Algebra
- If \(x + \frac{1}{x} = 3\), what is the value of \(x^4 + \frac{1}{x^4}\)? Answer: B. 47 Answer: (B) 47 x² + 1/x² = 9 - 2 = 7 . Then x⁴ + 1/x⁴ = 7² - 2 = 47 . 1
- What is the value of \(\frac{0.75^3 + 0.25^3}{0.75^2 - 0.75 \times 0.25 + 0.25^2}\)? Answer: C. 1 Answer: (C) 1 a³ + b³/a² - ab + b² = a + b = 0.75 + 0.25 = 1. 1
- If a = 11 and b = 9, what is the value of \(\frac{a^2 + ab + b^2}{a^3 - b^3}\)? Answer: D. 1/2 Answer: (D) 1/2 a³ - b³ = (a - b)(a² + ab + b²) , so the expression = 1/a - b = 1/2 . 1
- If \((x - 2)^2 + (y - 3)^2 + (z - 4)^2 = 0\), what is the value of x + y + z? Answer: B. 9 Answer: (B) 9 A sum of squares is 0 only if each square is 0. So x = 2, y = 3, z = 4 and x + y + z = 9. 1
- If a + b + c = 6, \(a^2 + b^2 + c^2 = 14\) and \(a^3 + b^3 + c^3 = 36\), what is the value of abc? Answer: C. 6 Answer: (C) 6 ab + bc + ca = 36 - 14/2 = 11. a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) = 6 × (14 − 11) = 1... 1
- If \(x - \frac{1}{x} = 3\), find the value of \(x^4 + \frac{1}{x^4}\). Answer: B. 119 Answer: (B) 119 x² + 1/x² = 3² + 2 = 11 . x⁴ + 1/x⁴ = 11² - 2 = 119 . 1
- If \(3x - 2y = 5\) and \(xy = 2\), find the value of \(27x^3 - 8y^3\). Answer: A. 305 Answer: (A) 305 a³ - b³ = (a - b)³ + 3ab(a - b) with a = 3x, b = 2y : ab = 6xy = 12 . 27x³ - 8y³ = 5³ + 3 × 12 × 5 = 125 + 18... 1
- If \(x + \frac{1}{x} = 5\), find the value of \(x^3 + \frac{1}{x^3}\). Answer: B. 110 Answer: (B) 110 x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) = 125 - 15 = 110 . 1
- If \(a + b + c = 6\), \(ab + bc + ca = 11\) and \(abc = 6\), find \(a^3 + b^3 + c^3\). Answer: D. 36 Answer: (D) 36 a³ + b³ + c³ = (a + b + c)³ - 3(a + b + c)(ab + bc + ca) + 3abc . = 216 - 3 × 6 × 11 + 3 × 6 = 216 - 198 + 18 =... 1