Mathmatics NDA/NA Solved Paper 2020-I (1-50)

11. Under which condition, are the points (a, b), (c, d) and (a - c, b - d) collinear ?

(a) ab = cd
(b) ac = bd
(c) ad = bc
(d) abc = d

Correct Answer: (c) ad = bc

Solution:

Given points (a,b), (c,d) and (a-c, b-d) are collinear, if a(d-b-d) + b(a-c-c) + 1(c(b-d) - d (a-c)) = 0
2ad - ab + ab - 2ac + bc -ad = 0
⇒ ad - bc = 0
∴ ad = bc.

12. Let ABC be a triangle. If D(2, 5) and E(5, 9) are the mid-points of the sides AB and AC respectively, then what is the length of the side BC?

(a) 8
(b) 10
(c) 12
(d) 14

Correct Answer: (b) 10

Solution:

13. If the foot of the perpendicular drawn from the point (0, k) to the line 3𝓍 - 4y - 5 = 0 is (3, 1), then what is the value of k ?

(a) 3
(b) 4
(c) 5
(d) 6

Correct Answer: (c) 5

Solution:

Slope of the line, 3𝓍 - 4y - 5 = 0 is m = 3/4
Slope of any perpendicular to 3𝓍 - 4y - 5 = 0 m' = -(3/4).
Now required line passes through the points (0,k) and (3,1)
∴ (k-1)/(0-3) = -(4/3) ⇒ K = 5.

14. What is the obtuse angle between the lines whose slopes are 2 - √3 and 2 + √3 ?

(a) 105°
(b) 120°
(c) 135°
(d) 150°

Correct Answer: (b) 120°

Solution:

15. If 3𝓍 - 4y - 5 = 0 and 3𝓍 - 4y + 15 = 0 are the equations of a pair of opposite sides of a square, then what is the area of the squares ?

(a) 4 square units
(b) 9 square units
(c) 16 square units
(d) 25 square units

Correct Answer: (c) 16 square units

Solution:

We know that pair of opposite sides of my square are parallel.
So, distance between two parallel sides
= Side length of the square
∴ Side length of the square