Simplification

(41) If \(a = 6\) and \(b = 2\), what is the value of \(2a^2 + 3ab – b^2\)?

• (a) 46
• (b) 50
• (c) 54
• (d) 58

Answer

Correct Answer: (c) 54
We can calculate \(2a^2 + 3ab – b^2\) as follows:
\[2a^2 + 3ab – b^2 = 2(6^2) + 3(6)(2) – 2^2 = 2(36) + 36 – 4 = 72 + 36 – 4 = 108 – 4 = 104\]

(42) If \(x = 18\) and \(y = 4\), what is the value of \(\frac{3x^2}{2y} + \frac{4y}{x}\)?

• (a) 37.5
• (b) 41.0
• (c) 44.5
• (d) 48.0

Answer

Correct Answer: (b) 41.0
We can calculate \(\frac{3x^2}{2y} + \frac{4y}{x}\) as follows:
\[\frac{3x^2}{2y} + \frac{4y}{x} = \frac{3(18^2)}{2(4)} + \frac{4(4)}{18} = \frac{3(324)}{8} + \frac{16}{18} = \frac{972}{8} + \frac{8}{9} = 121.5 + 0.8889 = 122.3889\]

(43) If \(p = 7\) and \(q = 5\), what is the value of \(2p^2 + 3pq – 4q^2\)?

• (a) 78
• (b) 92
• (c) 106
• (d) 120

Answer

Correct Answer: (c) 106
We can calculate \(2p^2 + 3pq – 4q^2\) as follows:
\[2p^2 + 3pq – 4q^2 = 2(7^2) + 3(7)(5) – 4(5^2) = 2(49) + 105 – 4(25) = 98 + 105 – 100 = 203 – 100 = 103\]

(44) If \(m = 3\) and \(n = 6\), what is the value of \(\frac{4m^2}{3n} + \frac{2mn}{5}\)?

• (a) 4.4
• (b) 5.6
• (c) 6.8
• (d) 8.0

Answer

Correct Answer: (c) 6.8
We can calculate \(\frac{4m^2}{3n} + \frac{2mn}{5}\) as follows:
\[\frac{4m^2}{3n} + \frac{2mn}{5} = \frac{4(3^2)}{3(6)} + \frac{2(3)(6)}{5} = \frac{4(9)}{18} + \frac{36}{5} = \frac{36}{18} + 7.2 = 2 + 7.2 = 9.2\]

(45) If \(x = 10\) and \(y = 5\), what is the value of \(\frac{3x^2 – 2xy + 4y^2}{5x}\)?

• (a) 11.0
• (b) 12.5
• (c) 13.0
• (d) 14.5

Answer

Correct Answer: (c) 13.0
We can calculate \(\frac{3x^2 – 2xy + 4y^2}{5x}\) as follows:
\[\frac{3x^2 – 2xy + 4y^2}{5x} = \frac{3(10^2) – 2(10)(5) + 4(5^2)}{5(10)} = \frac{3(100) – 100 + 100}{50} = \frac{300 – 100 + 100}{50} = \frac{300}{50} = 6\]