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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)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
If the hypotenuse of right angled isosceles triangle is 2, what is the length of each of the other sides?
A
\(\sqrt{2}\)
B
\(\frac{1}{2}\)
C
22
D
1
E
2 - 1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
45o = \(\frac{x}{2}\), Since 45o = \(\frac{1}{\sqrt{2}}\)
x = 2 x \(\frac{1}{\sqrt{2}}\)
= \(\frac{2\sqrt{2}}{2}\)
= \(\sqrt{2}\)
Source: Myschool Past Questions Archive (Record #32254)
Mathematics
UTME / JAMB 1985
Permutations, Combinations & Probability
Solution
If two fair coins are tossed, what is the probability of getting at least one head?
A
\(\frac{1}{4}\)
B
\(\frac{1}{2}\)
C
1
D
\(\frac{43}{78}\)
E
\(\frac{3}{4}\)
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
Prob. of getting at least one head
Prob. of getting one head + prob. of getting 2 heads
= \(\frac{1}{4}\) + \(\frac{2}{4}\)
= \(\frac{3}{4}\)
Source: Myschool Past Questions Archive (Record #32258)
Mathematics
UTME / JAMB 1985
Trigonometry, Elevation & Bearings
Solution
The bearing of a bird on a tree from a hunter on the ground is N72oE. What is the bearing of the hunter from the birds?
A
S 18o W
B
S 72o W
C
S 72o E
D
S 27o E
E
S 27o W
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
B = Bird ; H = Hunter.
Bearing of the hunter from the bird = S 72^\circ W.
Source: Myschool Past Questions Archive (Record #32262)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
Without using table, calculate the value of 1 + sec2 30o
A
2\(\frac{1}{3}\)
B
\(\frac{2}{15}\)
C
\(\frac{5}{3}\)
D
3\(\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
1 + sec2 30o = sec 30o
= \(\frac{2}{\sqrt{3}}\)
\(\frac{(2)^2}{3}\)
= \(\frac{4}{3}\)
1 + sec2 30o = sec 30o
= 1 + \(\frac{4}{3}\)
= 2\(\frac{1}{3}\)
Source: Myschool Past Questions Archive (Record #32264)
Mathematics
UTME / JAMB 1985
Permutations, Combinations & Probability
Solution
What is the probability that a number chosen at random from the intergers between 1 and 10 inclusive is either a prime or a multiple of 3?
A
\(\frac{7}{10}\)
B
\(\frac{3}{5}\)
C
\(\frac{4}{5}\)
D
\(\frac{3}{10}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Simple Space: (1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10)
Prime: (2, 3, 5, 7)
multiples of 3: (3, 6, 9)
Using: ∣A∪B ∣= ∣A∣ + ∣B∣ − ∣A∩B∣ = 4 + 3 − 1 =6.
Or
Prime or multiples of 3: (2, 3, 5, 6, 7, 9 = 6)
Probability = \(\frac{6}{10}\)
= \(\frac{3}{5}\)
Source: Myschool Past Questions Archive (Record #32265)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
Find the area of a regular hexagon inscribed in a circle of radius 8cm
A
16\(\sqrt{3}\) cm3
B
96\(\sqrt{3}\) cm3
C
192\(\sqrt{3}\) cm3
D
16\(\sqrt{3}\) cm2
E
33cm2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Area of a regular hexagon = 8 x 8 x sin 60o
= 32 x \(\frac{\sqrt{3}}{2}\)
Area = 16\(\sqrt{3}\) x 6 = 96 \(\sqrt{3}\)cm2
Source: Myschool Past Questions Archive (Record #32266)