Algebra
- 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
- If \(x^2 - 4x + 1 = 0\), what is the value of \(x^3 + \frac{1}{x^3}\)? Answer: D. 52 Answer: (D) 52 Divide by x: x - 4 + 1/x = 0 , so x + 1/x = 4 . x³ + 1/x³ = 4³ - 3 × 4 = 64 - 12 = 52 . 1
- If \(x^4 + \frac{1}{x^4} = 119\) and \(x > 1\), what is the value of \(x^3 - \frac{1}{x^3}\)? Answer: B. 36 Answer: (B) 36 (x² + 1/x²)² = x⁴ + 1/x⁴ + 2 = 121 , so x² + 1/x² = 11 . (x - 1/x)² = x² + 1/x² - 2 = 9 . Since x > 1 , x - 1/x... 1
- If \(x + \frac{1}{x} = 4\), what is the value of \(x^2 + \frac{1}{x^2}\)? Answer: A. 14 Answer: (A) 14 Shortcut: x² + 1/x² = (x + 1/x)² - 2 = 16 - 2 = 14 . 1
- If a + b = 10 and ab = 21, what is the value of \(a^2 + b^2\)? Answer: C. 58 Answer: (C) 58 a² + b² = (a + b)² - 2ab = 100 - 42 = 58 . 1
- What is \((a + b)^2 - (a - b)^2\) equal to? Answer: C. 4ab Answer: (C) 4ab (a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab . 1
- Solve: 3x − 7 = 2x + 5. Answer: A. 12 Answer: (A) 12 3x − 2x = 5 + 7, so x = 12. 1
- What is the value of 103 × 97? Answer: C. 9991 Answer: (C) 9991 Shortcut: (100 + 3)(100 − 3) = 100² − 3² = 10000 − 9 = 9991. 1
- If \(x - \frac{1}{x} = 3\), what is the value of \(x^2 + \frac{1}{x^2}\)? Answer: D. 11 Answer: (D) 11 x² + 1/x² = (x - 1/x)² + 2 = 9 + 2 = 11 . Note the + 2 here (it is − 2 when the sum x + 1/x is given). 1
- What is the value of \(999^2\)? Answer: A. 9,98,001 Answer: (A) 9,98,001 Shortcut: (1000 - 1)² = 1000000 - 2000 + 1 = 9,98,001. 1
- If 2x + 3y = 12 and 3x + 2y = 13, what is the value of x + y? Answer: A. 5 Answer: (A) 5 Shortcut: add the equations: 5x + 5y = 25, so x + y = 5. (Here x = 3, y = 2.) 1
- If a + b + c = 0 (with a, b, c non-zero), what is the value of \(\frac{a^3 + b^3 + c^3}{abc}\)? Answer: B. 3 Answer: (B) 3 Identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) . When a + b + c = 0, a³ + b³ + c³ = 3ab... 1
- If \(x + \frac{1}{x} = 3\), what is the value of \(x^3 + \frac{1}{x^3}\)? Answer: A. 18 Answer: (A) 18 Shortcut: x³ + 1/x³ = k³ - 3k where k = 3: 27 − 9 = 18. 1
- If a − b = 4 and ab = 12, what is the value of \(a^3 - b^3\)? Answer: B. 208 Answer: (B) 208 a³ - b³ = (a - b)³ + 3ab(a - b) = 64 + 3 × 12 × 4 = 208 . 1
- If \(x = 3 + 2\sqrt{2}\), what is the value of \(x^2 + \frac{1}{x^2}\)? Answer: D. 34 Answer: (D) 34 1/x = 1/3 + 2√(2) = 3 - 2√(2) (multiply by the conjugate; 9 − 8 = 1). So x + 1/x = 6 and x² + 1/x² = 36 - 2 = 3... 1
- If x + y + z = 12 and xy + yz + zx = 44, what is the value of \(x^2 + y^2 + z^2\)? Answer: D. 56 Answer: (D) 56 x² + y² + z² = (x + y + z)² - 2(xy + yz + zx) = 144 - 88 = 56 . 1
- If a + b = 8 and \(a^3 + b^3 = 152\), what is the value of ab? Answer: A. 15 Answer: (A) 15 a³ + b³ = (a + b)³ - 3ab(a + b) : 152 = 512 − 24ab, so ab = 360 ÷ 24 = 15. 1
- The sum of two numbers is 25 and their difference is 7. What is their product? Answer: C. 144 Answer: (C) 144 Shortcut: xy = (x + y)² - (x - y)²/4 = 625 - 49/4 = 144 . (The numbers are 16 and 9.) 1
- If x : y = 3 : 4, what is the value of \(\frac{7x - 4y}{3x + y}\)? Answer: A. 5/13 Answer: (A) 5/13 Put x = 3, y = 4: 21 - 16/9 + 4 = 5/13 . 1