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Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
simplify \(\frac{\frac{7}{9}-\frac{2}{3}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\)
A
\(\frac{11}{12}\)
B
\(\frac{5}{6}\)
C
\(\frac{1}{5}\)
D
\(\frac{2}{15}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{\frac{7}{9}-\frac{2}{3}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\\
=\frac{\frac{7-6}{9}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\\
=\frac{\frac{1}{9}}{\frac{1}{3}+\left(\frac{2}{5}\times \frac{5}{4}\right)}\\
=\frac{\frac{1}{9}}{\frac{1}{3}+\frac{1}{2}}\
=\frac{\frac{1}{9}}{\frac{2+3}{6}}\\
=\frac{\frac{1}{9}}{\frac{5}{6}}\\
=\frac{1}{9}\times\frac{6}{5}\\
=\frac{2}{15}\)
Source: Myschool Past Questions Archive (Record #8741)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
If m:n = 13:11, find m\(^2\) - n\(^2\) : (m + n)\(^2\)
A
1:11
B
1:13
C
1:10
D
1:12
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
m : n = 13 : 11
m\(^2\) - n\(^2\) : (m + n)2
= 13\(^2\) - 11\(^2\) : (13 + 11)\(^2\)
= 169 - 121 : 24\(^2\)
= 48 : 576
= 1 : 12
Source: Myschool Past Questions Archive (Record #8742)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Calculate the logarithm to base 9 of 3\(^{-4}\) x 9\(^2\) x (81)\(^{-1}\)
A
2
B
zero
C
-2
D
-4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(3^{-4}\times 9^2 \times 81^{-1}\)
=\(log_9 (3^{-4}\times 9^2 \times 81^{-1})\)
= \(log_9 \left(\frac{1}{3^4}\times 9^2 \times \frac{1}{81}\right)\)
= \(log_9 \left(\frac{1}{81}\times \frac{81}{1}\times \frac{1}{81}\right)\)
= \(log_9 \frac{1}{81}\)
= \(log_9 \frac{1}{9^2}\)
= \(log_9 9^{-2}\)
= -2\(log_9 9\)
- 2 \(\times 1\)
= -2
Source: Myschool Past Questions Archive (Record #8747)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
If (K2)\(_6\) * 3\(_6\) = 3\(_5\)(K4)\(_5\), what is the value of k?
A
1
B
4
C
3
D
2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
(K2)\(_6\) * 3\(_6\) = 3\(_5\)(K4)\(_5\)
(K*6\(^1\) + 2*6\(^0\)) 3*6\(^0\) = 3*5\(^0\)(K*5\(^1\) + 4*5\(^0\))
3(6K + 2) = 3(5K + 4)
6K + 2 = 5K + 4
6K - 5K = 4 - 2
K = 2
Source: Myschool Past Questions Archive (Record #8753)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
In a small village of 500 people, 350 speak the local language while 200 speak pidgin English. What percentage of the population speak both.
A
30%
B
10%
C
50%
D
14%
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Let n(L∩P) = x
n(L) only = 350 - x
n(P) only = 200 - x
∴350 - x + x + 200 - x = 500
550 - x = 500
x = 550 - 500
x = 50 speak both
∴x% = (50/500) * (100/1)
= 10%
Source: Myschool Past Questions Archive (Record #8827)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Find the tax on an income of N20,000 if no tax is paid on the first N10,000 and tax is paid at N50 in N1000 on the next N5000 and at N55 in N1000 on the remainder
A
N500
B
N552
C
N255
D
N525
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
N20,000 - N10,000 = N10,000
1st tax on N5000 at N50 per N1000
= 5*50 = N250
2nd tax on the remaining N5000 at N55 per N1000 = 5*55 = N275
Total tax paid = N250 + N275
= N525
Source: Myschool Past Questions Archive (Record #8828)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Evaluate \(\frac{2}{6-5\sqrt{3}}\)
A
\(-\left(\frac{12}{39}-\frac{10\sqrt{3}}{39}\right)\)
B
\(\frac{12}{39}-\frac{10\sqrt{3}}{39}\)
C
\(-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
D
\(\frac{12}{39}+\frac{10\sqrt{3}}{39}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{2}{6-5\sqrt{3}} = \frac{2}{6-5\sqrt{3}} \times \frac{6+5\sqrt{3}}{6+5\sqrt{3}}\\
=\frac{2(6+5\sqrt{3})}{(6-5\sqrt{3})(6+5\sqrt{3})}\\
=\frac{12+10\sqrt{3}}{36-25(3)}\\
=\frac{12+10\sqrt{3}}{36-75}\\
=\frac{12+10\sqrt{3}}{-39}\\
=-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
Source: Myschool Past Questions Archive (Record #8829)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Compute 1100112 + 111112
A
10010102
B
10100102
C
10001102
D
10001002
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
1100112 + 111112 = 10100102
Source: Myschool Past Questions Archive (Record #8830)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Simplify \((25)^{\frac{-1}{2}} \times (27)^{\frac{1}{3}} + (121)^{\frac{-1}{2}} \times (625)^{\frac{-1}{4}}\)
A
34/55
B
9/11
C
14/5
D
3/275
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(25)\(^{\frac{-1}{2}}\) x (27)\(^{\frac{1}{3}}\) + (121)\(^{\frac{-1}{2}}\) x (625)\(^{\frac{-1}{4}}\)
5\(^{2 \times \frac{-1}{2}}\) x 3\(^{3 \times \frac{1}{3}}\) + 11\(^{2 \times \frac{-1}{2}}\) x 5\(^{4 \times \frac{-1}{4}}\)
5\(^{-1}\) x 3\(^1\) x 11\(^{-1}\) x 5\(^{-1}\)
\(\frac{1}{5} \times \frac{3}{1} + \frac{1}{11} \times \frac{1}{5}\)
\(\frac{3}{5} + \frac{1}{55} = \frac{33+1}{55}\)
= \(\frac{34}{55}\)
Source: Myschool Past Questions Archive (Record #8831)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Convert 2232\(_4\) to base six
A
4506
B
2546
C
5536
D
5406
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
1st convert to base 10
2232\(_4\) = 2 x 4\(^3\) + 2 x 4\(^2\) + 3 x 4\(^1\) + 2 x 4\(^0\)
= 2 x 64 + 2 x 16 + 3 x 4 + 2 x 1
= 128 + 32 + 12 + 2
= 174 convert to base 6
6/174
6/29 R 0
6/4 R 5
6/0 R 4
= 450\(_6\)
Source: Myschool Past Questions Archive (Record #8832)
Mathematics
UTME / JAMB 2006
Number and Numeration
Diagram
Solution
The solution set of the shaded area above is
Attached Figure / Diagram
A
y ≥ 0, y ≥ x and y + x ≤ 4
B
y ≤ x, y + x ≤ 4
C
y + x ≥ 4, y ≤ x
D
y ≤ x, y + x ≤ 4 and y ≥ 0
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
y = x implies y \le x
y + x = 4 implies y = 4 – x
∴y = 4 – x
∴y \le x, y + x \le 4 and y \ge 0
Source: Myschool Past Questions Archive (Record #9008)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
A binary operation \(\oplus\) defined on the set of real number is such that x\(\oplus\)y = xy/6 for all x, y ∈ R. Find the inverse of 20 under this operation when the identity element is 6
A
1/12
B
10/3
C
1/20
D
9/5
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x ⊕ y = xy/6 = xe/6 = x
xe = 6x
e = 6
x ⊕ y = xx1/6 where x1 = 20
x x 20/6 = 6
x = 36/20
x = 9/5
Source: Myschool Past Questions Archive (Record #9014)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
If p varies inversely as the cube of q and q varies directly as the square of r, what is the relationship between p and r?
A
p varies directly as r3
B
p varies inversely as r6
C
p varies inversely as 6√P
D
p varies directly as r6
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
P ∝ 1/q3
P = K/q3
q3 = K/P
q = K/p1/3
But q ∝ r2
q = Kr2
K/p1/3 = Kr2
r2 = K/p1/3 x 1/K
r2 = 1/p1/3
r = (1/p1/3)2
r = 1/p1/6
r ∝ 1/p1/6 = p = \(\frac{1}{r^6}\)
∴ p varies inversely as \(r^6\)
Source: Myschool Past Questions Archive (Record #9035)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
A binary operation * on the set of rational numbers is defined as \(x \ast y = \frac{x^2 - y^2}{2xy}\). Find \(-5 \ast 3\)
A
\(\frac{-8}{15}\)
B
\(\frac{8}{15}\)
C
\(\frac{17}{15}\)
D
\(\frac{-17}{15}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(x \ast y = \frac{x^2 - y^2}{2xy}\)
=\(\frac{(x+y)(x-y)}{2xy}\)
-5 \(\ast 3 =\frac{(-5+3)(-5-3)}{2(-5\times3)}\)
=\(\frac{-2 \times -8}{2(-5\times3)}\)
=\(\frac{-8}{15}\)
Source: Myschool Past Questions Archive (Record #9036)
Mathematics
UTME / JAMB 2006
Number and Numeration
Solution
Find the value of k if the expression kx3 + x2 - 5x - 2 leaves a remainder 2 when it is divided by 2x + 1
A
10
B
8
C
-10
D
-8
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
F(x) = Q x D + R
Kx3 + x2 - 5x – 2 = Q(2x+1)+R
If 2x+1 = 0 implies x = -1/2
∴k(-1/2)3 + (-1/2)2 -5(-1/2) -2
= Q(2(-1/2) + 1) +2
K(-1/8) + 1/4 + 5/2 - 2 = Q(-1+1)+2
-k/8 + 1/4 + 5/2 - 2 = 0+2
(-k+2+20-16) / 8 = 2
(-k+6) / 8 = 2
-k+6 = 2*8
-k+6 = 16
-k = 16-6
-k = 10
∴k = -10
Source: Myschool Past Questions Archive (Record #9037)