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- The average monthly income of a family with four earning members was ₹15,000. When one of them retired and stopped earning, the average income of the remaining three became ₹13,000. What was the income of the member who retired? Answer: A. ₹21,000 Answer: (A) ₹21,000 4 × 15,000 − 3 × 13,000 = 60,000 − 39,000 = ₹21,000. 1
- The average weight of 8 people rises by 2.5 kg when a new person takes the place of one who weighs 65 kg. What is the weight of the new person? Answer: A. 85 kg Answer: (A) 85 kg Shortcut: new person = person replaced + (number of people × rise) = 65 + 8 × 2.5 = 85 kg. 1
- The average temperature from Monday to Wednesday was 36°C and from Tuesday to Thursday was 34°C. Thursday's temperature was \(\frac{4}{5}\) of Monday's. What was Thursday's temperature? Answer: D. 24°C Answer: (D) 24°C Mon + Tue + Wed = 108 and Tue + Wed + Thu = 102, so Mon − Thu = 6. Mon − 0.8 Mon = 6 gives Mon = 30°C, so T... 1
- The average of 21 results is 20. The average of the first 10 is 17 and the average of the last 10 is 21. What is the 11th result? Answer: C. 40 Answer: (C) 40 Total = 21 × 20 = 420. First 10 = 170, last 10 = 210. 11th result = 420 − 170 − 210 = 40. 1
- The average of five consecutive odd numbers A, B, C, D and E (in increasing order) is 45. What is the product of B and D? Answer: C. 2021 Answer: (C) 2021 The middle number C equals the average, 45. So B = 43 and D = 47. B × D = 43 × 47 = (45 − 2)(45 + 2) = 2025... 1
- The average age of a husband and wife was 26 years when they married. Six years later they have a two-year-old child. What is the average age of the family now? Answer: C. 22 years Answer: (C) 22 years At marriage the couple's total age = 52. After 6 years it is 52 + 12 = 64. Family total = 64 + 2 =... 1
- The average weight of A, B and C is 45 kg. The average weight of A and B is 40 kg and that of B and C is 43 kg. What is B's weight? Answer: D. 31 kg Answer: (D) 31 kg A + B + C = 135, A + B = 80 and B + C = 86. B = (A + B) + (B + C) − (A + B + C) = 80 + 86 − 135 = 31 kg. 1
- The average marks of 30 students in a test is 40. The average of those who passed is 45 and the average of those who failed is 30. How many students passed? Answer: B. 20 Answer: (B) 20 Shortcut (alligation): passed : failed = (40 − 30) : (45 − 40) = 10 : 5 = 2 : 1. Passed = 2/3 × 30 = 20 . Plain... 1
- What is the average of 12, 18, 25, 31 and 44? Answer: D. 26 Answer: (D) 26 Sum = 130, so average = 130 ÷ 5 = 26. 1
- What is the average of the first 20 natural numbers? Answer: B. 10.5 Answer: (B) 10.5 Shortcut: average of 1 to n = n+1/2 = 21/2 = 10.5 . 1
- What is the average of the first 15 odd natural numbers? Answer: C. 15 Answer: (C) 15 Shortcut: the average of the first n odd numbers is n. So the answer is 15. Plain: sum = 15² = 225, and 225 ÷ 1... 1
- The average of six numbers is 24. If the number 39 is removed, what is the average of the remaining five numbers? Answer: C. 21 Answer: (C) 21 Sum of six = 6 × 24 = 144. Remaining sum = 144 − 39 = 105, so average = 105 ÷ 5 = 21. Shortcut: 39 is 15 above... 1
- The average weight of 10 students is 50 kg. When the teacher joins them, the average becomes 52 kg. What is the teacher's weight? Answer: D. 72 kg Answer: (D) 72 kg Shortcut: teacher = new average + (number of old members × rise) = 52 + 10 × 2 = 72 kg. Plain: 11 × 52 −... 1
- The average of five consecutive even numbers is 36. What is the largest of them? Answer: B. 40 Answer: (B) 40 For five consecutive even numbers, the middle one is the average, i.e. 36. The numbers are 32, 34, 36, 38, 40,... 1
- A batsman scores 40, 60, 85 and 55 runs in four innings. How many runs must he score in the fifth innings to make his average 62? Answer: D. 70 Answer: (D) 70 Required total = 62 × 5 = 310. Runs so far = 240. Fifth innings = 310 − 240 = 70. 1
- The average of a and b is 40, and the average of b and c is 45. What is c − a? Answer: A. 10 Answer: (A) 10 a + b = 80 and b + c = 90. Subtracting, c − a = 10. Trap: 5 is the difference of the averages, not of the sums. 1
- The average age of 30 students in a class is 14 years. When the teacher's age is included, the average rises by 1 year. What is the teacher's age? Answer: A. 45 years Answer: (A) 45 years Shortcut: teacher = new average + 30 × rise = 15 + 30 × 1 = 45 years. Plain: 31 × 15 − 30 × 14 = 46... 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