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Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
If \(\cos^2 \theta + \frac{1}{8} = \sin^2 \theta\), find \(\tan \theta\).
A
3
B
\(\frac{3\sqrt{7}}{7}\)
C
3\(\sqrt{7}\)
D
\(\sqrt{7}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\cos^2 \theta + \frac{1}{8} = \sin^2 \theta\)..........(i)
from trigometric ratios for an acute angle, where \(cos^{2} \theta + \sin^2 \theta = 1\) ........(ii)
Substitute for equation (i) in (i) = \(\cos^2 \theta + \frac{1}{8} = 1 - \cos^2 \theta \)
= \(\cos^2 \theta + \cos^2 \theta = 1 - \frac{1}{8}\)
\(2\cos^2 \theta = \frac{7}{8}\)
\(\cos^2 \theta = \frac{7}{2 \times 8}\)
\(\frac{7}{16} = \cos \theta\)
\(\sqrt{\frac{7}{16}}\) = \(\frac{\sqrt{7}}{4}\)
but cos \(\theta\) = \(\frac{\text{adj}}{\text{hyp}}\)
\(opp^2 = hyp^2 - adj^2\)
\(opp^2 = 4^2 - (\sqrt{7})^{2}\)
= 16 - 7
opp = \(\sqrt{9}\) = 3
\(\tan \theta = \frac{\text{opp}}{\text{adj}}\)
= \(\frac{3}{\sqrt{7}}\)
\(\frac{3}{\sqrt{7}}\) x \(\frac{\sqrt{7}}{\sqrt{7}}\) = \(\frac{3\sqrt{7}}{7}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #31745)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
Simplify \(\frac{4a^2 - 49b^2}{2a^2 - 5ab - 7b^2}\)
A
\(\frac{a - b}{2a + b}\)
B
\(\frac{2a + 7b}{a - b}\)
C
\(\frac{2a - 7b}{a + b}\)
D
\(\frac{2a + 7b}{a + b}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{4a^2 - 49b^2}{2a^2 - 5ab - 7b^2}\) = \(\frac{(2a)^2 - (7b)^2}{(a + b)(2a - 7b)}\)
= \(\frac{(2a + 7b)(2a - 7b)}{(a + b)(2a - 7b)}\)
= \(\frac{2a + 7b}{a + b}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #31733)
Mathematics
WAEC / WASSCE 1988
Number and Numeration
Diagram
Which of the regions P, Q, R, S and T satisfies the following 0 < y < 1, y < x + 2, x < 0
Attached Figure / Diagram
A
P
B
Q
C
R
D
S
E
T
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option E.
Source: Myschool Past Questions Archive (Record #22185)
Mathematics
WAEC / WASSCE 1988
Number and Numeration
Solution
Cotonou and Niamey are on the same line of longitude and Niamey is 7o north of Cotonou. If the radius of the earth is 6400km, how far is Niamey north of Cotonou along the line of longitude, correct to the nearest kilometer? [Take π = 22/7]
A
391km
B
503km
C
782km
D
1006km
E
2012km
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
θ/360 * 2πR
= 7/360 * 2 * 22/7 * 6400 = 782km
Source: Myschool Past Questions Archive (Record #22226)
Mathematics
WAEC / WASSCE 1988
Number and Numeration
Solution
If Q = (all perfect squares less than 30) and P = (all odd numbers from 1 to 10). Find Q ∩ P.
A
(1, 4, 9, 16, 25)
B
(1, 3, 4, 5, 7, 9, 16, 25)
C
(1, 3, 5, 7, 9)
D
(1, 9)
E
Φ
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Q = {1, 4, 9, 16, 25} P = { 1, 3, 5, 7, 9}
Q ∩ P = { 1, 9}
Source: Myschool Past Questions Archive (Record #22227)
Mathematics
WAEC / WASSCE 1988
Number and Numeration
Diagram
Solution
The histogram below shows the number of candidates, in thousands, obtaining given ranges of marks in a State examination. How many candidates scored at most 30%?
Attached Figure / Diagram
A
15,000
B
20,000
C
25,000
D
35,000
E
60,000
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Those who scored at most 30% = ( 5 + 10 + 20)thousand = 35000
Source: Myschool Past Questions Archive (Record #22229)
Mathematics
WAEC / WASSCE 1988
Number and Numeration
Solution
The surnames of 40 children in a class arranged in alphabetical order. 16 of the surnames begins with O and 9 of the surname begins with A, 14, of the letters of the alphabet do not appear as the first letter of a surname
If more than one surname begins with a letter besides A and O, how may surnames begin with that letter
A
2
B
3
C
4
D
5
E
6
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
14 of the letters of the alphabet do not appear as the first letter of surname
∴ the no. of letters that appeared = 26-14 = 12 alphabets
15 surnames begin with 10 letters beside O and A
∴ 6 surnames begin with a letter
Source: Myschool Past Questions Archive (Record #22231)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
If \(1P03{_4} = 115_{10}\) find P
A
o
B
1
C
2
D
3
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(1 x 4^3 + P x 4^2 + 0 x 4 + 3 = 115_{10}\)
16p + 67 = 115 p = \(\frac{48}{16}\)
= 3
Source: Myschool Past Questions Archive (Record #30761)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
If \(\frac{1}{p}\) = \(\frac{a^2 + 2ab + b^2}{a - b}\) and \(\frac{1}{q}\) = \(\frac{a + b}{a^2 - 2ab + b^2}\) Find \(\frac{p}{q}\)
A
\(\frac{a + b}{a - b}\)
B
\(\frac{1}{a^2 - b^2}\)
C
\(\frac{a - b}{a + b}\)
D
a2 - b2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{1}{p} = \frac{a^{2} + 2ab + b^{2}}{a - b}\)
\(\frac{1}{q} = \frac{a + b}{a^{2} - 2ab + b^{2}}\)
\(\frac{1}{p} = \frac{(a + b)^{2}}{a - b}\)
\(\frac{1}{q} = \frac{a + b}{(a - b)^{2}}\)
\(\therefore p = \frac{a - b}{(a + b)^{2}}\)
\(\frac{p}{q} = p \times \frac{1}{q} = \frac{a - b}{(a + b)^{2}} \times \frac{a + b}{(a - b)^{2}}\)
= \(\frac{1}{(a + b)(a - b)}\)
= \(\frac{1}{a^{2} - b^{2}}\)
Source: Myschool Past Questions Archive (Record #31627)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
If x varies inversely as the cube root of y and x = 1 when y = 8, find y when x = 3
A
\(\frac{1}{3}\)
B
\(\frac{2}{3}\)
C
\(\frac{8}{27}\)
D
\(\frac{4}{9}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(x \propto \frac{1}{\sqrt[3]{y}} \implies x = \frac{k}{\sqrt[3]{y}}\)
When y = 8, x = 1
\(1 = \frac{k}{\sqrt[3]{8}} \implies 1 = \frac{k}{2}\)
\(k = 2\)
\(\therefore x = \frac{2}{\sqrt[3]{y}}\)
When x = 3,
\(3 = \frac{2}{\sqrt[3]{y}} \implies \sqrt[3]{y} = \frac{2}{3}\)
\(y = (\frac{2}{3})^{3} = \frac{8}{27}\)
Source: Myschool Past Questions Archive (Record #31629)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
if a = -3, b = 2, c = 4, evaluate \(\frac{a^3 - b^3 - c^{\frac{1}{2}}}{b - a - c}\)
A
37
B
\(\frac{-37}{5}\)
C
\(\frac{37}{5}\)
D
-37
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{a^{3} - b^{3} - c^{\frac{1}{2}}}{b - a - c} = \frac{(-3)^{3} - (2)^{3} - 4^{\frac{1}{2}}}{2 - (-3) - 4}\)
= \(\frac{-37}{1} = -37\)
Source: Myschool Past Questions Archive (Record #31630)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
If (g(y)) = \(\frac{y - 3}{11}\) + \(\frac{11}{y^2 - 9}\). what is g(y + 3)?
A
\(\frac{y}{11} + \frac{11}{y(y + 6)}\)
B
\(\frac{y}{11} + \frac{11}{y(y + 3)}\)
C
\(\frac{y + 30}{11} + \frac{11}{y(y + 3)}\)
D
\(\frac{y + 3}{11} + \frac{11}{y(y - 6)}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(g(y) = \frac{y - 3}{11} + \frac{11}{y^{2} - 9}\)
\(\therefore g(y + 3) = \frac{(y + 3) - 3}{11} + \frac{11}{(y + 3)^{2} - 9}\)
\(g(y + 3) = \frac{y}{11} + \frac{11}{y^{2} + 6y + 9 - 9}\)
\(g(y + 3) = \frac{y}{11} + \frac{11}{y^{2} + 6y}\)
= \(\frac{y}{11} + \frac{11}{y(y + 6)}\)
Source: Myschool Past Questions Archive (Record #31631)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
Factorize completely \((x^2 + x)^2 - (2x + 2)^2\)
A
(x + 1)(x + 2)(x - 2)
B
(x + 1) 2(x + 2) (x - 2)
C
(x + 1)2 (x + 2)2
D
(x + 1)2 (x + 2)(x - 2)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\((x^{2} + x)^{2} - (2x + 2)^{2}\)
= \((x^{2} + x + 2x + 2)(x^{2} + x - (2x + 2))\)
= \((x^{2} + 3x + 2)(x^{2} - x - 2)\)
= \(((x + 1)(x + 2))((x + 1)(x - 2))\)
= \((x + 1)^{2} (x + 2)(x - 2)\)
Source: Myschool Past Questions Archive (Record #31632)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
Simplify \(\frac{x - y}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\)
A
x2 + xy + y2
B
x\(\frac{2}{3}\) + x \(\frac{1}{3}\) + y\(\frac{2}{3}\)
C
x\(\frac{2}{3}\) - x\(\frac{1}{3}\)y\(\frac{2}{3}\)
D
y\(\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{x - y}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\)
( x - y ) = (\(x^{\frac{1}{3}})^3 - (y^{\frac{1}{3}})^3\) = (\(x^{\frac{1}{3}}\) - \(y^{\frac{1}{3}}\))(\(x^{\frac{2}{3}}\) + \(x^{\frac{1}{3}}\)\(y^{\frac{1}{3}}\) + \(y^{\frac{2}{3}}\))
SO, \(\frac{x - y}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\) = \(\frac{(x^{\frac{1}{3}} - y^{\frac{1}{3}})(x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}})}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\)
Common factor in numerator and denominator cancels out.
Final answer = (\(x^{\frac{2}{3}}\) + \(x^{\frac{1}{3}}\)\(y^{\frac{1}{3}}\) + \(y^{\frac{2}{3}}\))
Source: Myschool Past Questions Archive (Record #31633)
Mathematics
UTME / JAMB 1988
Number and Numeration
Solution
The solutions of x2 - 2x - 1 = 0 are the points of intersection of two graphs. if one of the graphs is y = 2 + x - x2, find the second graph
A
y = 1 - x
B
y = 1 + x
C
y = x - 1
D
y = 3x + 3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
The second graph is
\((x^{2} - 2x - 1) + (2 + x - x^{2})\)
= \(1 - x\)
Source: Myschool Past Questions Archive (Record #31639)