SSC CGL Mock – Tier II Paper I Mock Test 05 – Section I · Part A · Mathematical Abilities

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SSC CGL Mock – Tier II Paper I Mock Test 05 – Section I · Part A · Mathematical Abilities – 30 questions, subject: Full Mock Tests. 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.  
The mean of five numbers x, x + 2, x + 4, x + 6 and x + 8 is 15. What is the largest of them?

Correct Answer: D [19]

Explanation:

Mean = x + 4 = 15 → x = 11 → largest = 19.

2.  
The scores 1, 2, 3, 4 and 5 were obtained by 4, 6, 9, 7 and 4 students respectively. What is the median score?

Correct Answer: A [3]

Explanation:

N = 30; cumulative frequencies 4, 10, 19, 26, 30 → the 15th and 16th scores are both 3.

3.  
One of the integers 1, 2, 3, …, 50 is selected at random. Find the probability that the integer chosen is divisible by 3 or by 7.

Correct Answer: C [21/50]

Explanation:

Multiples of 3: 16, of 7: 7, of 21: 2 → 16 + 7 − 2 = 21 → 21/50.

4.  
A cube has a space diagonal of 6√3 cm. Find the total area of its six faces.

Correct Answer: B [216 cm²]

Explanation:

a√3 = 6√3 → a = 6 → 6a² = 216 cm².

5.  
A solid cylinder of radius 3 cm and height 8 cm is melted and made into spheres of radius 1 cm. How many spheres are formed?

Correct Answer: C [54]

Explanation:

π × 9 × 8 = 72π; one sphere = 4/3 π → 54.

6.  
An equilateral triangle encloses an area of 36√3 cm². How long is the boundary of the triangle?

Correct Answer: B [36 cm]

Explanation:

(√3/4)a² = 36√3 → a² = 144 → a = 12 → perimeter 36 cm.

7.  
2 men and 3 women together can finish a piece of embroidery in 12 days, while 1 man and 9 women can finish it in 9 days. How many days would 4 women alone take?

Correct Answer: D [27 days]

Explanation:

24m + 36w = 9m + 81w → 15m = 45w → 1 man = 3 women; work = 12 × (6 + 3) = 108 woman-days → 108/4 = 27 days.

8.  
If 7 sin²θ + 3 cos²θ = 4 and θ is acute, what is tan θ?

Correct Answer: A [1/√3]

Explanation:

3 + 4 sin²θ = 4 → sin θ = ½ → θ = 30° → tan θ = 1/√3.

9.  
Taking all values of θ for which the expression is defined, find the least value of 9 sec²θ + 4 cos²θ.

Correct Answer: D [13]

Explanation:

With t = cos²θ (0 < t ≤ 1): 9/t + 4t decreases on (0, 1] because 9/t² > 4 there, so the minimum is at t = 1: 9 + 4 = 13.

10.  
Two men stand on opposite sides of a tower 60 m high, in line with its foot, and see its top at elevations of 30° and 60°. How far apart are they?

Correct Answer: A [80√3 m]

Explanation:

60√3 + 60/√3 = 60√3 + 20√3 = 80√3 m.

11.  
The average weight of 45 students is 52 kg. When 5 students leave, the average of the rest falls by 0.5 kg. What is the average weight of the 5 who left?

Correct Answer: B [56 kg]

Explanation:

45 × 52 − 40 × 51.5 = 2,340 − 2,060 = 280 → 280/5 = 56 kg.

12.  
Diagram-basedFrom an external point P, a secant PAB and a tangent PT are drawn to a circle (see figure). If PA = 4 cm and AB = 5 cm, find PT.

Correct Answer: C [6 cm]

Explanation:

PT² = PA × PB = 4 × 9 = 36 → PT = 6 cm.

13.  
In an equilateral triangle of side 12 cm, how far is the centroid from each vertex?

Correct Answer: A [4√3 cm]

Explanation:

Height = 6√3; the centroid is 2/3 of the way down → 4√3 cm.

14.  
In triangle ABC, DE ∥ BC with D on AB and E on AC. If area(ADE) : area(trapezium DBCE) = 4 : 5, what is AD : DB?

Correct Answer: A [2 : 1]

Explanation:

area(ADE) : area(ABC) = 4 : 9 → AD : AB = 2 : 3 → AD : DB = 2 : 1.

15.  
A town's population rose by 10% in the first year, fell by 10% in the second and rose by 20% in the third, reaching 1,18,800. What was it at the start?

Correct Answer: D [1,00,000]

Explanation:

Factor = 1.1 × 0.9 × 1.2 = 1.188 → 1,18,800 ÷ 1.188 = 1,00,000.

16.  
₹ 4,890 is divided among A, B and C so that if ₹ 20, ₹ 30 and ₹ 40 are taken from their shares respectively, the remainders are in the ratio 3 : 4 : 5. What is C's share?

Correct Answer: B [₹ 2,040]

Explanation:

Remainders total 4,890 − 90 = 4,800 → one part = 400 → C = 5 × 400 + 40 = 2,040.

17.  
A shop marks its goods 50% above cost and then offers 'buy 4, get 1 free'. What profit per cent does it make?

Correct Answer: C [20%]

Explanation:

5 items cost 5C and fetch 4 × 1.5C = 6C → profit 1C on 5C = 20%.

18.  
A dealer bought two scooters for ₹ 10,000 in all. He sold one at a profit of 20% and the other at a loss of 10%, making an overall profit of ₹ 800. What was the cost of the scooter sold at a loss?

Correct Answer: D [₹ 4,000]

Explanation:

0.2a − 0.1(10,000 − a) = 800 → 0.3a = 1,800 → a = 6,000; the other cost 4,000.

19.  
If a + b = 7 and a³ + b³ = 133, find ab.

Correct Answer: A [10]

Explanation:

a³ + b³ = (a + b)³ − 3ab(a + b) → 133 = 343 − 21ab → ab = 10.

20.  
As x runs over all real numbers, what is the lowest value attained by the quadratic x² − 8x + 21?

Correct Answer: C [5]

Explanation:

x² − 8x + 21 = (x − 4)² + 5 → minimum 5 at x = 4.

21.  
Two-quantity comparison
x and y satisfy 3x + 2y = 18 and 2x − y = 5.
Quantity I: x
Quantity II: y

Correct Answer: A [Quantity I > Quantity II]

Explanation:

y = 2x − 5 → 3x + 4x − 10 = 18 → x = 4, y = 3 → Quantity I is greater.

22.  
Kiran walks to her office. At 5 km/h she arrives 6 minutes after time; at 6 km/h she arrives 6 minutes before time. What is the distance to her office?

Correct Answer: B [6 km]

Explanation:

d/5 − d/6 = 12 min = 1/5 h → d/30 = 1/5 → d = 6 km.

23.  
Find the remainder when 2¹⁰⁰ is divided by 101 (101 is prime).

Correct Answer: B [1]

Explanation:

By Fermat's little theorem 2¹⁰⁰ ≡ 1 (mod 101).

24.  
A number divided by 3 leaves remainder 2; the quotient, divided by 7, leaves remainder 5. What remainder does the number leave when divided by 21?

Correct Answer: A [17]

Explanation:

Number = 3(7k + 5) + 2 = 21k + 17 → remainder 17.

25.  
Simplify: (0.8 × 0.8 × 0.8 − 0.5 × 0.5 × 0.5) ÷ (0.8 × 0.8 + 0.8 × 0.5 + 0.5 × 0.5)

Correct Answer: C [0.3]

Explanation:

(a³ − b³)/(a² + ab + b²) = a − b = 0.8 − 0.5 = 0.3.

26.  

Directions (Q 26–28): The table gives, for five products of a shop, the cost price of one unit, the mark-up over cost and the discount allowed on the marked price (fictional data).

Product Cost price (₹) Mark-up (%) Discount (%)
P 800 50 20
Q 1,200 40 10
R 1,500 30 15
S 600 60 25
T 2,000 25 8
Data tableFor which of the five products is the profit, as a percentage of cost, the greatest?

Correct Answer: D [Q]

Explanation:

Profit %: P 20, Q 26, R 10.5, S 20, T 15 → Q.

27.  
(Use the directions above – Q 26–28)
Data tableWhat profit does the shop make on one unit of product S?

Correct Answer: A [₹ 120]

Explanation:

600 × 1.6 × 0.75 = 720 → profit ₹ 120.

28.  
(Use the directions above – Q 26–28)
Data tableWhat is the total selling price of 2 units of R and 3 units of S?

Correct Answer: C [₹ 5,475]

Explanation:

R: 1,500 × 1.3 × 0.85 = 1,657.5; S: 720 → 2 × 1,657.5 + 3 × 720 = 5,475.

29.  
₹ 23,000 is divided between A and B so that A's amount after 2 years equals B's amount after 4 years, simple interest being 5% per annum for both. What is A's part?

Correct Answer: B [₹ 12,000]

Explanation:

A × 1.1 = B × 1.2 → A : B = 12 : 11 → A = 12/23 × 23,000 = 12,000.

30.  
An alloy of 140 g contains 70% gold. How many grams of pure gold must be melted with it so that the new alloy is 79% gold?

Correct Answer: D [60 g]

Explanation:

98 + x = 0.79(140 + x) → 0.21x = 12.6 → x = 60 g.

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