Competive Exam Math 14
Q. If 5n = 2, then 25 2n + 1 = ?
(A) 100 (B) 200 (C) 225 (D) 400
Soln
252n+1 = (52)2n+1
Since 5n = 2,
54n = (5n)4 = 24 = 16
Therefore, 16 × 25 = 400
Ans: (D) 400
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Q. The difference between two numbers is 7. If their product is 330, what is the sum of the two numbers ?
(A) 37 (B) 43 (C) 45 (D) 49
Solution:
Let the two numbers be x and y.
x – y = 7, xy = 330
Using the identity: (x+y)2 = (x−y)2 + 4xy = (7)2 + 4(330) = 49 + 1320 = 1369
Therefore,
x + y = √1369 = 37
Ans: (A) 37
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Q. How many prime factors are included in the factorization of 67×353×1110
Trick Method
- 6 = 2 × 3, Therefore, 67 contains 7 × 2 = 14 prime factors.
- 35 = 5×7, Therefore, 353 contains 3 × 2 = 6 prime factors.
- 1110 contains 10 prime factors.
Total = 14 + 6 + 10 = 30
Ans: C) 30
Trick: If a number has n prime factors, then its mmth power has n × m prime factors, counting multiplicity.
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Q. A and B start a business with ₹12,000 and ₹18,000 respectively. After 4 months, A adds ₹6,000 more. Find their profit ratio after one year.
Solution
Rule: Profit Ratio = Capital × Time
1st: A's investment
First 4 months: 12000 × 4 = 48,000
Next 8 months, A's capital becomes: 12000 + 6000 = 18,000
18000 × 8 = 1,44,000
Total for A: 48,000 + 1,44,000 = 1,92,000
2nd: B's investment
B invests ₹18,000 for all 12 months: 18000 × 12 = 2,16,000
3rd: Profit Ratio
A : B = 1,92,000 : 2,16,000
Divide by 24,000:- 8 : 9
Ans: 8 : 9
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Q. A number when increased by 50% gives 3630. The number is: (RPF SI – 03/12/2024 (Shift-01))
(a) 7260 (b) 4840 (c) 1210 (d) 2420
Solution
Let the original number be x.
A 50% increase means:
x + 50/100x = 3630
1.5x = 3630
x = 3630/1.5 = 2420
Ans: 2420
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Percentage - Trick
Q. If the side of a square is increased by 20%, by what percentage will its perimeter increase ?
Solution:
Let the original side = 100
Original perimeter: 4 × 100 = 400
New side: 100 + 20% = 120
New perimeter: 4 × 120 = 480
Increase: 480 − 400 = 80
Percentage increase: 80/400 × 100 = 20%
Trick
For a square, perimeter is directly proportional to its side.
Side increases by 20% ⇒ Perimeter also increases by 20%
Ans: 20%
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