NDA/NA SOLVED PAPER 2016-I (MATHEMATICS)

Total Questions: 120

61. What is f(1) equal to?

Let f : R → R be a function such that
f(x)= x³ + x² f'(1) + xf''(2) + f'''(3)
for x ∈ R

Correct Answer: (d) 4
Solution:

62. What is f'(1) equal to?

Let f : R → R be a function such that
f(x)= x³ + x² f'(1) + xf''(2) + f'''(3)
for x ∈ R

Correct Answer: (b) -5
Solution:f'(1) = - 5

63. What is f'''(10) equal to?

Let f : R → R be a function such that
f(x)= x³ + x² f'(1) + xf''(2) + f'''(3)
for x ∈ R

Correct Answer: (c) 6
Solution:f''' (10) = 6

64. Consider the following:

Let f : R → R be a function such that
f(x)= x³ + x² f'(1) + xf''(2) + f'''(3)
for x ∈ R

1. f(2) = f(1) - f(0)
2. f''(2) - 2 f'(1) = 12

Which of the above is/are correct?

Correct Answer: (c) Both 1 and 2
Solution:1. f(1) - f(0) = 4 - 6 = -2
f(2) = 8 - 20 + 4 + 6 = - 2
Hence f(2) = f(1) - f(0)
∴ Statement (1) is correct.

2. f''(2) - 2f'(1) = 2 - 2(- 5)
f'' (2) - 2 f'(1) = 12
∴ Statement (2) is correct.

65. What are the direction ratios of the line of intersection of the given planes?

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2 , x + y - z = 1 and the point (1, 0, 1).

Correct Answer: (a) ⟨2, - 5, - 3⟩
Solution:

66. What is the equation of the plane P?

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2 , x + y - z = 1 and the point (1, 0, 1).

Correct Answer: (b) 5x + 2y - 5 = 0
Solution:Eq. of plane through two given planes is :
(2x - y + 3z - 2) + ʎ (x + y - z - 1) = 0
∴ It passes through (1, 0, 1)
∴ 3 - ʎ = 0 ⇒ ʎ = 3
∴ Eq. of plane is:
5x + 2y - 5 = 0

67. If the plane P touches the sphere x² + y² + z² = r² then what is r equal to?

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2 , x + y - z = 1 and the point (1, 0, 1).

Correct Answer: (c)
Solution:

68. What is f'(4) equal to?

Consider the function f(x) = |x² - 5x + 6|

Correct Answer: (c) 3
Solution:

69. What is f''(2.5) equal to?

Consider the function f(x) = |x² - 5x + 6|

Correct Answer: (b) -2
Solution:f"(2.5) = -2

70. Solve the following equation

Let f(x) be the greatest integer function and g(x) be the modulus function.

Correct Answer: (c) 1
Solution:f(x) → greatest integer function, f(x) = [x], g(x) → modulus function, g(x) = |x|