Mathematics (NDA/NA SOLVED PAPER 2023-I) (51 to 100)

Total Questions: 50

1. If Δ(a, b, c, α) = 0 for every α > 0 then which one of the following is correct?

Correct Answer: (b) a, b, c are in GP
Solution:

2. If Δ(7, 4, 2, α) = 0 then α is a root of which one of the following equation?

Correct Answer: (c) 7x² + 8x + 2 = 0
Solution:Δ(7, 4, 2, α) = 0
a = 7, b = 4, c = 2
aα² + 2bα+ c = 0
7α² + 8α + 2 = 0
α is a root of 7x² + 8x + 2 = 0

3. What is the least value of m (θ)?

Given that m(θ) = cot²θ + n² tan²θ+ 2n, where n is a fixed positive real number.

Correct Answer: (d) 4n
Solution:

4. Under what condition does m attain the least value?

Given that m(θ) = cot²θ + n² tan²θ+ 2n, where n is a fixed positive real number.

Correct Answer: (b) n = cot²θ
Solution:For minimum value,
cot²θ + n² tan²θ + 2n = 4n
(cotθ)²+ (n tanθ)² - 2(cotθ . n tan θ) = 0
(cotθ - n tanθ)² = 0
cotθ = n tanθ
cot²θ = n

5. What is the equation of diagonal through origin?

A quadrilateral is formed by the lines x = 0, y = 0.
x + y = 1 and 6x + y = 3.

Correct Answer: (c) 3x - 2y = 0
Solution:

6. What is the equation of other diagonal?

A quadrilateral is formed by the lines x = 0, y = 0.
x + y = 1 and 6x + y = 3.

Correct Answer: (d) 2x + y - 1 = 0
Solution:

7. What is PE + PF equal to?

P(x,y) is any point on the ellipse x² + 4y² = 1. Let E, F be the foci of the ellipse.

Correct Answer: (b) 2
Solution:

8. Consider the following points:

P(x,y) is any point on the ellipse x² + 4y² = 1. Let E, F be the foci of the ellipse.
Which of the following points lie on latus rectum of ellipse?

Correct Answer: (d) 1, 2 and 3
Solution:

9. What is the area of minor segment?

The line y = x partitions the circle
(x - a)² + y² = a² in two segments.

Correct Answer: (a)
Solution:

10. What is the area of major segment?

The line y = x partitions the circle
(x - a)² + y² = a² in two segments.

Correct Answer: (b)
Solution: