Mathematics (NDA/NA SOLVED PAPER 2022-II) (51 to 100)

11. An equilateral triangle is inscribed in a parabola x² = √3 y where one vertex of the triangle is at the vertex of the parabola. If p is the length of side of the triangle and q is the length of the latus rectum, then which one of the following is correct?

(a) p = q
(b) p = √3 q
(c) p = 2√3 q
(d) 2√3 p = q

Correct Answer: (c) p = 2√3 q

Solution:

12. Consider the points A(2, 4, 6), B(- 2, - 4, - 2), C(4, 6, 4) and D(8, 14, 12). Which of the following statements is/are correct?

1. The points are the vertices of a rectangle ABCD.
2. The mid-point of AC is the same as that of BD.
Select the correct answer using the code given below:

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Correct Answer: (b) 2 only

Solution:

13. Consider the equation of a sphere x² + y² + z² - 4x - 6y -8z - 16 = 0

Which of the following statements is/are correct?
1. z-axis is tangent to the sphere.
2. The centre of the sphere lies on the plane x + y + z - 9 = 0.
Select the correct answer using the code given below :

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 21.

Correct Answer: (b) 2 only

Solution:x² + y² + z²- 4x - 6y - 8z - 16 = 0
⇒ (x² - 4x + 4) + (y² - 6y + 9) + (z² - 8z + 16)
⇒ (x - 2)² + (y - 3)² + (z - 4)² = (3√5)² Centre of sphere = (2, 3, 4)
Radius = 3√5
Since, radius 3√5 is not equal to √2² + 3²= √13
Z-axis is not tangent to the sphere, so, statement 1 is wrong.
2 + 3 + 4 - 9 = 0
Centre of the sphere lies on the plane x + y + z - 9 = 0
So, statement 2 is correct.

14. A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?

Correct Answer: (c)

Solution:

15. Consider the following statements:

1. The direction ratios of y-axis can be < 0 , 4, 0 >
2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>
Which of the statements given above is/are correct?