Mathematics (Solved Paper 2024 – II) (51 to 100)

Total Questions: 50

11. What is <BAC equal to?

A triangle ABC is inscribed in the circle x² + y² = 100. B and C have coordinates (6, 8) and (-8, 6) respectively.

Correct Answer: (c) π/4 or 3π/4
Solution:

12. What are the coordinates of A?

A triangle ABC is inscribed in the circle x² + y² = 100. B and C have coordinates (6, 8) and (-8, 6) respectively.

Correct Answer: (d) Cannot be determined due to insufficient data
Solution:Since, the point A can be taken at any point on the circumference of circle which subtends π/4 to the chord so coordinates of A cannot be determined due to insufficient data.

13. What are the coordinates of vertex D?

ABCD is an isosceles trapezium and AB is parallel to DC. Let A (2, 3) B(4, 3) C(5,1) be the vertices.

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

14. What is the point of intersection of the diagonals of the trapezium?

ABCD is an isosceles trapezium and AB is parallel to DC. Let A (2, 3) B(4, 3) C(5,1) be the vertices.

Correct Answer: (c) (7/2, 2)
Solution:Since Diagonal of Isosceles trapezium bisect each other then let intersecting point of diagonal is O.

Coordinates of O is (2 + 5/2 , 3 + 1/2) i.e (7/2 , 2)

15. What is the diameter of the sphere?

Let 2x² + 2y² + 2z² + 3x + 3y + 3z - 6 = 0 be a sphere.

Correct Answer: (b) 5√3/2
Solution:

16. The centre of the sphere lies on the plane

Let 2x² + 2y² + 2z² + 3x + 3y + 3z - 6 = 0 be a sphere.

Correct Answer: (d) 4x + 8y + 8z + 15 = 0
Solution:Centre of sphere is C(- 3/4, - 3/4, - 3/4)

4x + 8y + 8z + 15 = 0 is the only plane which satisfy the centre of sphere

- 3 + - 6 - 6 + 15 =0 ⇒ 0 = 0 Option (d) is correct.

17. Which of the following are the direction ratios of S?

Let S be the line of intersection of two planes x + y + z = 1 and 2x + 3y - 4z = 8

Correct Answer: (b)
Solution:2 planes are x + y + z = 1 and 2 + 3y - 4z = 8 .

Let a, b, c are dr's is of lines of formed by intersection of planes then the lines is perpendicular to normal of both planes.

a + b + c = 0                   ...(i)

and 2a + 3b - 4c = 0   ... (ii)

a/(- 4 - 3) = b/(2 + a) = c/(3 - 2) = λ (Solving (i) and (ii)

a = 7λ,  b = 6λ, c = λ

Dr's or < - 7 , 6, 1 >

18. If (l, m, n) are direction cosines of S, then what is the value of 43 (l² - m² - n²) ?

Let S be the line of intersection of two planes x + y + z = 1 and 2x + 3y - 4z = 8

Correct Answer: (a) 6
Solution:

19. What are the direction ratios of the line?

Let L / x + y + z + 4 = 0 = 2x - y - z + 8 be a line and P : x + 2y + 3z + 1 = 0 be a plane

Correct Answer: (c)
Solution:L : x + y + z + 4 = 0 = 2x - y - z - 8 and

P : x + 2y + 3z + 1 = 0 is a plane.

Let Dr's of line < a, b, c > Since, line L obtained by intersection of planes

x + y + z +4=0 and 2x - y - z - 8 = 0

So, a + b + c = 0    ..(i)

2a - b - c = 0         ...(ii)

(line is perpendicular to normal of both plane)

a/(- 1 + 1) = b/(2 + 1) = c/(- 1 - 2) = λ (solving (i) and (ii)

a = 0,  b = 3λ b = 3λ, c = - 3 λ

Dr's of line is < 0 , 3, - 3 > or < 0 1, - 1 >

20. What is the point of intersection of L and P?

Let L / x + y + z + 4 = 0 = 2x - y - z + 8 be a line and P : x + 2y + 3z + 1 = 0 be a plane

Correct Answer: (d) (-4,-3, 3)
Solution:To find point of intersection of L and P.

The point must satisfy both line L and P.

From option (d) (-4, -3, 3), will satisfy both line L and plane P.

So, option (d) is correct answer.