∫[0,2] π(20-t^2) dt = π[20t - (t^3/3)] from 0 to 2 Plugging in the values, we get: π[20(2) - (2^3/3)] - π[20(0) - (0^3/3)] = π[40 - 8/3] = π[120/3 - 8/3] = π[112/3] = 37.33π Therefore, the volume of the balloon after 2 seconds is approximately 37.33π cubic centimeters.

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Integration

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Question 34 Report

Find the sum to infinity to the following series $3 + 2 + \frac{4}{3} + \frac{8}{9} + \frac{16}{17} + .....$

1270

190

18

9

Answer Details

3 + 2 +

\frac{4}{3}

+

\frac{8}{9}

+

\frac{16}{17}

+ .....
a = 3
r =

\frac{2}{3}

s

α

=

\frac{a}{1 - r}

=

\frac{3}{1 - \frac{2}{3}}

=

\frac{3}{\frac{1}{3}}

= 3 x 3
= 9

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Sequence And Series

Progression

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Question 35 Report

If U = (1, 2, 3, 6, 7, 8, 9, 10) is the universal set. E = (10, 4, 6, 8, 10) and F = is odd. Find (E ∩ F), where means x 1x2 = 26, x the complement of a set

(0)

U

(8)

ϕ

Answer Details

U = (1, 2, 3, 6, 7, 8, 9, 10)
E = (10, 4, 6, 8, 10)
F = (x : x2 = 26, x is odd)
∴ F =

ϕ

Since x2 = 26 = 64
x = + which is even
∴ E ∩ F =

ϕ

Since there are no common elements

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Sets

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Question 36 Report

Find all values of x satisfying the inequality $ -11 \le 4 - 3x \le 28 $

-5

\leq

x v 8

5

\leq

x

\leq

8

-8

\leq

x

\leq

5

-5 < x

\leq

8

Answer Details

To solve the inequality -11 ≤ 4 - 3x ≤ 28, we need to isolate x in the middle of the inequality. First, we subtract 4 from all parts of the inequality: -15 ≤ -3x ≤ 24. Then, we divide by -3, remembering to flip the direction of the inequality signs when dividing by a negative number: 5 ≥ x ≥ -8. Therefore, the correct option is -8 ≤ x ≤ 5.

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Variation

Linear Inequalities

Inequalities

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Question 37 Report

No. of children	0	1	2	3	4	5	6
No. of families	7	11	6	7	7	5	3

Find the mode and median respectively of the distribution above

2, 1

1, 2

1, 5

5, 2

Probability

Statistics

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Question 38 Report

In the figure, if XZ is 10cm, calculate RY in cm.

1010(1 -

\frac{1}{\sqrt{2}}

)

10(1 -

\frac{1}{\sqrt{3}}

)

10(1 -

\frac{1}{\sqrt{2}}

)

10[1 -

\sqrt{3}

]

Circles

Triangles And Polygons

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Question 39 Report

Evaluate the integral $ \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} 2 \cos 2x\, dx $

-

\frac{1}{2}

-1

\frac{1}{2}

1

Answer Details

let I =

\int_{\frac{π}{12}}^{\frac{π}{4}}

2 cos 2x dx
= 2(sin 2x)

\frac{π}{4}

(sin 2x)

\frac{π}{4}

\frac{π}{12}

(2)

\frac{π}{12}

= -1 -

\frac{1}{2}

=

\frac{1}{2}

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Integration

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Question 40 Report

If x + 1 is a factor of x3 + 3x2 + kx + 4, find the value of k

6

-6

8

-8

Answer Details

To find the value of k, we need to use the factor theorem which states that if x - a is a factor of a polynomial, then the polynomial is zero when x = a. Since x + 1 is a factor of x^3 + 3x^2 + kx + 4, we know that x = -1 makes the polynomial zero. Substituting -1 for x in the polynomial, we get: (-1)^3 + 3(-1)^2 + k(-1) + 4 = 0 -1 + 3 - k + 4 = 0 6 - k = 0 Solving for k, we get: k = 6 Therefore, the value of k is 6.

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Algebraic Expressions

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Question 41 Report

Evaluate logban if b = an

nn

n

\frac{1}{n}

\frac{1}{n^{2}}

Answer Details

If b = a, then log base b of a^n is equal to n. To explain this in simple terms, let's first define what a logarithm is. A logarithm is simply the power to which a number (called the base) must be raised in order to get another number. For example, if we have the logarithm base 2 of 8, this means "what power must we raise 2 to in order to get 8?" The answer is 3, because 2^3 = 8. Now, in the given expression log base b of a^n, we have a base of b and an exponent of n. If b = a, this means that the base and the exponent are the same number. So we can rewrite the expression as log base a of a^n. What power must we raise a to in order to get a^n? The answer is n, because a^n is simply a multiplied by itself n times. So the logarithm base a of a^n is equal to n. Therefore, if b = a, then log base b of a^n is equal to n. The answer is (a) n.

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Read lesson note on Logarithms (WAEC)

Surds (radicals)

Logarithms

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Question 42 Report

In the figure, PQr is a semicircle while PQ and Qr are chords. QS is the perpendicular from Q to the diameter PR. What is the expression for QS?

QS = PS

\sqrt{S R}

QS =

\sqrt{(P S S R)}

QS = 2

\sqrt{(P S S R)}

QS =

\frac{1}{\sqrt{2}}

\sqrt{(P S S R)}

Polynomials

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Question 43 Report

If $y = x \sin x$, Find $\dfrac{d^2y}{d^2x}$

2 cosx - x - x sinx

sinx + x cosx

sinx - x cosx

x sinx - 2 cosx

Answer Details

To find the second derivative of y = x sin x, we need to differentiate the function twice with respect to x. First, let's find the first derivative: y' = (x cos x) + (sin x) Using the product rule and the derivative of sin x. Next, we can find the second derivative: y'' = [(x cos x) + (sin x)]' = (cos x - x sin x) + cos x Using the product rule and the derivative of cos x. Therefore, the second derivative of y = x sin x is y'' = 2 cos x - x sin x.

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Fractions, Decimals, Approximations And Percentage

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Question 44 Report

A student has 5 courses to take from Mathematics and physics. There are 4 courses in Mathematics and 3 in Physics which he can choose his courses so that he takes exactly two courses in Physics?

48

12

10

7

Answer Details

To solve this problem, we need to use the combination formula, which is nCr = n! / r!(n-r)!. Here, we have 4 Mathematics courses and 3 Physics courses. The student needs to select 2 Physics courses out of 3, which can be done in 3C2 ways (i.e., 3 different ways). For each of the 3 ways the student selects 2 Physics courses, he must also select 3 Mathematics courses out of 4, which can be done in 4C3 ways (i.e., 4 different ways). Therefore, the total number of ways the student can select exactly two courses in Physics is 3C2 * 4C3 = 3 * 4 = 12. Hence, the correct answer is 12.

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Sets

Progression

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Question 45 Report

Simplify $5\sqrt{18} - 3\sqrt{72} + 4\sqrt{50}$

17

\sqrt{4}

4

\sqrt{17}

17

\sqrt{2}

12

\sqrt{4}

Answer Details

To simplify these expressions, we need to simplify the numbers under the radical sign first by factoring out their perfect square factors. For the first expression 5√18, we can factor out the perfect square factor of 9, which leaves us with 5√2√2√2 or 10√2. For the second expression, we can simplify √72 to √(36*2), and since 36 is a perfect square, we can factor it out, which leaves us with 3√2*6, or 3√2*2√3, which simplifies to 6√6. For the third expression, we can simplify √50 to √(25*2), and since 25 is a perfect square, we can factor it out, which leaves us with 2√2*5, or 2√2*√5, which simplifies to 2√10. Putting it all together, we have: 5√18 = 10√2 - 3√72 = 6√6 - 4√50 = 2√10 Now we can substitute these simplified expressions back into the original expressions: 10√2 - 6√6 + 2√10 To simplify this expression, we can group like terms. The coefficients of √6 are -6 and 0, since there are no other terms with √6. The coefficients of √10 are 2 and 0, since there are no other terms with √10. The coefficient of √2 is 10. So our simplified expression is: 10√2 - 6√6 + 2√10 = 10√2 - 6√6 + 2√10 = 10√2 - 6√(2*3) + 2√(2*5) = 10√2 - 6√2√3 + 2√2√5 = (10-6√3+2√5)√2 Therefore, the answer is 17√2.

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Read lesson note on Areas (WAEC)

Sequence And Series

Surds (radicals)

Areas

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Question 46 Report

The diagram is a circle with centre O. Find the area of the shaded portion.

9(

π - 2) c m^{2}

18

π c m^{2}

9

π c m^{2}

36

π c m^{2}

Answer Details

To find the area of the shaded portion, we need to subtract the area of the triangle from the area of the sector. First, we need to find the radius of the circle. Since the diameter is given as 6cm, the radius is half of it which is 3cm. Next, we need to find the angle of the sector. We can do this by using the properties of the isosceles triangle. The angle at the centre of the circle is twice the angle at the circumference of the circle. So, the angle at the centre of the circle is: 2 x (180 - 90) = 180 degrees This means that the angle of the sector is 180 degrees. Now we can calculate the area of the sector: Area of sector = (angle/360) x πr^2 = (180/360) x π(3)^2 = 4.5π cm^2 Next, we need to find the area of the triangle. Since the triangle is isosceles, the base angles are equal, and we can use trigonometry to find the height. The base of the triangle is equal to the diameter of the circle which is 6cm. Using trigonometry, we know that: tan(45) = height/base height = base x tan(45) height = 6 x 1 height = 6cm Now we can calculate the area of the triangle: Area of triangle = (1/2) x base x height = (1/2) x 6 x 6 = 18 cm^2 Finally, we can calculate the area of the shaded portion by subtracting the area of the triangle from the area of the sector: Area of shaded portion = Area of sector - Area of triangle = 4.5π - 18 = (9/2)(π - 2) cm^2 Therefore, the answer is 9(π−2)cm2.

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Circles

Areas

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Question 47 Report

What is the n-th term of the sequence 2, 6, 12, 20...?

4n - 2

2(3n - 1)

n2 + n

n2 + 3n + 2

Answer Details

The given sequence is formed by adding consecutive even numbers starting from 2. The first term is 2, the second term is 2 + 4, the third term is 2 + 4 + 6, and so on. Thus, we can observe that the n-th term is obtained by adding the first n even numbers. The sum of the first n even numbers is n(n+1), which can be proved using mathematical induction. Therefore, the n-th term of the sequence is n(n+1). Hence, the correct option is: n2 + n.

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Sequence And Series

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Question 48 Report

The sketch is the curve of y = ax2 + bx + c. Find a, b and c respectively

1, 0, -4

-2, 2, -4

0, 1, -4

2, -2, -4

-1, 0, 4

Answer Details

Given the graph and the curve y = ax2 + bx + c the roots are x - 2 and 2 while its equation (x + 2)(x - 2) = y