Algebra
Top questions
See all →- If \(x - \frac{1}{x} = 4\), find the value of \(x^3 - \frac{1}{x^3}\). Answer: A. 76 Answer: (A) 76 x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x) = 64 + 12 = 76 . 1
- If \(2x + 3y = 13\) and \(3x - 2y = 0\), what is the value of \(x + y\)? Answer: D. 5 Answer: (D) 5 From 3x = 2y , y = 3x/2 . Then 2x + 9x/2 = 13 ⇒ 13x/2 = 13 ⇒ x = 2,\ y = 3 . So x + y = 5 . 1
- If \(x + \frac{1}{x} = 6\), find the value of \(x^2 + \frac{1}{x^2}\). Answer: A. 34 Answer: (A) 34 x² + 1/x² = (x + 1/x)² - 2 = 36 - 2 = 34 . 1
- If \(a + b + c = 9\) and \(ab + bc + ca = 23\), find the value of \(a^3 + b^3 + c^3 - 3abc\). Answer: D. 108 Answer: (D) 108 a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)] = 9(81 - 69) = 9 × 12 = 108 . 1
- If \(x = \sqrt{3} + \sqrt{2}\), what is the value of \(x^3 + \frac{1}{x^3}\)? Answer: B. 18√(3) Answer: (B) 18√(3) 1/x = √(3) - √(2) , so x + 1/x = 2√(3) . x³ + 1/x³ = (2√(3))³ - 3(2√(3)) = 24√(3) - 6√(3) = 18√(3) . 1
- If x + y + z = 0 and xy + yz + zx ≠ 0, what is the value of \(\frac{x^2 + y^2 + z^2}{xy + yz + zx}\)? Answer: D. -2 Answer: (D) -2 (x + y + z)² = 0 gives x² + y² + z² = -2(xy + yz + zx) . So the ratio is −2. 1