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- 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
- 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
- What is the remainder when \(2^{31}\) is divided by 5? Answer: D. 3 Answer: (D) 3 Powers of 2 mod 5 repeat in a cycle of 4: 2, 4, 3, 1. 31 = 4 × 7 + 3 , so 2³¹ equiv 2³ = 8 equiv 3 ±od 5 . 1
- What is the area of an equilateral triangle of side 8 cm? Answer: A. 16√(3) cm² Answer: (A) 16√(3) cm² Area = √(3)/4a² = √(3)/4 × 64 = 16√(3) cm². 1
- In a row of 40 students, Meena is 15th from the left end. What is her position from the right end? Answer: D. 26th Answer: (D) 26th Position from right = total − position from left + 1 = 40 − 15 + 1 = 26th. 1
- In a certain code language, FRIEND is written as HUMJTK. How will CANDLE be written in that language? Answer: B. EDRIRL Answer: (B) EDRIRL The letters move forward by +2, +3, +4, +5, +6, +7: F→H, R→U, I→M, E→J, N→T, D→K. CANDLE: C+2 = E, A+3... 1