SSC CGL Quant – Speed Rounds – Tier II Level · Speed Round 5 · Marathon

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SSC CGL Quant – Speed Rounds – Tier II Level · Speed Round 5 · Marathon – 25 questions, subject: Quantitative Aptitude. Every question has the correct answer with a one-line explanation; tap "Show Answer". Free practice for TNPSC, UPSC, SSC, Bank and RRB exams – Shanmugam IAS Academy, Coimbatore.

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1.  
Simplify: (0.75³ − 0.25³) ÷ (0.75² + 0.75 × 0.25 + 0.25²)

Correct Answer: B [0.5]

Explanation:

Algebra – (a³ − b³)/(a² + ab + b²) = a − b = 0.5.

2.  
What is the sum of all the divisors of 36 (including 1 and 36)?

Correct Answer: C [91]

Explanation:

Factors – 36 = 2² × 3² → (1 + 2 + 4)(1 + 3 + 9) = 7 × 13 = 91.

3.  
What is the value of 1/(√5 + √4) + 1/(√6 + √5) + 1/(√7 + √6) + 1/(√8 + √7) + 1/(√9 + √8)?

Correct Answer: A [1]

Explanation:

Surds – each term rationalises to √(n + 1) − √n; the sum telescopes to √9 − √4 = 1.

4.  
A number leaves remainder 5 when divided by 7. What remainder does its cube leave when divided by 7?

Correct Answer: D [6]

Explanation:

Remainder – cube the remainder: 5³ = 125 = 7 × 17 + 6 (or 5 ≡ −2, (−2)³ = −8 ≡ 6).

5.  
A man spends 75% of his income. His income rises by 20% and his spending by 10%. By what percentage do his savings rise?

Correct Answer: A [50%]

Explanation:

Percentage – income 100 → 120, spending 75 → 82.5, savings 25 → 37.5: rise 12.5/25 = 50%.

6.  
P and Q earn in the ratio 7 : 5 and spend in the ratio 3 : 2. Each of them saves ₹ 3,000 a month. What is P's monthly income?

Correct Answer: C [₹ 21,000]

Explanation:

Ratio – 7x − 3y = 3000 and 5x − 2y = 3000 → y = 2x, x = 3000; P earns 7 × 3000 = 21,000.

7.  
Vessel P has milk and water in the ratio 5 : 3 and vessel Q in the ratio 3 : 1. Equal volumes from P and Q are mixed. What is the ratio of milk to water in the mixture?

Correct Answer: B [11 : 5]

Explanation:

Mixture – milk 5/8 + 6/8 = 11/8, water 3/8 + 2/8 = 5/8 → 11 : 5.

8.  
A trader marks goods 60% above cost and allows two successive discounts of 25% and 10%. What is his profit percentage?

Correct Answer: D [8%]

Explanation:

Profit – 1.6 × 0.75 × 0.9 = 1.08 → 8%.

9.  
What is the compound interest on ₹ 20,000 for 1 year at 8% per annum, compounded half-yearly?

Correct Answer: C [₹ 1,632]

Explanation:

Compound interest – 4% for 2 half-years: 20000 × 1.04² = 21632 → CI 1632.

10.  
Pipe A fills a tank in 20 minutes and pipe B in 30 minutes. Both are opened together and B is closed after 6 minutes. In how many minutes in all is the tank filled?

Correct Answer: A [16]

Explanation:

Pipes – LCM 60 units; 6 min × (3 + 2) = 30 units; the other 30 units by A alone take 10 min → 16 min.

11.  
Walking at 5 km/h a clerk reaches office 6 minutes late; walking at 6 km/h he reaches 4 minutes early. How far is the office?

Correct Answer: D [5 km]

Explanation:

Speed – d/5 − d/6 = 10 min = 1/6 h → d/30 = 1/6 → d = 5 km.

12.  
Two trains start at the same time from towns P and Q, 390 km apart, towards each other at 60 km/h and 70 km/h. How far from P do they meet?

Correct Answer: B [180 km]

Explanation:

Trains – they meet after 390/130 = 3 h; the train from P has gone 60 × 3 = 180 km.

13.  
If x = √13 + √12, what is the value of x² + 1/x²?

Correct Answer: C [50]

Explanation:

Surds – 1/x = √13 − √12 (13 − 12 = 1), so x² + 1/x² = 2(13 + 12) = 50.

14.  
If (x − 3)(x + 5) = x² + kx − 15 for all x, what is k?

Correct Answer: A [2]

Explanation:

Algebra – (x − 3)(x + 5) = x² + 2x − 15 → k = 2.

15.  
For real numbers, a² + b² + c² − ab − bc − ca = 0 and a = 4. What is a + b + c?

Correct Answer: D [12]

Explanation:

Algebra – the expression = ½[(a − b)² + (b − c)² + (c − a)²] = 0 → a = b = c = 4 → 12.

16.  
In triangle ABC, D and E lie on AB and AC with DE ∥ BC and AD : DB = 3 : 4. What is the ratio of the area of triangle ADE to that of triangle ABC?

Correct Answer: B [9 : 49]

Explanation:

Similarity – AD : AB = 3 : 7; areas are in the square ratio 9 : 49 (9 : 40 is ADE : trapezium).

17.  
Two circles of radii 10 cm and 4 cm have their centres 10 cm apart. What is the length of a direct common tangent?

Correct Answer: B [8 cm]

Explanation:

Common tangent – √(d² − (r₁ − r₂)²) = √(100 − 36) = 8 cm.

18.  
ABCD is a cyclic quadrilateral in which ∠A = 3∠C. What is ∠A?

Correct Answer: D [135°]

Explanation:

Cyclic quadrilateral – A + C = 180° and A = 3C → C = 45°, A = 135°.

19.  
A metal cuboid 12 cm × 9 cm × 8 cm is melted and recast into cubes of side 2 cm. How many cubes are made?

Correct Answer: C [108]

Explanation:

Volume – 864 ÷ 8 = 108 cubes.

20.  
What is the total surface area of a solid hemisphere of radius 7 cm? (Take π = 22/7)

Correct Answer: A [462 cm²]

Explanation:

Hemisphere – TSA = 3πr² = 3 × 22/7 × 49 = 462 cm² (308 is the curved surface only).

21.  
For 0° < θ < 90°, (sec² θ − 1)(1 − sin² θ) equals:

Correct Answer: C [sin² θ]

Explanation:

Trigonometry – tan² θ × cos² θ = sin² θ (check with θ = 45°: 1 × ½ = ½).

22.  
What is the value of cos 1° × cos 2° × cos 3° × … × cos 100°?

Correct Answer: D [0]

Explanation:

Trigonometry – the product contains cos 90° = 0, so the whole product is 0.

23.  
From two points on the same side of a tower and in line with its foot, 20 m apart, the angles of elevation of its top are 30° and 45°. What is the height of the tower?

Correct Answer: B [10(√3 + 1) m]

Explanation:

Heights – h = d × tan 30° × tan 45°/(tan 45° − tan 30°) = 20/(√3 − 1) = 10(√3 + 1) m.

24.  
The mean of 4, 7, x, 10 and 9 is 8. What is the mode of the data?

Correct Answer: A [10]

Explanation:

Statistics – x = 40 − 30 = 10; data 4, 7, 10, 10, 9 → mode 10.

25.  
Four fair coins are tossed together. What is the probability of getting exactly two heads?

Correct Answer: B [3/8]

Explanation:

Probability – ⁴C₂ = 6 favourable outcomes out of 2⁴ = 16 → 6/16 = 3/8.

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