1. Suppose cos A is given. If only one value of cos (A/2) is possible, then A must be

(a) An odd multiple of 90°

(b) A multiple of 90°

(c) An odd multiple of 180°

(d) A multiple of 180°

2. If cos α + cos β + cos γ = 0, where 0 < α ≤ π/2, 0 < β ≤ π/2, 0 < γ ≤ π/2, then what is the value of sin α + sin β + sin γ?

(a) 0

(b) 3

(c) 5√2/2

(d) 3√2/2

3. The maximum value of sin (x + π/5) + cos (x + π/5), where x ∈ (0, π/2), is attained at

(a) π/20

(b) π/15

(c) π/10

(d) π/2

4. What is the distance between the points which divide the line segment joining (4, 3) and (5, 7) internally and externally in the ratio 2 : 3?

(a) 12√17/5

(b) 13√17/5

(c) √17/5

(d) 6√17/5

5. What is the angle between the straight lines (m2 – mn) y = (mn + n2) x + n3 and (mn + m2) y = (mn – n2) x + m3, where m > n?

(a) tan-1 (2mn/m2 + n2)

(b) tan-1 (4m2n2/m4 – n4)

(c) tan-1 (4m2n2/m4 + n4)

(d) 45°

6. What is the equation of the straight line cutting off an intercept 2 from the negative direction of y-y-axis and inclined at 30° with the positive direction of x-axis?

(a) x – 2 √3 y – 3√2 = 0

(b) x + 2 √3 y – 3√2 = 0

(c) x + √3 y – 2√3 = 0

(d) x – √3 y – 2√3 = 0

7. What is the equation of the line passing through the point of intersection of the lines x + 2y – 3 = 0 and 2x – y + 5 = 0 and parallel to the line y – x + 10 = 0?

(a) 7x – 7y + 18 = 0

(b) 5x – 7y + 18 = 0

(c) 5x – 5y + 18 = 0

(d) x – y + 5 = 0

8. Consider the following statements:

1. The length p of the perpendicular from the origin to the line ax + by = c satisfies the relation p2 = c2/a2 + b2.

2. The length p of the perpendicular from the origin to the line x/a + y/b = 1 satisfies the relation 1/p2 = 1/a2 + 1/b2.

3. The length p of the perpendicular from the origin to the line y = mx + c satisfies the relation 1/p2 = 1 + m2 + c2/c2.

Which of the above is/are correct?

(a) 1, 2 and 3

(b) 1 only

(c) 1 and 2 only

(d) 2 only

9. What is the equation of the ellipse whose vertices are (± 5, 0) and foci are at (± 4, 0)?

(a) x2/25 + y2/9 = 1

(b) x2/16 + y2/9 = 1

(c) x2/25 + y2/16 = 1

(d) x2/9 + y2/25 = 1

10. What is the equation of the straight line passing through the point (2, 3) and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis?

(a) 2x + y = 5

(b) 2x + y = 7

(c) x + 2y = 7

(d) 2x – y = 1

11. Let the coordinates of the points A, B, C be (1, 8, 4), (0, – 11, 4) and (2, – 3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?

(a) (3, 4, -2)

(b) (4, -2, 5)

(c) (4, 5, -2)

(d) (2, 4, 5)

12. What is the equation of the plane passing through the points (- 2, 6, – 6), (- 3, 10, – 9)
and (- 5, 0, – 6)?

(a) 2x – y – 2z = 2

(b) 2x + y + 3z = 3

(c) x + y + z = 6

(d) x – y – z = 3

13. A sphere of constant radius r through the origin intersects the coordinate axes in A, B and C. What is the locus of the centroid of the triangle ABC?

(a) x2 + y2 + z2 = r2

(b) x2 + y2 + z2 = 4r2

(c) 9 (x2 + y2 + z2) = 4r2

(d) 3(x2 + y2 + z2) = 2r2

14. The coordinates of the vertices P, Q and R of a triangle PQR are (1, -1, 1), (3, -2, 2) and (0, 2, 6) respectively. If ∠RQP = θ, then what is ∠PRQ equal to?

(a) 30° + θ

(b) 45° – θ

(c) 60° – θ

(d) 90° – θ

15. The perpendiculars that fall from any point or the straight line 2x + 11y = 5 upon the two straight lines 24x + 7y = 20 and 4x – 3y = 2 are

(a) 12 and 4 respectively

(b) 11 and 5 respectively

(c) Equal to each other

(d) Not equal to each other

16. The equation of the line, when the portion of it intercepted between the axes is divided by the point (2, 3) in the ratio of 3 : 2, is

(a) Either x + y = 4 or 9x + y = 12

(b) Either x + y = 5 or 4x + 9y = 30

(c) Either x + y = 4 or x + 9y = 120

(d) Either x + y = 5 or 9x + 4 y = 30

17. What is the distance between the straight lines 3x + 4y = 9 and 6x + 8y = 15?

(a) 3/2

(b) 3/10

(c) 6

(d) 5

18. What is the equation to the sphere whose centre is at (- 2, 3, 4) and radius is 6 units?

(a) x2 + y2 + z2 + 4x – 6y – 8z = 7

(b) x2 + y2 + z2 + 6x – 4y – 8z = 7

(c) x2 + y2 + z2 + 4x – 6y – 8z = 4

(d) x2 + y2 + z2 + 4x + 6y + 8z = 4

19. Suppose f : R → R is defined by f(x) = x2/1 + x2. What is the range of the function?

(a) [10, 1)

(b) [10, 1]

(c) (0, 1]

(d) (0, 1)

20. if f(x) = |x| + |x – 1|, then which one of the following is correct?

(a) f(x) is continuous at x = 0 and x = 1

(b) f(x) is continuous at x = 0 but not at x = 1

(c) f(x) is continuous at x = 1 but not at x = 0

(d) f(x) is neither continuous at x = 0 nor at x = 1