Quantitative aptitude
- The average of 11 numbers is 50. The average of the first six numbers is 49 and the average of the last six numbers is 52. What is the sixth number? Answer: D. 56 Answer: (D) 56 The sixth number is counted in both groups. Sixth number = (6 × 49 + 6 × 52) − 11 × 50 = 294 + 312 − 550 = 56. 1
- The angles of a triangle are in the ratio 2 : 3 : 4. What is the largest angle? Answer: A. 80° Answer: (A) 80° 2x + 3x + 4x = 180^° , so x = 20^° . Largest angle = 4 × 20 = 80^° . 1
- A chord of length 16 cm is at a distance of 6 cm from the centre of a circle. What is the radius of the circle? Answer: C. 10 cm Answer: (C) 10 cm The perpendicular from the centre bisects the chord, so half-chord = 8 cm. r = √(8² + 6²) = √(100) = 10 c... 1
- In triangle ABC, D and E are points on AB and AC such that DE ∥ BC. If AD = 4 cm and DB = 6 cm, what is the ratio of the area of triangle ADE to the area of trapezium DBCE? Answer: D. 4 : 21 Answer: (D) 4 : 21 Triangles ADE and ABC are similar with ratio AD : AB = 4 : 10 = 2 : 5, so their areas are in the ratio... 1
- The HCF and LCM of two numbers are 12 and 360 respectively. If one of the numbers is 72, what is the other? Answer: A. 60 Answer: (A) 60 Product of the numbers = HCF × LCM. Other number = 12 × 360/72 = 60 . 1
- From a point on the ground, the angle of elevation of the top of a tower is 30°. On walking 20 m towards the tower, the angle of elevation becomes 60°. What is the height of the tower? Answer: A. 10√(3) m Answer: (A) 10√(3) m Let the nearer point be x m from the foot. h = xtan 60^° = x√(3) and h = (x + 20)tan 30^° = x + 20/... 1
- Find the curved surface area of a cylinder of radius 7 cm and height 10 cm. (Take \(\pi = \frac{22}{7}\).) Answer: A. 440 cm² Answer: (A) 440 cm² CSA = 2π r h = 2 × 22/7 × 7 × 10 = 440 cm². 1
- A right circular cone has base radius 6 cm and height 8 cm. What is its curved surface area? Answer: B. 60π cm² Answer: (B) 60π cm² Slant height l = √(6² + 8²) = 10 cm. CSA = π r l = π × 6 × 10 = 60π cm². 1
- What is the largest four-digit number which, when divided by 12, 15 and 18, leaves a remainder of 7 in each case? Answer: C. 9907 Answer: (C) 9907 LCM(12, 15, 18) = 180. The largest 4-digit multiple of 180: 9999 ÷ 180 = 55 remainder 99, so it is 180 × 55... 1
- A's salary is 25% more than B's salary. By what percentage is B's salary less than A's salary? Answer: D. 20% Answer: (D) 20% Let B = 100, so A = 125. B is less than A by 25 on a base of 125: 25/125 × 100 = 20% . 1
- A shopkeeper marks his goods 40% above the cost price and allows a discount of 15% on the marked price. What is his profit percentage? Answer: B. 19% Answer: (B) 19% Let CP = 100. MP = 140. SP = 140 × 0.85 = 119 . Profit = 19%. 1
- If A : B = 2 : 3 and B : C = 4 : 5, and A + B + C = 700, find C. Answer: C. 300 Answer: (C) 300 Make B equal: A : B = 8 : 12, B : C = 12 : 15, so A : B : C = 8 : 12 : 15 (total 35). C = 15/35 × 700 = 300 . 1
- What is the difference between the compound interest (compounded annually) and the simple interest on ₹10,000 at 10% per annum for 3 years? Answer: C. ₹310 Answer: (C) ₹310 For 3 years, CI − SI = P(R/100)²(3 + R/100) = 10000 × 0.01 × 3.1 = ₹310 . Check: CI = 13310 − 10000 = 3310,... 1
- Find the value of \(18 \div 3 \times 2 + 4^2 - 5 \times 3\). Answer: A. 13 Answer: (A) 13 BODMAS: 4² = 16 . Then left to right for ÷ and ×: 18 ÷ 3 = 6 , 6 × 2 = 12 ; 5 × 3 = 15 . 12 + 16 - 15 = 13 . 1
- A can finish a piece of work in 12 days and B can finish it in 18 days. In how many days will they finish it working together? Answer: C. 71/5 days Answer: (C) 71/5 days Work per day = 1/12 + 1/18 = 3 + 2/36 = 5/36 . Time = 36/5 = 71/5 days. 1
- A train 240 m long crosses a pole in 12 seconds. How long will it take to cross a platform 360 m long? Answer: D. 30 s Answer: (D) 30 s Speed = 240/12 = 20 m/s. To cross the platform the train covers 240 + 360 = 600 m. Time = 600/20 = 30 s. 1
- If \(\sin\theta = \frac{3}{5}\) and θ is acute, find \(\tan\theta + \sec\theta\). Answer: D. 2 Answer: (D) 2 cosθ = 4/5 , so tanθ = 3/4 , secθ = 5/4 . Sum = 3 + 5/4 = 2 . 1
- Find the value of \(\sin^2 30^\circ + \cos^2 60^\circ + \tan^2 45^\circ\). Answer: A. 3/2 Answer: (A) 3/2 sin 30^° = cos 60^° = 1/2 , tan 45^° = 1 . 1/4 + 1/4 + 1 = 3/2 . 1
- If \(\tan\theta + \cot\theta = 2\), where \(0^\circ < \theta < 90^\circ\), find \(\tan^{10}\theta + \cot^{10}\theta\). Answer: A. 2 Answer: (A) 2 Let t = tanθ . t + 1/t = 2 gives (t - 1)² = 0 , so t = 1 (θ = 45°). Then tan¹⁰θ + cot¹⁰θ = 1 + 1 = 2 . 1