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

Total Questions: 100

21. If α and β are the roots of the equation x² - 7x + 1 = 0 , then what is the value of α⁴ + β⁴ ?

Correct Answer: (a) 2207
Solution:

22. Consider the following statements in respect of all factors of 360

1. The number of factors is 24.
2. The sum of all factors is 1170.

Which of the above statements is/are correct?

Correct Answer: (c) Both 1 and 2
Solution:All factors of 360 are mentioned below
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20,
24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360
Clearly, we can see that,
there are 24 factors of 360
Sum of all factors = 1170.

23. Consider a 6-digit number of the form XYXYXY. The number is divisible by

Correct Answer: (d) 3, 7, 13 and 37
Solution:Number = XYXYXY
= XY × 10000 + XY × 100 + XY
= XY(10000 + 100 + 1)
= XY(10101)
= XY[3 × 7 × 13 × 37]
Hence, number XYXYXY is
divisible by 3, 7, 13 and 37.

24. What is the HCF of 3²⁹ - 9 and 3³⁸ - 9 ?

Correct Answer: (d) 3¹¹ - 9
Solution:


25. Solved

Correct Answer: (b) 4
Solution:

26. Let m and n be natural numbers. What is the minimum value of (m + n) such that, 33m + 22n is divisible by 121?

33m + 22n = 11(3m + 2n)
And given 33m + 22n is divisible by
121 So, 11(3m + 2n) is multiple of 11 × 11.
and minimum value of
(3m + 2n) = 11
Put m = 3 and n = 1
3m + 2n = 3(3) + 2(1)
9 + 2 = 11
∴ m + n = 3 + 1 = 4
Minimum value of (m + n) = 4

Correct Answer: (b) 4

27. The product of two numbers is 2160 and their HCF is 12. If the sum of the squares of the two numbers is 4896, then what is the mean of the two numbers ?

Correct Answer: (c) 48
Solution:

28. The age of Q exceeds the age of P by 3 yr. The age of R is twice the age of P and the age of Q is twice the age of S. Further, the age difference of R and S is 30 yr. What is the sum of the ages of P and Q?

Correct Answer: (d) 45 year
Solution:Let Age of S = x year
∴ Age of Q = 2x year
And
Age of Q = Age of P + 3
⇒ 2x = of P + 3
⇒ Age of P = (2x - 3) year
Age of R = 2(2x - 3) = 4x - 6
According to the question,
4x - 6 - x = 30 ⇒ 3x = 36
⇒ x = 12
∴ Age of P + Age of Q
= 2x - 3 + 2x = 4x - 3
=  4 × 12 - 3 = 48 - 3
= 45 year

29. If a, b and c are the sides of a triangle ABC, then √a+ √b - √c is always

Correct Answer: (b) Positive
Solution:Let a = 3
b = 4
c = 5 are the sides of a ΔABC
Now, √a + √b - √c
= √3 + √4 - √5
= 1.732 + 2 - 2.2
= 3.7 - 2.2
= 1.5 (positive)

30. There are four bells which ring at an interval of 15 min, 25 min, 35 min and 45 min respectively. If all of them ring at 9 am, how many more times will they ring together in the next 72 h?

Correct Answer: (c) 2
Solution: