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Mathematics
UTME / JAMB 1986
Fractions, Decimals, Percentages & Approximations
Solution
A number of pencils were shared out among Bisi, Sola and Tunde in the ratio of 2 : 3 : 5 respectively. If Bisi got 5, how many were share out?
A
15
B
25
C
30
D
50
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Let x r3epresent total number of pencils shared
B : S : T = 2 + 3 + 5 = 10
2 : 3 : 5
= \(\frac{2}{10}\) x y
= 5
2y =5
2y = 50
∴ y = \(\frac{50}{2}\)
= 25
Source: Myschool Past Questions Archive (Record #32026)
Mathematics
UTME / JAMB 1986
Plane Geometry, Polygons & Circle Theorems
Solution
The ages of Tosan and Isa differ by 6 and the product of their ages is 187. Write their ages in the form (x, y), where x > y.
A
(12, 6)
B
(23, 17)
C
(17, 11)
D
(18, 12)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
x - y = 6.......(i)
xy = 187.......(ii)
From equation (i), x(6 + y)
sub. for x in equation (ii) = y(6 + y)
= 187
y2 + 6y = 187
y2 + 6y - 187 = 0
(y + 17)(y - 11) = 0
y = -17 or y = 11
y cannot be negative, y = 11
Sub. for y in equation(i) = x - 11
= 16
x = 6 + 11
= 17
∴(x, y) = (17, 11)
Source: Myschool Past Questions Archive (Record #32030)
Mathematics
UTME / JAMB 1986
Plane Geometry, Polygons & Circle Theorems
Solution
In 1984, Ike was 24 yrs old and his father was 45 yrs old. In what year was Ike exactly half his father's age?
A
1981
B
1979
C
1982
D
1978
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Let the no. of years be y
24 - y = \(\frac{1}{2}\)(45 - y)
45 - y = 2(24 - y)
45 - y = 48 - 2y
2y - y = 48 - 45
∴ y = 3
The exact year = 1984
1984 - 3 = 1981
Source: Myschool Past Questions Archive (Record #32031)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Simplify \((\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}}\)
A
\(\frac{\sqrt{3}}{\sqrt{5}}\)
B
\(\frac{2 \sqrt{3}}{7}\)
C
-2
D
-1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\((\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}}\)
\(\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}\)
\(\frac{(\sqrt{5} - \sqrt{3}) - (\sqrt{5} + \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}\)
= \(\frac{\sqrt{5} - \sqrt{3} - \sqrt{5} - \sqrt{3}}{5 - \sqrt{15} + \sqrt{15} - 3}\)
= \(\frac{-2\sqrt{3}}{2}\)
= \(- \sqrt{3}\)
\(\therefore (\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}} = - \sqrt{3} \times \frac{1}{\sqrt{3}}\)
= \(-1\)
Source: Myschool Past Questions Archive (Record #32036)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Find n if lo\(g_2\) 4 + lo\(g_2\) 7 - lo\(g_2\) n = 1
A
10
B
14
C
27
D
28
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
log\(_2\) 4 + log\(_2\) 7 - log\(_2\) n = 1
= log\(_2\) (4 x 7) - log\(_2\) n = 1
\(\therefore\) log\(_2\) 28 - log\(_2\) n = 1
= \(\frac{28}{n} = 2^1\)
\(\frac{28}{n}\) = 2
2n = 28
∴ n = 14
Source: Myschool Past Questions Archive (Record #32041)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Simplify \(\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{-\frac{1}{6}} \times 3^{\frac{-2}{3}}}\)
A
\(\frac{1}{3}\)
B
1
C
3
D
9
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\) = \(\frac{(3^2)^{\frac{1}{3}} \times (3^3)^{-\frac{1}{3}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\)
= \(\frac{3^{\frac{2}{3}} \times 3^{-\frac{3}{2}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\)
= \(\frac{3^{-\frac{5}{6}}}{3^{-\frac{5}{6}}}\)
= 1
Source: Myschool Past Questions Archive (Record #32057)
Mathematics
UTME / JAMB 1986
Variations
Solution
If x varies directly as y3 and x = 2 when y = 1, find x when y = 5
A
2
B
10
C
125
D
250
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x \(\alpha\) y3
x = ky3
k = \(\frac {x}{y^3}\)
when x = 2, y = 1
k = 2
Thus x = 2y3 - equation of variation
= 2(5)3
= 250
Source: Myschool Past Questions Archive (Record #32063)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Factorize completely 8a + 125ax\(^3\)
A
(2a + 5x2)(4 + 26ax)
B
a(2 + 5x)(4 - 10x + 25x2)
C
(2a + 5x)(4 - 10ax + 25x2)
D
a(2 + 5x)(4 + 10ax + 25x2)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(8a + 125ax^{3} = (2^{3})a + 5^{3} ax^{3}\)
= \(a(2^3 + 5^3 x^3)\)
∴\(a[2^3 + (5x)^3]\)
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
∴ \(a(2^3 + (5x)^3)\)
= \(a(2 + 5x)(4 - 10x + 25x^2)\)
Source: Myschool Past Questions Archive (Record #32076)
Mathematics
UTME / JAMB 1986
Plane Geometry, Polygons & Circle Theorems
Solution
If y = \(\frac{x}{x - 3}\) + \(\frac{x}{x + 4}\) find y when x = -2
A
-\(\frac{3}{5}\)
B
\(\frac{3}{5}\)
C
-\(\frac{7}{5}\)
D
\(\frac{2}{5}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
y = \(\frac{x}{x - 3}\) + \(\frac{x}{x + 4}\) when x = -2
y = \(\frac{-2}{-5}\) + \(\frac{(-2)}{-2 + 4}\)
= \(\frac{2}{5}\) + \(\frac{-2}{2}\)
= \(\frac{4 -10}{10}\)
= \(\frac{-6}{10}\)
= -\(\frac{3}{5}\)
Source: Myschool Past Questions Archive (Record #32079)
Mathematics
UTME / JAMB 1986
Inequalities & Linear Programming
Solution
Find all real numbers x which satisfy the inequality \(\frac{1}{3}\)(x + 1) - 1 > \(\frac{1}{5}\)(x + 4)
A
x < 11
B
x < -1
C
x > 6
D
x > 11
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{1}{3}\)(x + 1) - 1 > \(\frac{1}{5}\)(x + 4)
= \(\frac{x + 1}{3}\) - 1 > \(\frac{x + 4}{5}\)
\(\frac{x + 1}{3}\) - \(\frac{x + 4}{5}\) - 1 > 0
= \(\frac{5x + 5 - 3x - 12}{15}\)
= 2x - 7 > 15
= 2x > 22
= x > 11
Source: Myschool Past Questions Archive (Record #32080)
Mathematics
UTME / JAMB 1986
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize \(x^2 + 2a + ax + 2x\)
A
(x + 2a)(x + 1)
B
(x + 2a)(x - 1)
C
(x2 - 1)(x - a)
D
(x + 2)(x + a)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(x^{2} + 2a + ax + 2x\)
\(x^{2} + 2a + ax + 2x\)
\(x(x + 2) + a(x + 2)\)
\((x + 2)(x + a)\)
Source: Myschool Past Questions Archive (Record #32081)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Solve the equation 3x\(^2\) + 6x - 2 = 0
A
x = -1 \(\pm\) \(\frac{\sqrt{3}}{3}\)
B
x = -1 \(\pm\) \(\frac{\sqrt{15}}{3}\)
C
x = -2 \(\pm\) 2
D
x = 3 \(\pm\) \(\frac{\sqrt{3}}{15}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
3x2 + 6x - 2 = 0
Using almighty formula i.e. x = \(\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\)
a = 3, b = 6, c = -2
x = \(\frac{-6 \pm \sqrt{6^2 - 4(3)(-2)}}{2(3)}\)
x = \(\frac{-6 \pm \sqrt{36 + 24}}{6}\)
x = \(\frac{-6 \pm \sqrt{60}}{6}\)
x = \(\frac{-6 \pm \sqrt{4 \times 15}}{6}\)
x = \(-1 \pm \frac{\sqrt{15}}{3}\)
Source: Myschool Past Questions Archive (Record #32086)
Mathematics
UTME / JAMB 1986
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify \(\frac{1}{5x + 5}\) + \(\frac{1}{7x+ 7}\)
A
\(\frac{12}{35x + 1}\)
B
\(\frac{1}{35(x + 1)}\)
C
\(\frac{12x}{35(x + 7)}\)
D
\(\frac{12}{35x + 35}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{1}{5x + 5}\) + \(\frac{1}{7x+ 7}\) = \(\frac{1}{5(x + 1)}\) + \(\frac{1}{7(x + 1)}\)
= \(\frac{7 + 5}{35(x + 1)}\)
= \(\frac{12}{35(x + 1)}\)
Source: Myschool Past Questions Archive (Record #32090)
Mathematics
UTME / JAMB 1986
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize (4a + 3)2 - (3a - 2)2
A
(a + 1)(a + 5)
B
(a - 5)(7a - 1)
C
(a + 5)(7a + 1)
D
a(7a + 1)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(4a + 3)2 - (3a - 2)2 = a2 - b2
= (a + b)(a - b)
= [(4a + 3) + (3a - 2)][(4a + 3) + (3a - 2)]
= [(4a + 3 + 3a - 2)][(4a + 3 - 3a + 2)]
= (7a + 1)(a + 5)
∴ (a + 5)(7a + 1)
Source: Myschool Past Questions Archive (Record #32105)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
If \(5^{(x + 2y)} = 5\) and \(4^{(x + 3y)} = 16\), find \(3^{(x + y)}\).
A
7
B
1
C
3
D
27
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(5^{(x + 2y)} = 5\)
∴ x + 2y = 1.....(i)
\(4^{(x + 3y)} = 16 = 4^2\)
x + 3y = 2 .....(ii)
x + 2y = 1.....(i)
x + 3y = 2......(ii)
y = 1
Substitute y = 1 into equation (i)
\(x + 2y = 1 \implies x + 2(1) = 1\)
\(x + 2 = 1 \implies x = -1\)
\(\therefore 3^{(x + y)} = 3^{(-1 + 1)}\)
\(3^{0} = 1\)
Source: Myschool Past Questions Archive (Record #32111)