SOLVED PAPER 2024 (CDS) (I) (Mathematics)

Total Questions: 100

11. If the equation x cosθ = x² + p has a real solution for every θ where 0 ≤ θ ≤ π/4 , then which one of the following is correct?

Correct Answer: (b) p ≤ 1/8
Solution:

12. What is the difference between the greatest value and the least value of cos²θ+ 3sin²θ + 2 ?

Correct Answer: (c) 2
Solution:

13. ABC is a right angled triangle, right - angled at B such that AB = 6 cm and BC = 8cm What is the perimeter of the square inscribed in the triangle ABC with maximum area?

Correct Answer: (b) 96/7 cm
Solution:

14. What is the greatest value of K for which 2x² - 4x + K = 0 has real roots?

Correct Answer: (b) 2
Solution:Given equation,
2x² - 4x + k = 0
Here, a = 2, b = - 4 and c = k
For real roots,
b²- 4ac = 0
⇒ (- 4)² - 4 × 2 × k = 0
⇒ 16 - 8k = 0
∴ k = 16/8 = 2

15. Consider the following data :

110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements :
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0.1
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct?

Correct Answer: (b) 2 and 3 only
Solution:Arranging the data in ascending order,
36, 41, 43, 61, 71, 92, 93, 95, 99, 110, 110, 127
n = 12 (even)

16. What is the digit at the 100th place of number (225)⁴⁰ ?

Correct Answer: (a) 6
Solution:Last three digit of (225)ⁿ = ....625,
where n is a natural number.
[∵ (225)² = 50625, (225)³ = 11390625 ,...]
Hence, digit of hundred's place = 6

17. If a, b, c, d are natural numbers, then how many possible remainders are there when 1ᵃ + 2ᵇ + 3ᶜ + 4ᵈ is divided by 10?

Correct Answer: (c) 5
Solution:

18. If n is a natural number, then what is the sum of all distinct remainders of 4ⁿ + 6ⁿ + 9ⁿ + 11ⁿ when divided by 10 for various values of n ?

Correct Answer: (b) 4
Solution:

19. When the number (12345678910111213 ....... 99100) is divided by 16, what will be the remainder?

Correct Answer: (b) 12
Solution:According to the question,
Number (1 2 3 4 5 6 7 8 9 10 11 12 13 99 100)
is divided by 16 then,
9100/12 = 568¹²⁄₁₆
Hence, remainder = 12

20. A, B, C, D can complete a work in 3, 6, 9, 12 h, respectively. Further, only one person can work at a time in each hour and nobody can work for two consecutive hours. It is not necessary to engage all. What is the minimum number of hours that they will take to finish the work?

Correct Answer: (c) 4
Solution:According to the given information,

LCM of 3, 6, 9 and 12 = 36
Efficiency of A = 36/3 = 12
Efficiency of B = 36/6 = 6
Efficiency of C = 36/9 = 4
and Efficiency of D = 36/12 = 3
According to question,
for completing the work in minimum time we have to need men of highest efficiency.
Two hours work by A and B together = 12 + 6 = 18
Again two hours work of A and B = 12 + 6 =18
∴ Total time taken to complete the work = 2 + 2 = 4h