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Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
22\(\frac{1}{2}\)% of the Nigerian Naira is equal to 17\(\frac{1}{10}\)% of a foreign currency M. What is the conversion rate of the M to the Naira?
A
1M = 1\(\frac{15}{57}\)N
B
1M = 38\(\frac{1}{4}\)N
C
1M = 1\(\frac{18}{57}\)N
D
1M = 384\(\frac{3}{4}\)N
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
N = 22\(\frac{1}{2}\)%, M = 17\(\frac{1}{10}\)%
M = \(\frac{171}{10}\)%, N = \(\frac{45}{2}\)
\(\frac{45}{2}\) x \(\frac{10}{171}\)
= \(\frac{225}{171}\)
= 1 \(\frac{54}{171}\)
= 1 \(\frac{18}{57}\)
Source: Myschool Past Questions Archive (Record #32218)
Mathematics
UTME / JAMB 1985
Linear, Simultaneous & Quadratic Equations
Solution
Find the values of p for which the equation x\(^2\) - (p - 2)x + 2p + 1 = 0
A
(21, 0)
B
(0, 12)
C
(1, 2)
D
(3, 4)
E
(4, 5)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Equal roots implies b\(^2\) - 4ac = 0
a = 1, b = - (p - 2), c = 2p + 1
[-(p - 2)]\(^2\) - 4 x 1 x (2p + 1) = 0
p\(^2\) - 4p + 4 - 4(2p + 1) = 0
p\(^2\) - 4p = 4 - 8p - 4 = 0
p\(^2\) - 12p = 0
p(p - 12) = 0
p = 0 or 12
Source: Myschool Past Questions Archive (Record #32219)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
If \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ... \), find \(\frac{1}{e^{\frac{1}{2}}}\)
A
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
B
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{2.4.3}\) + ..
C
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2}\) + \(\frac{x^3}{1.2.3}\) + \(\frac{x^4}{1.23.4}\) + .........
D
1 - x + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
E
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{1.2.6}\) + .........
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ...\)
\(\frac{1}{e^{\frac{1}{2}}} = e^{-\frac{1}{2}}\)
\(e^{-\frac{1}{2}} = 1 - \frac{x}{2} + \frac{x^{2}}{1.2^{3}} - \frac{x^{3}}{1.2^{4}.3} + ... \)
Source: Myschool Past Questions Archive (Record #32220)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) is equal to
A
1
B
(\(\sqrt{2} + 4\sqrt{2}\))
C
(6\(\sqrt{2}\)
D
8
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\)
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2}) = \sqrt{3} - \sqrt[4]{6} + \sqrt[4]{6} - \sqrt{2}\)
= \(\sqrt{3} - \sqrt{2}\)
\((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 + \sqrt{6} - \sqrt{6} - 2\)
= \(3 - 2 = 1\)
Source: Myschool Past Questions Archive (Record #32221)
Mathematics
UTME / JAMB 1985
Variations
Solution
In a restaurant, the cost of providing a particular type of food is partly constant and partially inversely proportional to the number of people. If cost per head for 100 people is 30k and the cost for 40 people is 60k, Find the cost for 50 people?
A
15k
B
20k
C
50k
D
40k
E
45k
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
C = a + k
\(\frac{1}{N}\) = c
= \(\frac{aN + k}{N}\)
CN = aN + K
30(100) = a(100) + k
3000 = 100a + k.......(i)
60(40) = a(40) + k
2400 = 40a + k.......(ii)
eqn (i) - eqn (ii)
600 = 60a
a = 10
subt. for a in eqn (i) 3000 = 100(10) + K
3000 - 1000 = k
k = 2000
CN = 10N + 2000. when N = 50,
50C = 10(50) + 2000
50C = 500 + 2000
C = \(\frac{2500}{50}\)
= 50k
Source: Myschool Past Questions Archive (Record #32222)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
The factors of 9 - (x2 - 3x - 1)2 are
A
-(x - 4)(x + 1) (x - 1)(x - 2)
B
(x - 4)(x - 2) (x - 1)(x + 1)
C
-(x - 2)(x + 1) (x - 2) (x - 1)
D
(x - 2)(x + 2) (x - 1)(x + 1)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
9 - (x2 - 3x - 1)2 = [3 - (x2 - 3x - 1)] [3 + (x2 - 3x - 1)]
= (3 - x2 + 3x + 1)(3 + x2 - 3x - 1)
= (4 + 3x - x2)(x2 - 3x + 2)
= (4 - x)(1 + x)(x - 1)(x - 2)
= -(x - 4)(x + 1) (x - 1)(x - 2)
Source: Myschool Past Questions Archive (Record #32223)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
If 32y + 6(3y) = 27. Find y
A
3
B
-1
C
2
D
-3
E
1
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
32y + 6(3y) = 27
This can be rewritten as (3y)2 + 6(3y) = 27
Let 3y = x
x2 + 6x - 27 = 0
(x + 9)(x - 3) = 0
when x - 3 = 0, x = 3
sub. for x in 3y = x
3y = 3
log33 = y
y = 1
Source: Myschool Past Questions Archive (Record #32224)
Mathematics
UTME / JAMB 1985
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize abx2 + 8y - 4bx - 2axy
A
(ax - 4)(bx - 2y)
B
(ax + b)(x - 8y)
C
(ax - 2y)(bx - 4)
D
(bx - 4)(ax - 2y)
E
(abx - 4)(x - 2y)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
abx2 + 8y - 4bx - 2axy = (abx2 - 4bx) + (8y - 2axy)
= bx(ax - 4) 2y(ax - 4) 2y(ax - 4)
= (bx - 2y)(ax - 4)
Source: Myschool Past Questions Archive (Record #32225)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
At what real value of x do the curves whose equations are y = x3 + x and y = x2 + 1 intersect?
A
-2
B
2
C
-1
D
9
E
1
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
y = x3 + x and y = x2 + 1
\(\begin{array}{c|c} x & -2 & -1 & 0 & 1 & 2 \\ \hline Y = x^3 + x & -10 & -2 & 0 & 2 & 10 \\ \hline y = x^2 + 1 & 5 & 2 & 1 & 2 & 5\end{array}\)
The curves intersect at x = 1
Source: Myschool Past Questions Archive (Record #32226)
Mathematics
UTME / JAMB 1985
Linear, Simultaneous & Quadratic Equations
Solution
If the quadratic function 3x2 - 7x + R is a perfect square, find R
A
\(\frac{49}{24}\)
B
\(\frac{49}{12}\)
C
\(\frac{49}{13}\)
D
\(\frac{49}{3}\)
E
\(\frac{49}{36}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
3x2 - 7x + R. Computing the square, we have
x2 - \(\frac{7}{3}\) = -\(\frac{R}{3}\)
(\(\frac{x}{1} - \frac{7}{6}\))2 = -\(\frac{R}{3}\) + \(\frac{49}{36}\)
\(\frac{-R}{3}\) + \(\frac{49}{36}\) = 0
R = \(\frac{49}{36}\) x \(\frac{3}{1}\)
= \(\frac{49}{12}\)
Source: Myschool Past Questions Archive (Record #32227)
Mathematics
UTME / JAMB 1985
Linear, Simultaneous & Quadratic Equations
Solution
Solve for (x, y) in the equation 2x + y = 4: x^2 + xy = -12
A
(6, -8): (-2, 8)
B
(3, -4): (-1, 4)
C
(8, -4): (-1, 4)
D
(-8, 6): (8, -2)
E
(-4, 3): (4, -1)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
2x + y = 4......(i)
x^2 + xy = -12........(ii)
from eqn (i), y = 4 - 2x
= x2 + x(4 - 2x)
= -12
x2 + 4x - 2x2 = -12
4x - x2 = -12
x2 - 4x - 12 = 0
(x - 6)(x + 2) = 0
sub. for x = 6, in eqn (i) y = -8, 8
=(6,-8); (-2, 8)
Source: Myschool Past Questions Archive (Record #32228)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
If f(x - 2) = 4x2 + x + 7, find f(1)
A
12
B
27
C
7
D
46
E
17
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
f(x - 2) = 4x2 + x + 7
x - 2 = 1, x = 3
f(x - 2) = f(1)
= 4(3)2 + 3 + 7
= 36 + 10
= 46
Source: Myschool Past Questions Archive (Record #32231)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
In \(\bigtriangleup\) XYZ, XY = 13cm, YZ = 9cm, XZ = 11cm and XYZ = \(\theta\). Find cos\(\theta\)o
A
\(\frac{4}{39}\)
B
\(\frac{43}{39}\)
C
\(\frac{209}{3}\)
D
\(\frac{43}{78}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
cos\(\theta\) = \(\frac{13^2 + 9^2 - 11^2}{2(13)(9)}\)
= \(\frac{169 + 81 - 21}{26 \times 9}\)
cos\(\theta\) = \(\frac{129}{26 \times 9}\)
= \(\frac{43}{78}\)
Source: Myschool Past Questions Archive (Record #32244)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
Find the missing value in the table below
\(\begin{array}{c|c} x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y = 4 - 3x - x^3 & 80 & & 18 & 8 & 4 & 0 & -10 & -32 \end{array}\)
A
-32
B
-14
C
40
D
22
E
37
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
When x = -3, y = 4 - 3(3) - (-3)3
= 4 + 9 + 27
= 13 + 27
= 40
Source: Myschool Past Questions Archive (Record #32248)
Mathematics
UTME / JAMB 1985
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
The number of goals scored by a football team in 20 matches is shown below
\(\begin{array}{c|c} \text{No. of goals} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text{No. of matches} & 3 & 5 & 7 & 4 & 1 & 0 \end{array}\)
What are the values of the mean and the mode respectively?
A
(1.75, 5)
B
(1.75, 2)
C
(1.75, 1)
D
(65, 74)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\begin{array}{c|c}x & f & fx\\ \hline 0 & 3 & 0\\ 1 & 5 & 5\\ 2 & 7 & 14\\ 3 & 4 & 12\\4 & 1 & 4\\5 & 0 & 0\end{array}\)
\(\sum{f}\) = 10
\(\sum fx\) = 35
Mean = \(\frac{\sum fx}{\sum f}\)
= \(\frac{35}{20}\)
= \(\frac{7}{4}\)
= 1.75
Mode = 2
= 1.75, 2
Source: Myschool Past Questions Archive (Record #32252)