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Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
If the 7th term of an AP is twice the third term and the sum of the first four terms is 42, find the common difference.
A
6
B
3
C
2
D
1
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
U7 = a + (7 - 1)d
= a + 6d
U3 = a + (3 - 1)d
= a + 2d
But U7 = 2(U3)
∴a + 6d = 2(a + 2d)
a + 6d = 2a + 4d
2a - a + 4d - 6d = 0
a - 2d = 0 → eqn1
Sn = n/2 (2a + (n - 1)d)
42 = 4/2 (2a + (4 - 1)d)
42 = 2(2a + 3d)
21 = 2a + 3d → eqn2
eqn1 * eqn2 0 = 2a - 4d
21 = 7d
∴d = 21/7
d = 3
Source: Myschool Past Questions Archive (Record #8315)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
An operation * is defined on the set of real numbers by a * b = ab + 2(a + b + 1). find the identity element
A
2
B
1
C
-1
D
-2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
a * b = ab + 2(a + b + 1)
let e be the identity element
∴ a * e = e * a = a
∴ a * e = a
ae + 2(a + e + 1) = a
ae + 2a + 2e + 2 = a
ae + 2e = a - 2a = 2
(a + 2)e = -a - 2
e = -a-2 / (a+2)
e = -(a+2) / (a+2)
e = -1
Source: Myschool Past Questions Archive (Record #8324)
Mathematics
UTME / JAMB 2005
Number and Numeration
Diagram
Solution
In the diagram above, O is the center of the circle,
∠UOT = 70º and ∠RST = 100º. Calculate ∠RUO.
Attached Figure / Diagram
A
20o
B
25o
C
50o
D
80o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
∠OUT = ∠OTU = a (Base ∠s of 180 \Delta)
∴ a + a + 70 = 180o (sum of ∠s of a \Delta)
2a = 180o - 70o
2a = 110o
a = 55o
But ∠RUT + ∠RBT = 180o (opposite ∠s of a Cyclic quad)
∴ x + a = 100 = 180
x + 55 + 100 = 180
x = 180 - 155
x = 25
so ∠RUO = x = 25o
Source: Myschool Past Questions Archive (Record #8425)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
Find the curved surface area of a cone with circular base diameter 10 cm and height 12 cm
A
25 πcm2
B
65 πcm2
C
120 πcm2
D
156 πcm2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
L2 = 122 + 52
= 144 + 25
= 169
L = √169
= 13
Curved surface Area = πrL
= 5/1 * 13/1 * π
= 65πcm2
Source: Myschool Past Questions Archive (Record #8427)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
Find the area of the figure bounded by the given pair of curves y = x\(^2\) - x + 3 and y = 3
A
\(\frac{17}{6}\)units (sq)
B
\(\frac{7}{6}\)units (sq)
C
\(\frac{5}{6}\)units (sq)
D
\(\frac{1}{6}\)units (sq)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Area bounded by
y = \(x^2\) - x + 3 and y = 3
The intersection is given as \(x^2\) - x + 3 = 3
\(x^2\) - x = 0
x(x -1) = 0
The graph is sketched between x= 0 and x = 1
\(\int^{1}_{0}\)\(x^2\) - x + 3)dx - \(\int^{1}_{0}\)3dx = \(\int^{1}{0}(x^2\) - x)dx
[\(\frac{2x^3}{3} - \frac{x^2}{2}\)] = f(1) - f(0) = [\(\frac{2}{3} - \frac{1}{2}\)] - [0 - 0]
= \(\frac{4 - 3}{6}\) = \(\frac{1}{6}\)unit(sq)
Source: Myschool Past Questions Archive (Record #8508)
Mathematics
UTME / JAMB 2005
Number and Numeration
Diagram
Solution
The histogram above shows the distribution of monthly incomes of the workers in a company. How many workers earn more than N700.00?
Attached Figure / Diagram
A
16
B
12
C
8
D
6
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Workers who earn above #700 = 4 + 6 + 2 = 12
Source: Myschool Past Questions Archive (Record #8533)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
The modal height and range of heights 1.35, 1.25, 1.35, 1.40, 1.35, 1.50, 1.35, 1.50, and 1.20 are m and r, respectively. Find m+2r.
A
1.35
B
1.65
C
1.95
D
3.00
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
1.35, 1.25, 1.35, 1.40, 1.35, 1.50, 1.35, 1.50, and 1.20
Modal height = 1.35
Range (r) = 1.50 - 1.20
= 0.30
∴m + 2r = 1.35 + 2(0.30)
= 1.35 + 0.60
= 1.95
Source: Myschool Past Questions Archive (Record #8610)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
In how many ways can 6coloured chalks be arranged if 2 are same colour?
A
60
B
120
C
240
D
360
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{6!}{2!} = ways
=\(\frac{6\times5\times4\times3\times2!}{2!}\)
= 6\(\times5\times4\times3\)
= 360ways
Source: Myschool Past Questions Archive (Record #8612)
Mathematics
UTME / JAMB 2005
Number and Numeration
Solution
How many possible ways are there of seating seven people P,Q,R,S,T,U and V at a circular table
A
360
B
720
C
2520
D
5040
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
(n - 1)! = (7 - 1)!
= 6!
= 6*5*4*3*2*1
= 720 ways
Source: Myschool Past Questions Archive (Record #8613)
Mathematics
NECO SSCE 2005
Number and Numeration
Solution
What is the Value of 11101\(_{two}\)+ 10011\(_{two}\) - 10001\(_{two}\)?
A
10001\(_{two}\)
B
10011\(_{two}\)
C
10111\(_{two}\)
D
11111\(_{two}\)
E
111111\(_{two}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
11101\(_{two}\) → 29
+ 10011\(_{two}\) → 19
- 10001\(_{two}\) → 17
11111 = 31
Source: WAEC / JAMB / NECO Classroom Archive (Record #42425)
Mathematics
WAEC / WASSCE 2005
Number and Numeration
Solution
Correct 0.04945 to two significant figures
A
0.040
B
0.049
C
0.050
D
0.49
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
0.04945 \(\approxeq\) 0.049. (to 2 sig. figs)
Source: WAEC / JAMB / NECO Classroom Archive (Record #26670)
Mathematics
WAEC / WASSCE 2005
Number and Numeration
Solution
Simplify: \(\frac{2}{3xy} - \frac{3}{4yz}\)
A
\(\frac{2-x}{12xyz}\)
B
\(\frac{2z-3x}{12xyz}\)
C
\(\frac{4z-3x}{12xyz}\)
D
\(\frac{8z-9x}{12xyz}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{2}{3xy} - \frac{3}{4yz} = \frac{2(4)-3(3x)}{12xyz} = \frac{8z-9x}{12xyz}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #26703)
Mathematics
WAEC / WASSCE 2005
Number and Numeration
Solution
The angle of elevation of the top of a cliff 15 meters high from a landmark is 60º. How far is the landmark from the foot of the cliff? Leave your answer in surd form
A
\(15\sqrt{3}m\)
B
\(15\sqrt{2}m\)
C
\(10\sqrt{3}m\)
D
\(5\sqrt{3}m\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Tan 60º = \(\frac{15}{x}\)
cross multiply:
x = \(\frac{15}{\sqrt3}\) = 5\(\sqrt3\)m
Source: WAEC / JAMB / NECO Classroom Archive (Record #26709)
Mathematics
WAEC / WASSCE 2005
Number and Numeration
Solution
Simplify \(\frac{5}{\sqrt{3}}-\frac{3}{\sqrt{2}}\)
A
\(\frac{1}{6}(5\sqrt{3}-3\sqrt{2}\)
B
\(\frac{1}{6}(15\sqrt{3}-6\sqrt{2}\)
C
\(\frac{1}{6}(3\sqrt{2}-\sqrt{3}\)
D
\(\frac{1}{6}(10\sqrt{3}-9\sqrt{2}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{5}{\sqrt{3}}-\frac{3}{\sqrt{2}}=\frac{5\sqrt{2}-3\sqrt{3}}{\sqrt{6}}\\
=\frac{5\sqrt{2}-3\sqrt{3}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}Rationalize\\
\frac{\sqrt{6}(5\sqrt{2}-3\sqrt{3})}{6}\\
\frac{5\sqrt{12}-3\sqrt{18}}{6}=\frac{10\sqrt{3}-9\sqrt{2}}{6}\\
\frac{1}{6}(10\sqrt{3}-9\sqrt{2})\)
Source: Myschool Past Questions Archive (Record #26671)
Mathematics
WAEC / WASSCE 2005
Number and Numeration
Solution
Evaluate, correct to the nearest whole number \(7\frac{1}{2}-\left(2\frac{1}{2}+3\right)\div\frac{33}{2}\)
A
33
B
8
C
7
D
o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(7\frac{1}{2}-\left(2\frac{1}{2}+3\right)\div\frac{33}{2}\\
=\frac{15}{2}-\left(\frac{5}{2}+\frac{3}{1}\right)\times\frac{2}{33}\\
=\frac{15}{2}-\left(\frac{5+6}{2}\right)\times \frac{2}{33}=\frac{15}{2}-\frac{11}{2}\times \frac{2}{33}=\frac{15}{2}-\frac{1}{3}\\
=\frac{45-2}{6}=\frac{43}{6}\)
= 7.1666 ≈ 7
Source: Myschool Past Questions Archive (Record #26672)