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Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
Find the value of log10 r + log10 r2 + log10 r4 + log10 r8 + log10 r16 + log10 r32 = 63
A
10-1
B
10o
C
10
D
102
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
log10 r + log10 r2 + log10 r4 + log10 r8 + log10 r16 + log10 r32 = 63
log10r63 = 63
63 = 1063
∴ r = 10
Source: Myschool Past Questions Archive (Record #31014)
Mathematics
UTME / JAMB 1994
Sequences & Series (AP & GP)
Solution
Find the nth term of the sequence 3, 6, 10, 15, 21.....
A
\(\frac{n(n - 1)}{2}\)
B
\(\frac{n(n + 1)}{2}\)
C
\(\frac{(n + 1)(n + 2)}{2}\)
D
n(2n + 1)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{(n + 1)(n + 2)}{2}\)
If n = 1, the expression becomes 3
n = 2, the expression becomes 6
n = 4, the expression becomes 15
n = 5, the expression becomes 21
Source: Myschool Past Questions Archive (Record #31015)
Mathematics
UTME / JAMB 1994
Binary Operations, Matrices & Determinants
Solution
A binary operation \(\oplus\) is defines on the set of all positive integers by a \(\oplus\) b = ab for all positive integers a, b. Which of the following properties does NOT hold?
A
closure
B
identity
C
positive
D
inverse
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
a \(\oplus\) b = ab
The set of all national rules Q, is closed under the operations, additions, subtraction, multiplication and division. Since a \(\oplus\) b = ab; b \(\oplus\) a = ba = ab
The number 1 is the identity element under multiplication
Source: Myschool Past Questions Archive (Record #31016)
Mathematics
UTME / JAMB 1994
Number Bases & Modular Arithmetic
Solution
\(\begin{array}{c|c} \oplus mod 10 & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 8 & 2 & 6 \\4 & 8 & 6 & 4 & 2\\ 4 & 8 & 6 & 4 & 2\\ 6 & 2 & 4 & 6 & 8\\ 8 & 6 & 2 & 8 & 4\end{array}\)
The multiplication table above has modulo 10 on the set S = (2, 4, 6, 8). Find the inverse of 2
A
2
B
4
C
6
D
8
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
The inverse of 2 is 6 since 2 x 6 = 12; under mod 10
12 = 2 which is also the value required
Source: Myschool Past Questions Archive (Record #31017)
Mathematics
UTME / JAMB 1994
Linear, Simultaneous & Quadratic Equations
Solution
Solve for x and y \(\begin{pmatrix} 1 & 1 \\ 3 & y \end{pmatrix}\)\(\begin{pmatrix} x \\ 1 \end{pmatrix}\) = \(\begin{pmatrix} 4 \\ 1\end{pmatrix}\)
A
x = 3, y = 8
B
x = 3, y = -8
C
x = -8, y = 3
D
x = 8, y = -3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\begin{pmatrix} 1 & 1 \\ 3 & y \end{pmatrix}\)\(\begin{pmatrix} x \\ 1 \end{pmatrix}\) = \(\begin{pmatrix} 4 \\ 1\end{pmatrix}\) = x + 1 = 4
x = 4 - 1
= 3
3x + y =1
3(3) = y = 1
= 9 + y = 1
y = 1 - 9
= -8
Source: Myschool Past Questions Archive (Record #31018)
Mathematics
UTME / JAMB 1994
Permutations, Combinations & Probability
Solution
Age in years
10
11
12
Number of pupils
6
27
7
The table above shows the number of pupils in each age group in a class. What is the probability that a pupil chosen at random is at least 11 years old?
A
\(\frac{27}{40}\)
B
\(\frac{17}{20}\)
C
\(\frac{33}{40}\)
D
\(\frac{3}{20}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Total number of pupils = 7 + 27 + 6 = 40
n(at least 11 years old) = 27 + 7 = 34.
P(picking a pupil who is at least 11 years old) = \(\frac{34}{40} = \frac{17}{20}\).
Source: Myschool Past Questions Archive (Record #31020)
Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
If three angles of a quadrilateral are (3y - x - z)o, 3xo, (2z - 2y - x)o find the fourth angle in terms of x, y and z
A
(360 - x- y - z)o
B
(360 + x + y - z)o
C
(180 - x + y + z)o
D
(180 - x + y + z)o
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
The sum of angles of a quadrilateral is 360o
∴ (3y - x - z)o + 3xo + (2z - 2y - x)o + po = 360o
Where P is the fourth angle
3y - x - z + 3x + 2z - 2y - x + p = 360o
p = 360 - (x + y + z)
∴ p = (360 - x - y - z)o
Source: Myschool Past Questions Archive (Record #31021)
Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
An open rectangular box is made of wood 2cm thick. If the internal dimensions of the box are 50cm long, 36cm wide and 20cm deep, the box volume of wood in the box is
A
11 520cm3
B
36 000cm3
C
38 200cm3
D
47 520cm3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Internal dimension are 50cm, 36cm and 20cm
internal volume = 50 x 36 x 20cm3
1000 x 36cm3
= 36000cm3
External dimension are 54cm x 40cm x 22cm
= 2160cm2 x 22cm = 47520cm3
Volume of wood = Ext. volume - Int. volume
= 47,520cm3 - 36,000cm3
= 11,520cm3
Source: Myschool Past Questions Archive (Record #31024)
Mathematics
UTME / JAMB 1994
Mensuration (Areas, Perimeters & Volumes)
Solution
Calculate the perimeter, in cm, of a sector of a circle of radius 8cm and angle 45o
A
2\(\pi\)
B
8 + 2\(\pi\)
C
16 + 2\(\pi\)
D
16 + 16\(\pi\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Perimeter = OP + OQ + PQ
= 8 + 8 + PQ
length PQ = \(\frac{\theta}{360 \times 2\pi r}\)
= \(\frac{45}{360}\) x 2 x \(\pi\) x 8
= 2\(\pi\)
Perimeter of sector 2r + L
Where l = length of arc and r = radius
∴ P = 2(8) + 2\(\pi\)
= 16 + 2\(\pi\)
Source: Myschool Past Questions Archive (Record #31038)
Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
What is the locus of a point P which moves on one side of a straight line XY, so that the angle XPY is always equal to 90o?
A
the perpendicular bisector of XY
B
a right-angled triangle
C
a circle
D
a semi circle
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Since XY is a fixed line and
XPY = 90o P is on one side of XY
P1P2P3......Pn are all possible cases where
XPY = 90o the only possible tendency is a semicircle because angles in semicircle equals 90o
Source: Myschool Past Questions Archive (Record #31040)
Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
If M(4, q) is the mid-point of the line joining L(p, -2) and N(q, p). Find the values of p and q
A
P = 2, q = 4
B
p = 3, q = 1
C
p = 5, q = 3
D
p = 6, q = 2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{p + q}{2}\) = 4
p + q = 8 ....(i)
\(\frac{p - 2}{2}\) = q
p - 2q 2 ....(ii)
q = 2, p = 6
3q = 6
Source: Myschool Past Questions Archive (Record #31041)
Mathematics
UTME / JAMB 1994
Trigonometry, Elevation & Bearings
Solution
The angle of depression of a boat from the top of a cliff 10m high is 30. How far is the boat from the foot of the cliff?
A
\(\frac{5√3m}{3}\)
B
5√3m
C
10√3m
D
\(\frac{10√3m}{3}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
RQP = 90o
PQS = 30o but
RQS = 90o = \(\frac{RS}{10}\)
RS = 10 tan 60o
RS = 10 tan 60o
RS = 10√3m
Source: Myschool Past Questions Archive (Record #31042)
Mathematics
UTME / JAMB 1994
Trigonometry, Elevation & Bearings
Solution
What is the value of sin(-690)?
A
\(\frac{\sqrt{3}}{2}\)
B
-\(\frac{\sqrt{3}}{2}\)
C
\(\frac{-1}{2}\)
D
\(\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Sin(-690o) = Sin(-360 -3300
sin - 360 = sin 0
∴ sin(-690o) = sin(330o)
Negative angles are measured in clockwise direction
The acute angle equivalent of sin(-330o) = sin(30o)
sin(-330o) = sin(30o)
= \(\frac{1}{2}\)
= 0.5
Source: Myschool Past Questions Archive (Record #31044)
Mathematics
UTME / JAMB 1994
Plane Geometry, Polygons & Circle Theorems
Solution
If y = 3t3 + 2t2 - 7t + 3, find \(\frac{dy}{dt}\) at t = -1
A
-1
B
1
C
-2
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
y = 3t3 + 2t2 - 7t + 3
\(\frac{dy}{dt}\) = 9t2 + 4t - 7
When t = -1
\(\frac{dy}{dt}\) = 9(-1)2 + 4(-1) - 7
= 9 - 4 -7
= 9 - 11
= -2
Source: Myschool Past Questions Archive (Record #31047)
Mathematics
UTME / JAMB 1994
Differentiation & Applications
Solution
Find the point (x, y) on the Euclidean plane where the curve y = 2x2 - 2x + 3 has 2 as gradient
A
(1, 3)
B
(2, 7)
C
(0, 3)
D
(3, 15)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Equation of curve;
y = 2x2 - 2x + 3
gradient of curve;
\(\frac{dy}{dx}\) = differential coefficient
\(\frac{dy}{dx}\) = 4x - 2, for gradient to be 2
∴ \(\frac{dy}{dx}\) = 2
4x - 2 = 2
4x = 4
∴ x = 1
When x = 1, y = 2(1)2 - 2(1) + 3
= 2 - 2 + 3
= 5 - 2
= 3
coordinate of the point where the curve; y = 2x2 - 2x + 3 has gradient equal to 2 is (1, 3)
Source: Myschool Past Questions Archive (Record #31056)