Mathematics NDA/NA SOLVED PAPER 2021-II (01 to 50)

Total Questions: 50

1. What is the value of the following?

Correct Answer: (a) 6/17
Solution:

2. equal to?

Let 4 sin²x = 3, where 0 < x <= π. What is tan 3x equal to?

Correct Answer: (c) 0
Solution:

3. Let p, q and 3 be respectively the first, third and fifth terms of an A.P. Let d be the common difference. If the product (pq) is minimum, then what is the value of d?

Correct Answer: (c) 9/8
Solution:

4. Consider the following statements in respect of the roots of the equation x³ - 8 = 0:

1. The roots are non-collinear.
2. The roots lie on a circle of unit radius.
Which of the above statements is/are correct?

Correct Answer: (a) 1 only
Solution:x³ - 8 = 0
⇒ (x - 2) (x² + 2x + 4) = 0
x = 2, -1 + √3 i, -1 - √3i
So, roots non - collinear and lie on a circle of radius 2 units.

5. Let the equation sec x. cosec x = p have a solution, where p is a positive real number. What should be the smallest value of p?

Correct Answer: (c) 2
Solution:

6. For what value of θ, where 0 < θ < π/2, does sin θ + sin θ cos θ attain maximum value?

Correct Answer: (b) π/3
Solution:

7. Consider the following statements in respect of sets

1. The union over intersection of sets is distributive.
2. The complement of union of two sets is equal to intersection of their complements.
3. If the difference of two sets is equal to empty set, then the two sets must be equal.
Which of the above statements are correct?

Correct Answer: (a) 1 and 2 only
Solution:A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) in called distributive.
and (A ∪ B)' = A' ∩ B'
So statements 1 and 2 are true
If A - B = φ then A ⊆ B and A = φ
So, statement 3 is not true.

8. Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct, Let S =(X - Y ) ∪ Z. How many proper subsets does S have?

Correct Answer: (c) 1023
Solution:Given that n(x) = 6,. n(y) = 5 and n(z) = 4.
and all these 15 elements are distinct
x - y = x and x ∩ z = φ.
n(s) = [(x - y) ∪ z] = n(x ∪ z)
= n(x) + n(z) = 6 + 4 = 10
Number of proper subsets of S = 20¹⁰ -1 = 1023.

9. Consider the following statements in respect of relations and functions:

1. All relations are functions but all functions are not relations.
2. A relation from A to B is a subset of Cartesian product A × B
3. A relation in A is a subset of Cartesian product A × A
Which of the above statements are correct?

Correct Answer: (b) 2 and 3 only
Solution:Since a radiation R from a non - empty set A to a non - empty set B is a subset of the cartesian product A × B. so statements 2 and 3 are true.
Some special type of relation is called function. So, statement 1 is wrong.

10. If log₁₀ 2log₂ 10 + log₁₀(10ˣ) = 2, then what is the value of x?

Correct Answer: (b) 1
Solution: