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Mathematics
UTME / JAMB 1992
Plane Geometry, Polygons & Circle Theorems
Solution
If the angles of quadrilateral are (p + 10)^\circ, (2p - 30)^\circ, (3p + 20)^\circ and 4p^\circ, find p.
A
63
B
40
C
36
D
28
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
The sum of angles in a quadrilateral = 360^\circ
\(\therefore (p + 10) + (2p - 30) + (3p + 20) + 4p = 360\)
\(10p = 360^\circ \implies p = \frac{360}{10} = 36^\circ\)
Source: Myschool Past Questions Archive (Record #31335)
Mathematics
UTME / JAMB 1992
Trigonometry, Elevation & Bearings
Solution
Determine the distance on the earth's surface between two town P (lat 60^\circN, Long 20^\circE) and Q(Lat 60^\circN, Long 25^\circW) (Radius of the earth = 6400km)
A
\(\frac{800\pi}{9}\)km
B
\(\frac{800\sqrt{3\pi}}{9}\)km
C
800\(\pi\) km
D
800\(\sqrt{3\pi}\) km
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Angular difference (\(\theta\))= 25^\circ + 20^\circ = 45^\circ
\(\alpha\) = common latitude = 60^\circ
\(S = \frac{\theta}{360^\circ} \times 2\pi R \cos \alpha\)
\(S = \frac{45^\circ}{360^\circ} \times 2 \pi \times 6400 \times \cos 60^\circ\)
= \(\frac{6400\pi}{8} = 800\pi km\)
Source: Myschool Past Questions Archive (Record #31337)
Mathematics
UTME / JAMB 1992
Plane Geometry, Polygons & Circle Theorems
Diagram
If in the diagram, FG is parallel to KM, find the value of x
Attached Figure / Diagram
A
75o
B
95o
C
105o
D
125o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option B.
Source: Myschool Past Questions Archive (Record #31338)
Mathematics
UTME / JAMB 1992
Plane Geometry, Polygons & Circle Theorems
The locus of a point which is equidistant from two given fixed points is the
A
perpendicular bisector of the straight line joining them
B
parallel line to the straight line joining them
C
transverse to the straight line joining them
D
angle bisector of 90o which the straight line joining them makes with the horizontal
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option A.
Source: Myschool Past Questions Archive (Record #31341)
Mathematics
UTME / JAMB 1992
Indices, Logarithms & Surds
Solution
What is the perpendicular distance of a point (2, 3) from the line 2x - 4y + 3 = 0?
A
\(\frac{\sqrt{5}}{2}\)
B
\(\frac{\sqrt{5}}{20}\)
C
\(\frac{5}{\sqrt{13}}\)
D
6
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
2x - 4y + 3 = 0
Required distance = \(\frac{(2 \times 2) + 3(-4) + 3}{\sqrt{2^2} + (-4)^2}\)
= \(\frac{4 - 12 + 3}{\sqrt{20}}\)
= \(\frac{-5}{-2\sqrt{5}}\)
= \(\frac{\sqrt{5}}{2}\)
Source: Myschool Past Questions Archive (Record #31342)
Mathematics
UTME / JAMB 1992
Plane Geometry, Polygons & Circle Theorems
Solution
find then equation line through (5, 7) parallel to the line 7x + 5y = 12
A
5x + 7y = 120
B
7x + 5y = 70
C
x + y = 7
D
15x + 17y = 90
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Equation (5, 7) parallel to the line 7x + 5y = 12
5Y = -7x + 12
y = \(\frac{-7x}{5}\) + \(\frac{12}{5}\)
Gradient = \(\frac{-7}{5}\)
∴ Required equation = \(\frac{y - 7}{x - 5}\) = \(\frac{-7}{5}\) i.e. 5y - 35 = -7x + 35
5y + 7x = 70
Source: Myschool Past Questions Archive (Record #31347)
Mathematics
UTME / JAMB 1992
Trigonometry, Elevation & Bearings
Solution
Given that \(\theta\) is an acute angle and sin \(\theta\) = \(\frac{m}{n}\), find cos \(\theta\)
A
\(\frac{\sqrt{n^2 - m^2}}{m}\)
B
\(\frac{\sqrt{(n + m)(n - m)}}{n}\)
C
\(\frac{m}{\sqrt{n^2 - m^2}}\)
D
\(\sqrt{\frac{n}{n^2 - m^2}}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
sin \(\theta\) = \(\frac{m}{n}\)
Opp = m; Hyp = n
Adj = \(\sqrt{n^{2} - m^{2}}\)
\(\cos \theta = \frac{\sqrt{n^{2} - m^{2}}}{n}\)
= \(\frac{\sqrt{(n + m)(n - m)}}{n}\)
Source: Myschool Past Questions Archive (Record #31349)
Mathematics
UTME / JAMB 1992
Linear, Simultaneous & Quadratic Equations
Solution
Evaluate \(\lim \limits_{x \to 2} \frac{(x - 2)(x^2 + 3x - 2)}{x^2 - 4}\)
A
7
B
2
C
3
D
4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\lim \limits_{x \to 2} \frac{(x - 2)(x^2 + 3x - 2)}{x^2 - 4}\)
\(\frac{(x - 2)(x^{2} + 3x - 2)}{x^{2} - 4} = \frac{(x - 2)(x^{2} + 3x - 2)}{(x - 2)(x + 2)}\)
= \(\frac{(x^{2} + 3x - 2)}{x + 2}\)
\(\therefore \lim \limits_{x \to 2} \frac{(x - 2)(x^2 + 3x - 2)}{x^2 - 4} = \lim \limits_{x \to 2} \frac{x^{2} + 3x - 2}{x + 2}\)
= \(\frac{2^{2} + 3(2) - 2}{2 + 2}\)
= \(\frac{4 + 6 - 2}{4}\) = 2
Source: Myschool Past Questions Archive (Record #31350)
Mathematics
UTME / JAMB 1992
Trigonometry, Elevation & Bearings
Solution
If y = x sin x, Find \(\frac{d^2 y}{d^2 x}\)
A
2 cosx - x sinx
B
sinx + x cosx
C
sinx - x cosx
D
x sinx - 2 cosx
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(y = x \sin x\)
\(\frac{\mathrm d y}{\mathrm d x} = x \cos x + \sin x\)
\(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = x (- \sin x) + \cos x + \cos x\)
= \(2 \cos x - x \sin x\)
Source: Myschool Past Questions Archive (Record #31352)
Mathematics
UTME / JAMB 1992
Integration & Applications
Solution
Ice forms on a refrigerator ice-box at the rate of (4 - 06t)g per minute after t minutes. If initially there are 2g of ice in the box, find the mass of ice formed in 5 minutes
A
19.5
B
17.0
C
14.5
D
12.5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{dm}{dt}\) = 4 - 0.6t
\(\int\)dm = \(\int\)(4 - 0.6t)dt
m = \(4t - 0.3t^2 + c\), when t = 0, m = 2g
∴ c = 2
m = \(4t - 0.3t^2 + 2\), when t = 5 minutes
m = \(4(5) - 0.3(5)^2 + 2 = 20 - 7.5 + 2\)
= 14.5
Source: Myschool Past Questions Archive (Record #31353)
Mathematics
UTME / JAMB 1992
Differentiation & Applications
Solution
Obtain a maximum value of the function f(x) x\(^3\) - 12x + 11
A
-5
B
-2
C
2
D
27
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
f(x) = x\(^3\) - 12x + 11
\(\frac{df(x)}{dx)}\) = 3x\(^2\) - 12 = 0
∴ 3x\(^2\) - 12 = 0 \(\to\) x\(^2\) = 4
x = \(\pm\)2, f(+2) = 8 - 24 + 11 = -15
= f(-2) = (-8) + 24 + 11
= 35 - 8 = 27
∴ maximum value = 27
Source: Myschool Past Questions Archive (Record #31354)
Mathematics
UTME / JAMB 1992
Integration & Applications
Solution
a student blows a balloon and its volume increases at a rate of \(\pi\)(20 - t2)cm3S-1 after t seconds. If the initial volume is 0 cm3, find the volume of the balloon after 2 seconds
A
37.00\(\pi\)
B
37.33\(\pi\)
C
40.00\(\pi\)
D
42.67\(\pi\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{dv}{dt}\) = \(\pi\)(20 - t2)cm2S-1
\(\int\)dv = \(\pi\)(20 - t2)dt
V = \(\pi\) \(\int\)(20 - t2)dt
V = \(\pi\)(20 \(\frac{t}{3}\) - t3) + c
when c = 0, V = (20t - \(\frac{t^3}{3}\))
after t = 2 seconds
V = \(\pi\)(40 - \(\frac{8}{3}\)
= \(\pi\)\(\frac{120 - 8}{3}\)
= \(\frac{112}{3}\)
= 37.33\(\pi\)
Source: Myschool Past Questions Archive (Record #31355)
Mathematics
UTME / JAMB 1992
Integration & Applications
Solution
Evaluate the integral \(\int^{\frac{\pi}{4}}_{\frac{\pi}{12}} 2 \cos 2x \mathrm {d} x\)
A
-\(\frac{1}{2}\)
B
-1
C
\(\frac{1}{2}\)
D
1
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\int_{\frac{\pi}{12}} ^{\frac{\pi}{4}} 2 \cos 2x \mathrm {d} x\)
= \([\frac{2 \sin 2x}{2}]|_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\)
= \(\sin 2x |_{\frac{\pi}{12}} ^{\frac{\pi}{4}}\)
= \(\sin 2(\frac{\pi}{4}) - \sin 2(\frac{\pi}{12})\)
= \(\sin \frac{\pi}{2} - \sin \frac{\pi}{6}\)
= \(1 - \frac{1}{2} = \frac{1}{2}\)
Source: Myschool Past Questions Archive (Record #31356)
Mathematics
UTME / JAMB 1992
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
\(\begin{array}{c|c} x & 2 & 4 & 6 & 8\\ \hline f & 4 & y & 6 & 5 \end{array}\)
If the mean of the above frequency distribution is 5.2, find y
A
6.0
B
5.2
C
5.0
D
4.0
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Mean \(\bar{x}\) = \(\frac{\sum fx}{\sum f}\)
= \(\frac{5.2}{1}\)
= \(\frac{8 + 4y + 36 + 40}{4 + y + 6 + 5}\)
= \(\frac{5.2}{1}\) = \(\frac{84 + 4y}{15 + y}\)
= 5.2(15 + y) = 84 + 4y
= 84 + 4y = 78 + 5.2y
= 5.2y - 4y = 84 - 78
1.2y = 6
y = \(\frac{6}{1.2}\)
= \(\frac{60}{12}\)
= 5
Source: Myschool Past Questions Archive (Record #31358)
Mathematics
UTME / JAMB 1992
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
\(\begin{array}{c|c} \text{No. of children} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text{No. of families} & 7 & 11 & 6 & 7 & 7 & 5 & 3 \end{array}\)
Find the mode and median respectively of the distribution above
A
2, 1
B
1, 2
C
1, 5
D
5, 2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
From the table, the mode = 1.
The median = 2.
Source: Myschool Past Questions Archive (Record #31359)