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Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
If P = \(\frac{\frac{2}{3}({1 - r^2})}{n^2}\), find n when r = \(\sqrt{\frac{1}{3}}\) and p = 1
A
\(\frac{3}{2}\)
B
\(\frac{1}{3}\)
C
3
D
\(\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
If P = \(\frac{\frac{2}{3}({1 - r^2})}{n^2}\), find n when r = \(\sqrt{\frac{1}{3}}\) and p = 1
p = \(\frac{\frac{2}{3}({1 - r^2})}{n^2}\) when r = \(\sqrt{\frac{1}{3}}\) and p = 1
1 = \(\frac{\frac{2}{3}({1 - (\sqrt{\frac{1}{3}})^2})}{n^2}\)
\(n^2\) = \(\frac{2}{3}(1 - \frac{1}{3}\))
\(n^2\) = \(\frac{2 \times 2}{3 \times 3}\)
= \(\frac{4}{9}\)
n = \(\sqrt{\frac{4}{9}}\)
= \(\frac{2}{3}\)
Source: Myschool Past Questions Archive (Record #31884)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
If a = u\(^2\) - 3v\(^2\) and b = 2uv + v\(^2\) evaluate (2a - b)(a - b\(^2\)), when u = 1 and v = -1
A
9
B
15
C
27
D
33
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
a = u\(^2\) - 3v\(^2\) = (1)2 - 3(-1) = -2
b = b = 2uv + v\(^2\)
= 2(1)(-1) + (-1)2 = -1
∴ 2(2a - b)(a - b\(^2\)) = [2(-2) + 1](-2 - (-1)\(^2\))
= (-3)( - 3)
= 9
Source: Myschool Past Questions Archive (Record #31885)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
Make y the subject of the formula Z = x\(^2\) + \(\frac{1}{y^3}\)
A
y = \(\frac{1}{(Z - x^2)^3}\)
B
y = \(\frac{1}{(Z + x^2)^{\frac{1}{3}}}\)
C
y = \(\frac{1}{(Z - x^2)^{\frac{1}{3}}}\)
D
y = \(\frac{1}{\sqrt[3]{Z} - \sqrt[3]{x^2}}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Z = x\(^2\) + \(\frac{1}{y^3}\)
Z - x\(^2\) = \(\frac{1}{y^3}\)
y\(^3\) = \(\frac{1}{Z - x^2}\)
y = \(\sqrt[3]{\frac{1}{Z - x^2}}\)
∴ y = \(\frac{1}{\sqrt[3]{Z - x^2}}\)
y = \(\frac{1}{(Z - x^2)^{\frac{1}{3}}}\)
Source: Myschool Past Questions Archive (Record #31893)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
Factorize 62x + 1 + 7(6x) - 5
A
[3(6x) - 5][2(6x)m + 1]
B
[3(6x) + 5][2(6x) - 1]
C
[(6x) - 5][3(6x) + 1]
D
[2(6x) + 5][3(6x) - 1]
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
62x + 1 + 7(6x) - 5
Let 6x = y....(i)
6yx + 7y - 5 = 0
(3y + 5)(2y - 1) = 0.....(i)
Subt. for y from eqn. (i) in eqn. (ii)
= [3(6x) + 5][2(6x) - 1]
Source: Myschool Past Questions Archive (Record #31900)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
An (n - 2)2 sided figure has n diagonals. Find the number n diagonals for 25-sided figure
A
7
B
8
C
9
D
10
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(n - 2)2 = 25
n - 2 = 25 = 5
n = 5 + 2
= 7
Source: Myschool Past Questions Archive (Record #31903)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
The sine, cosine and tangent of 210o are respectively
A
\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\), \(\frac{\sqrt{3}}{2}\)
B
\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\), \(\frac{\sqrt{3}}{3}\)
C
\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\), \(\frac{\sqrt{3}}{2}\)
D
\(\frac{-1}{2}\), \(\frac{\sqrt{-3}}{2}\), \(\frac{\sqrt{3}}{3}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
210o = 180o - 210o = - 30o
From ratio of sides, sin -30o = -\(\frac{1}{2}\)
Cos 210o = 180o - 210o = -30o
= cos -30o = \(\frac{-3}{2}\)
But tan 30o = \(\frac{1}{\sqrt{3}}\), rationalizing this
= \(\frac{1}{\sqrt{3}}\) x \(\frac{\sqrt{3}}{\sqrt{3}}\) = \(\frac{\sqrt{3}}{3}\)
∴ = \(\frac{-1}{2}\), \(\frac{\sqrt{-3}}{2}\), \(\frac{\sqrt{3}}{3}\)
Source: Myschool Past Questions Archive (Record #31933)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
The base of a pyramid is a square of side 8cm. If its vertex is directly above the centre, find the height, given that the edge is 4\(\sqrt3\)cm
A
6cm
B
5cm
C
4cm
D
3cm
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Base of pyramid of a square of side 8cm vertex directly above the centre edge = \(4\sqrt{3}\)cm
From the diagram, the diagonal of one base is AC\(^2\) = 8\(^2\) + 8\(^2\)
AC\(^2\) = 64 + 64 = 128
AC = \(8\sqrt{2}\)
but OC = \(\frac{1}{2}\)AC = \(\frac{1}{2}\) x 8\(\sqrt{2}\) = \(4\sqrt{2}\)cm
OE = h = height
h\(^2\) = EC\(^2\) - OC\(^2\)
h\(^2\) = (\(4\sqrt{3}\))\(^2\) - (\(4\sqrt{2}\))\(^2\)
h\(^2\) = 16 x 3 - 16 x 2
48 - 32 = 16
h = \(\sqrt{16}\)
= 4cm
Source: Myschool Past Questions Archive (Record #31975)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
Below are the scores of group of students in a music test
\(\begin{array}{c|c}scores & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text{No, of students} & 3 & 6 & 10 & 8 & 6 & 5 & 2 & 4 & 12\end{array}\)
If CF(x) is the number of students with scores less than or equal to x, find CF(6)
A
40
B
38
C
33
D
5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
CF (6) = 5 + 6 + 8 + 10 + 6 + 3
= 38
Source: Myschool Past Questions Archive (Record #31992)
Mathematics
UTME / JAMB 1987
Number and Numeration
Solution
An (\(n - 2)^2\) sided figure has n diagonals. Find the number n diagonals for a 25-sided figure
A
7
B
8
C
9
D
10
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(n-2)\(^2\) = 25
n - 2 = 5
n = 5 + 2 = 7
Source: Myschool Past Questions Archive (Record #32820)
Mathematics
UTME / JAMB 1987
Number and Numeration
Diagram
Solution
In the diagram above, QR // TS, QR : TS = 2.3, Find the ratio of the area of triangle PQR to the area of the trapezium QRST.
Attached Figure / Diagram
A
4:9
B
4:5
C
1:3
D
2:3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
h1 = \(\frac{h_1 + h_2}{1.5}\) = \(\frac{h_1}{h_2} = 2\)
\(\frac{A_{\bigtriangleup PQR}}{A_{TP(QRST)}} = \frac{\frac{1}{2} \times 2 \times h_1}{\frac{1}{2} \times h_2 (2 + 3)}\)
\(\frac{2}{5} \times \frac{h_1}{h_2} = \frac{2}{5} \times 2\)
= \(\frac{4}{5}\)
Source: Myschool Past Questions Archive (Record #32824)
Mathematics
UTME / JAMB 1987
Number and Numeration
Diagram
Solution
In the figure, find the value of x
Attached Figure / Diagram
A
110o
B
100o
C
90o
D
80o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Z = X = Y = 180o .........(i)
2Z + Y + Y = 190 = 2Z + 2Y = 180
Z + Y = 90o........(ii)
hence x + (z + y) = 180
x + 90o = 180o
x = 180o - 90o
= 90o
Source: Myschool Past Questions Archive (Record #32828)
Mathematics
UTME / JAMB 1987
Number and Numeration
Diagram
Solution
The figure is an example of the construction of a
Attached Figure / Diagram
A
perpendicular bisector of a given straight line QR
B
perpendicular from a given point to a given line QR
C
perpendicular to a line from a given point P on that line
D
given angle
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
QR is a given line and P is a given point. The construction is the perpendicular from a given point P to a given line QR
Source: Myschool Past Questions Archive (Record #32830)
Mathematics
UTME / JAMB 1986
Number and Numeration
Solution
Find the probability that a number selected at random from 40 to 50 is a prime
A
\(\frac{2}{11}\)
B
\(\frac{5}{11}\)
C
\(\frac{3}{11}\)
D
\(\frac{4}{11}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
From 40 to 50 = these number are prime i.e. 41, 43, 47
prob. of selecting a prime No. is \(\frac{3}{11}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #32176)
Mathematics
UTME / JAMB 1986
Number and Numeration
Solution
Evaluate (212)3 - (121)3 + (222)3
A
(313)3
B
(1000)3
C
(1020)3
D
(1222)3
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Evaluate (212)3 - (121)3 + (222)3
(221)3+
(222)3
1211
121
------
(1020)3
Source: Myschool Past Questions Archive (Record #31994)
Mathematics
UTME / JAMB 1986
Number and Numeration
Solution
If Musa scored 75 in biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?
A
282
B
240
C
222
D
210
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Let x represent Musa's total mark when he scores 57 in biology and Let Y represent Musa's total mark when he now scored 75 in biology, if he scored 75 in biology his new total mark will be \(\frac{Y}{4}\) = 60, y = 4 x 60 = 240
To get his total mark when he scored 57, subtract 57 from 75 to give 18, then subtract this 18 from the new total mark(ie. 240)
= 240 - 18
= 222
Source: Myschool Past Questions Archive (Record #32001)