SOLVED PAPER 2022 (CDS) (I) (Elementary Mathematics)

Total Questions: 100

41. Consider a question and two statements:

Question
Is 3x + 2y positive?

Statements
I. x³ = - 29.8
II. y³ = 3x

Which one of the following is correct in respect of the question and the statements?

Correct Answer: (c) Both Statement I and Statement II are together sufficient to answer the question
Solution:Statement I :  x³ = - 29.8
⇒ x = - ve
Now using equation in Statement II,
we get
⇒ y³ = 3x
⇒ y³ = 3(- ve) = - ve
So, y is also negative.
Now, if both the terms x and y are
negative.
Sum of negative terms is always
negative
⇒ 3(- ve) + 2(- ve) = - ve
⇒ 3x + 2y is negative.
∴ Both Statement I and Statement II
are together sufficient to answer the
question.

42. Consider a question and two statements:

Question
Does the equation ax² + bx + c = 0
have real roots of opposite sign ?

Statements
I. The discriminant D > 0
II. (c/a) > 0

Which one of the following is correct in respect of the question and the statements?

Correct Answer: (b) Statement II alone is sufficient to answer the question
Solution:

43. Consider a question and two statements:

Question
Is a² + b² + c² - ab - bc - ca
(a, b, c are distinct real numbers)
always positive?

Correct Answer: (d) Neither Statement I nor Statement II is required to answer the question
Solution:


44. Solved


Correct Answer: (d) Neither Statement I nor Statement II is required to answer the question
Solution:


45. How many quadratic equations have the sum of their roots equal to the product of their roots?

Correct Answer: (d) Infinitely many
Solution:

46. Solved

Correct Answer: (b)
Solution:

47. Solved

Correct Answer: (a)
Solution:


48. What is the minimum value of

sin⁴θ + cos⁴θ - 2sin²θ cos²θ ?

Correct Answer: (a) 0
Solution:

49. If cosθ + secθ - 2 = 0, where

0 ≤ θ < π/2, then what is the value of
cos⁴θ + sec⁴θ - 2

Correct Answer: (c) 0
Solution:

50. If y = cos² x + sec² x where

0 ≤ θ ≤ π/2, then which one of the
following is correct?

Correct Answer: (d) y ≥ 2
Solution: