(Mathematics) NDA/NA SOLVED PAPER 2023 – II (01 to 50)

Total Questions: 50

31. What is the number of real roots of the equation?

Consider the equation (1 - x)⁴ + (5 - x)⁴ = 82.

Correct Answer: (b) 2
Solution:

32. What is the sum of all the roots of the equation?

Consider the equation (1 - x)⁴ + (5 - x)⁴ = 82.

Correct Answer: (b) 12
Solution:Since, sum of all roots
= 4 + 2 + 3 - 5i + 3 + 5i = 12

33. What are the roots of equation - I ?

Correct Answer: (b) -1, ω, ω²
Solution:

34. Which one of the following is a root of equation -II?

Correct Answer: (d) ω
Solution:

35. What is the number of common roots of equation -I and equation -II?

Correct Answer: (c) 2
Solution:Since, the common roots of equation - I and equation - II
are ω, ω²
⇒ Number of common roots = 2

36. If k = c/2, ( c ≠ 0), then the roots of the equation are:

A quadratic equation is given by
(a + b)x² - (a + b + c)x + k = 0 , where a, b, c are real.

Correct Answer: (b) Real and unequal
Solution:

37. If k = c, then the roots of the equation are:

A quadratic equation is given by
(a + b)x² - (a + b + c)x + k = 0 , where a, b, c are real.

Correct Answer: (c)
Solution:(c)

38. What is T₁ + 2T₂ + 3T₃ +______+nTₙ equal to?

Let (1+x)ⁿ = 1 + T₁x + T₂x² + T₃x³ +____+Tₙxⁿ

Correct Answer: (d) n2ⁿ⁻¹
Solution:

39. What is 1 - T₁ + 2T₂ - 3T₃+_____+(-1)ⁿ nTₙ equal to?

Let (1+x)ⁿ = 1 + T₁x + T₂x² + T₃x³ +____+Tₙxⁿ

Correct Answer: (d) 1
Solution:

40. What is T₁ + T₂ + T₃+_____+Tₙ equal to?

Let (1+x)ⁿ = 1 + T₁x + T₂x² + T₃x³ +____+Tₙxⁿ

Correct Answer: (b) 2ⁿ⁻¹
Solution: