SOLVED PAPER 2023 (CDS) (I) (Elementary Mathematics)

Total Questions: 100

1. What is the largest number which divides both 2³⁵ - 1 and 2⁹¹ - 1 ?

Correct Answer: (c) 127
Solution:

2. What is the largest power of 10 that divides the product 29 × 28 × 27 × ... 2 × 1 ?

Correct Answer: (c) 6
Solution:29! 29x28x27x... x 2x1
Number of multiples of
5 = 5, 10, 15, 20, 25
∴ Power of 5 in the prime
factorization of the given term = 6
∴ Largest power of 10 that divides
the given product = 6

3. What is the remainder when 65⁹⁹ is divided by 11?

Correct Answer: (d) 10
Solution:65⁹⁹ = 65⁹⁹ + 1⁹⁹ - 1⁹⁹
∵ For odd positive integer value of
n, xⁿ + aⁿ is completely divisible by (x + a)
∴ 65⁹⁹ + 1⁹⁹ is divisible by
(65 + 1 = 66) i.e. it is also divisible by 11.
Hence, required remainder = 11 - 1 = 10

4. If the roots of the equation x² - bx + c = 5 differ by 5, then which one of the following is correct?

Correct Answer: (a) b² = 4c + 5
Solution:

5. In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only?

Correct Answer: (c) 30
Solution:

Let, the number of persons taking
milk only be x
and number of persons taking milk
and coffee only = 50 - x
∴ Number of person taking coffee
only = 60 - 15 - (50 - x)
= 45 - 50 + x = x - 5
According to the question,
∴ Total persons = 150
⇒ 60 + 15 + (x - 5) + (50 - x) + x = 150
⇒ x + 120 = 150
⇒ x =150-120 ⇒ x = 30
The number of persons taking milk only = 30

6. A, B, C, D and E enter into a business. They invest money in the ratio 2 : 3 : 4 : 5 : 6. However, the time invested by them is in the ratio 6 : 5 : 4 : 3 : 2. If the profit distributed is directly proportional to time and money invested, then who receives the highest amount of profit

Correct Answer: (a) C
Solution:Ratio of profit of A, B, C, D
and E = Ratio of their investments.
=2 × 6 : 3 × 5 : 4 × 4 : 5 × 3 : 6 × 2
= 12 : 15 : 16 : 15 : 12
Hence, C receives the heighest
amount of profit.

7. Consider the following numbers

1. 437,  2. 797, 3. 1073

How many of the above numbers are prime?

Correct Answer: (a) Only one
Solution:A prime number is a number
that is divisible by only itself and 1.
∴ 437 = 1 × 19 × 23
Since, 437 has more than two factors
∴ 437 is not a prime number.
797 = 1 × 797
797 is divisible by only itself and 1.
Hence, 797 is a prime number.
1073 = 1 × 29 × 37
Hence, 1073 is not a prime number.
Hence, only one number is prime number.

8. A can do a certain work at twice the speed of B. Further, B can do the same work at 1.5 times the speed of C. All of them together can finish the work in 12 days. In how many days can C alone finish the work?

Correct Answer: (d) 66 days
Solution:Let work done by C in 1 day = x
and work done by B in 1 day = 1.5x
and work done by A in 1 day = 3x
Now, number of days A, B and C
together can finish the work = 12 days

Work done by A, B and C together in 1 day = ¹⁄₁₂
∴ x + 1.5x + 3x = ¹⁄₁₂ ⇒ 5.5x = ¹⁄₁₂
⇒ x = 10 ÷ (12×55) = ⅙
∴ C can finish the same work in. 66 days.

9. The sum of digits of a 2-digit number is 12. When the digits are reversed, the number becomes greater by eighteen. What is the difference between the digits in the number?

Correct Answer: (b) 2
Solution:Let one and ten digits of a
2-digit number be x and y.
∴ Original number = 10x + y
According to the question,
x + y = 12  .....(i)
and 10y + x - 10x - y = 18
⇒ - 9x + 9y = 18
⇒ - x + y = 2  .....(ii)
From Equation (i) and (ii), x = 5 y = 7
∴ The difference between the digits of number
= y - x = 7 - 5 = 2

10. The time taken by a train to cross a man travelling in another train is 10 sec, when the other train is travelling in the opposite direction. However, it takes 20 sec, if both the trains are travelling in the same direction. The length of the first train is 200 m and that of the second train is 150 m. What is the speed of the first train ?

Correct Answer: (c) 54 km/h
Solution: