Quantitative aptitude
- Each exterior angle of a regular polygon is 24°. How many sides does it have? Answer: D. 15 Answer: (D) 15 Number of sides = 360^°/24^° = 15 . 1
- A point is 13 cm from the centre of a circle of radius 5 cm. What is the length of the tangent drawn from the point to the circle? Answer: A. 12 cm Answer: (A) 12 cm The radius is perpendicular to the tangent: tangent = √(13² - 5²) = √(144) = 12 cm. 1
- In triangle ABC, I is the incentre and ∠A = 70°. Find ∠BIC. Answer: D. 125° Answer: (D) 125° ∠ BIC = 90^° + ∠ A/2 = 90^° + 35^° = 125^° . Reason: ∠ IBC + ∠ ICB = 1/2(180^° - 70^°) = 55^° , so ∠ BIC =... 1
- Three bells ring at intervals of 12, 18 and 30 minutes. If they ring together at 9:00 a.m., when will they next ring together? Answer: B. 12:00 noon Answer: (B) 12:00 noon LCM(12, 18, 30) = 180 minutes = 3 hours. 9:00 a.m. + 3 h = 12:00 noon. 1
- From the top of a lighthouse 75 m high, the angle of depression of a ship is 30°. How far is the ship from the foot of the lighthouse? Answer: C. 75√(3) m Answer: (C) 75√(3) m tan 30^° = 75/d , so d = 75√(3) m (about 129.9 m). 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
- Find the surface area of a sphere of radius 21 cm. (Take \(\pi = \frac{22}{7}\).) Answer: B. 5544 cm² Answer: (B) 5544 cm² Surface area = 4π r² = 4 × 22/7 × 441 = 5544 cm². (1386 is π r² ; 38808 is the volume.) 1
- A cuboid of dimensions 12 cm × 8 cm × 6 cm is melted and recast into cubes of side 2 cm. How many cubes are formed? Answer: C. 72 Answer: (C) 72 Volume of cuboid = 12 × 8 × 6 = 576 cm³. Volume of one cube = 8 cm³. Number = 576 ÷ 8 = 72. 1
- A 60-litre mixture contains milk and water in the ratio 7 : 3. How much water must be added to make the ratio 7 : 5? Answer: D. 12 litres Answer: (D) 12 litres Milk = 42 L, water = 18 L. For 7 : 5 with 42 L milk, water must be 42 × 5/7 = 30 L. Water to add... 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
- A invests ₹15,000 for 12 months and B invests ₹20,000 for 6 months in a business. Out of a total profit of ₹10,000, what is B's share? Answer: D. ₹4,000 Answer: (D) ₹4,000 Ratio = 15000 × 12 : 20000 × 6 = 180000 : 120000 = 3 : 2. B's share = 2/5 × 10000 = ₹4,000 . 1
- The price of sugar rises by 25%. By what percentage must a family reduce its consumption so that its expenditure on sugar stays the same? Answer: A. 20% Answer: (A) 20% Reduction = 25/100 + 25 × 100 = 20% . Check: 1.25 × 0.8 = 1. 1
- Pipes A and B can fill a tank in 20 minutes and 30 minutes respectively, and pipe C can empty it in 15 minutes. If all three are opened together, in how much time will the empty tank be filled? Answer: A. 60 minutes Answer: (A) 60 minutes Net rate = 1/20 + 1/30 - 1/15 = 3 + 2 - 4/60 = 1/60 . Time = 60 minutes. 1
- A trader sells two articles for ₹990 each. On one he gains 10% and on the other he loses 10%. What is his overall gain or loss percentage? Answer: B. 1% loss Answer: (B) 1% loss CP of first = 990/1.1 = 900 ; CP of second = 990/0.9 = 1100 . Total CP = 2000, total SP = 1980. Loss... 1
- Find the compound interest on ₹8,000 at 10% per annum for 3 years, compounded annually. Answer: C. ₹2,648 Answer: (C) ₹2,648 Amount = 8000 × (1.1)³ = 8000 × 1.331 = 10648 . CI = 10648 − 8000 = ₹2,648. (₹2,400 is the simple inter... 1
- Find the value of \(\sqrt{1764} - \sqrt[3]{2744}\). Answer: B. 28 Answer: (B) 28 √(1764) = 42 and sqrt[3]2744 = 14 . 42 - 14 = 28 . 1
- If \(\sec\theta + \tan\theta = 3\), find \(\sin\theta\). Answer: D. 4/5 Answer: (D) 4/5 Since sec²θ - tan²θ = 1 , secθ - tanθ = 1/3 . Adding: secθ = 5/3 ; subtracting: tanθ = 4/3 . sinθ = tanθ/secθ... 1
- Find the value of \(\frac{\sin 35^\circ}{\cos 55^\circ} + \frac{\cos 28^\circ}{\sin 62^\circ} - 2\cos^2 60^\circ\). Answer: A. 3/2 Answer: (A) 3/2 cos 55^° = sin 35^° and sin 62^° = cos 28^° (complementary angles), so the first two terms are 1 each. 2cos²... 1
- The table shows the production (in thousand units) of a company over five years.Year20212022202320242025Production120150135180165What is the average annual production over the five years? Answer: B. 150 thousand Answer: (B) 150 thousand Total = 120 + 150 + 135 + 180 + 165 = 750 . Average = 750/5 = 150 thousand units. 1
- The table shows the production (in thousand units) of a company over five years.Year20212022202320242025Production120150135180165By what percentage did production increase from 2023 to 2024? Answer: A. 331/3 % Answer: (A) 331/3 % Increase = 180 − 135 = 45. Percentage = 45/135 × 100 = 331/3% . (25% would wrongly use 180 as the bas... 1
- The table shows the production (in thousand units) of a company over five years.Year20212022202320242025Production120150135180165What is the ratio of the total production in 2021 and 2022 to the total production in 2024 and 2025? Answer: C. 18 : 23 Answer: (C) 18 : 23 2021 + 2022 = 270; 2024 + 2025 = 345. 270 : 345 ; divide by 15: 18 : 23. 1
- The mean of 10 observations was found to be 20. Later it was noticed that one observation, 25, had been wrongly copied as 52. What is the correct mean? Answer: A. 17.3 Answer: (A) 17.3 Wrong sum = 10 × 20 = 200. Correct sum = 200 − 52 + 25 = 173. Correct mean = 173/10 = 17.3 . 1
- If \(x + \frac{1}{x} = 5\), find the value of \(x^3 + \frac{1}{x^3}\). Answer: B. 110 Answer: (B) 110 x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) = 125 - 15 = 110 . 1
- If \(a + b + c = 6\), \(ab + bc + ca = 11\) and \(abc = 6\), find \(a^3 + b^3 + c^3\). Answer: D. 36 Answer: (D) 36 a³ + b³ + c³ = (a + b + c)³ - 3(a + b + c)(ab + bc + ca) + 3abc . = 216 - 3 × 6 × 11 + 3 × 6 = 216 - 198 + 18 =... 1