Quantitative aptitude
- 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
- The table shows the percentage distribution of 2,400 students among five colleges.CollegePQRSTStudents (%)1525203010How many more students are there in college S than in college P? Answer: B. 360 Answer: (B) 360 S − P = 30% − 15% = 15% of 2400 = 360. 1
- The table shows the percentage distribution of 2,400 students among five colleges.CollegePQRSTStudents (%)1525203010What is the ratio of the number of students in college Q to the total number of students in colleges R and T together? Answer: B. 5 : 6 Answer: (B) 5 : 6 Q : (R + T) = 25 : (20 + 10) = 25 : 30 = 5 : 6. (The total cancels out.) 1
- Two fair dice are thrown together. What is the probability that the sum of the numbers is 8? Answer: B. 5/36 Answer: (B) 5/36 Favourable pairs: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 out of 36. Probability = 5/36 . 1
- The present ages of A and B are in the ratio 5 : 7. After 6 years, the ratio will be 3 : 4. What is B's present age? Answer: D. 42 years Answer: (D) 42 years 5x + 6/7x + 6 = 3/4 ⇒ 20x + 24 = 21x + 18 ⇒ x = 6 . B = 7 × 6 = 42 years (A = 30). Check: 36 : 48 =... 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
- A boat takes 3 hours more to travel 45 km upstream than to travel 45 km downstream. If the speed of the stream is 2 km/h, what is the speed of the boat in still water? Answer: D. 8 km/h Answer: (D) 8 km/h Let the boat's speed be b. 45/b - 2 - 45/b + 2 = 3 ⇒ 180/b² - 4 = 3 ⇒ b² = 64 , b = 8 km/h. Check: upst... 1