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

Total Questions: 100

21. Consider the following statements

I. If (a + b) is directly proportional to (a - b) then (a² + b²) is directly proportional to ab.
II. If a is directly proportional to b, then, (a² - b²) is directly proportional to ab.

Which of the statements given above is/are correct?

Correct Answer: (c) Both I and II
Solution:

22. If (3a + 6b + c + 2d) × (3a - 6b - c + 2d)

= (3a - 6b + c - 2d) × (3a + 6b - c - 2d),
then which one of the following is correct?

Correct Answer: (c) ad = bc
Solution:

23. If 3sinθ + 5cosθ = 5, then what is the value of 5sinθ - 3cosθ equal to?

Correct Answer: (b) -3
Solution:

24. The combined age of a man and his wife is 6 times the combined age of their children. 2 yr ago their combined age was 10 times the combined age of their children; and 6 yr later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 yr old ?

Correct Answer: (b) 3
Solution:

25. What is 3 (sin x - cos x)⁴+ 6

(sin x + cos x)² + 4 (sin x)⁶
+ 4 (cos x)⁶ equal to?

Correct Answer: (c) 13
Solution:

26. What is the value of sinθ + cosθ if θ satisfies the equation

cot² θ - (√3) + 1) cot θ + √3 = 0;
0 < θ < π ÷ 4 ?

Correct Answer: (c) (√3+1) ÷ 2
Solution:

27. Which one of the following is a value of θ if θ satisfies the equation tan 2θ tan 4θ-1 = 0 ; 0 < θ < π÷2 ?

Correct Answer: (a) π ÷ 12
Solution:

28. Solved

Correct Answer: (a)
Solution:

29. How many values of theta will satisfy

the equation (sin² θ - 4sin θ + 3)
(4 - cos² θ + 4sin θ) = 0, where 0 < θ < π÷2 ?

Correct Answer: (a) None
Solution:

30. If x sin³ θ + y cos³ θ = sin θ cos θ

and x sin θ - y cos θ = 0 for every
θ ∈ (0, π/2), then what is x² + y² equal to?

Correct Answer: (b) 1
Solution: