JEE Main - Direction ratios and direction cosines Online Test (MCQ Quiz)

1. What are direction ratios of a line?

A) The squares of the direction cosines B) The direction cosines themselves C) Any three numbers proportional to the direction cosines D) The sum of the squares of the direction cosines

2. If l, m, n are the direction cosines of a line, what is the relationship between them?

A) l + m + n = 1 B) l^2 + m^2 + n^2 = 0 C) l^2 + m^2 + n^2 = 1 D) l^2 + m^2 + n^2 = -1

3. What are the direction cosines of a line passing through the origin and the point (x, y, z)?

A) (x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x^2+y^2+z^2)) B) (x, y, z) C) (x^2, y^2, z^2) D) (sqrt(x^2+y^2+z^2)/x, sqrt(x^2+y^2+z^2)/y, sqrt(x^2+y^2+z^2)/z)

4. If a, b, c are direction ratios of a line, what are its direction cosines?

A) (a/sqrt(a^2+b^2+c^2), b/sqrt(a^2+b^2+c^2), c/sqrt(a^2+b^2+c^2)) B) (a^2, b^2, c^2) C) (sqrt(a^2+b^2+c^2)/a, sqrt(a^2+b^2+c^2)/b, sqrt(a^2+b^2+c^2)/c) D) (a, b, c)

5. What are the direction ratios of the x-axis?

A) (1, 0, 0) B) (0, 1, 0) C) (0, 0, 1) D) (1, 1, 1)

6. What are the direction cosines of the y-axis?

A) (1, 0, 0) B) (0, 1, 0) C) (0, 0, 1) D) (0, -1, 0)

7. If a line makes angles alpha, beta, gamma with the positive x, y, and z axes respectively, then what are its direction cosines?

A) (cos(alpha), cos(beta), cos(gamma)) B) (sin(alpha), sin(beta), sin(gamma)) C) (tan(alpha), tan(beta), tan(gamma)) D) (alpha, beta, gamma)

8. A line has direction ratios -18, 12, -4. What are its direction cosines?

A) (-9/11, 6/11, -2/11) B) (9/11, -6/11, 2/11) C) (-18/11, 12/11, -4/11) D) (18/11, -12/11, 4/11)

9. If the direction ratios of a line are 2, 3, 6, what is the magnitude of its direction cosine vector?

A) 1 B) 7 C) sqrt(7) D) 49

10. What are the direction cosines of a line parallel to the vector 2i - 3j + 6k?

A) (2/7, -3/7, 6/7) B) (2/sqrt(7), -3/sqrt(7), 6/sqrt(7)) C) (2/49, -3/49, 6/49) D) (7/2, -7/3, 7/6)

11. If a line has direction ratios proportional to 1, -1, 2, what are its direction cosines?

A) (1/sqrt(6), -1/sqrt(6), 2/sqrt(6)) B) (1/2, -1/2, 2/2) C) (1/3, -1/3, 2/3) D) (1/sqrt(2), -1/sqrt(2), 2/sqrt(2))

12. Two lines have direction ratios (a1, b1, c1) and (a2, b2, c2). Under what condition are they perpendicular?

A) a1*a2 + b1*b2 + c1*c2 = 0 B) a1*a2 + b1*b2 + c1*c2 = 1 C) a1*a2 = b1*b2 = c1*c2 D) a1/a2 = b1/b2 = c1/c2

13. What are the direction cosines of a line perpendicular to both the x and y axes?

A) (0, 0, 1) or (0, 0, -1) B) (1, 0, 0) or (-1, 0, 0) C) (0, 1, 0) or (0, -1, 0) D) (1, 1, 0)

14. If a line has direction ratios (0, 0, k) where k is non-zero, what is its direction?

A) Parallel to the x-axis B) Parallel to the y-axis C) Parallel to the z-axis D) Along the origin

15. The direction cosines of a line are (l, m, n). If the line makes an angle of 60 degrees with the positive x-axis, what is the value of l?

A) 1/2 B) -1/2 C) sqrt(3)/2 D) -sqrt(3)/2

16. If a line has direction ratios (1, 2, 3), what is the cosine of the angle it makes with the x-axis?

A) 1/sqrt(14) B) 2/sqrt(14) C) 3/sqrt(14) D) 1/14

17. What is the sum of the products of the direction cosines of two perpendicular lines?

A) 0 B) 1 C) -1 D) Depends on the lines

18. A line passes through the origin and has direction ratios (1, 1, 1). What is the angle it makes with the x-axis?

A) 30 degrees B) 45 degrees C) 60 degrees D) 90 degrees

19. If the direction ratios of two lines are (1, 2, -2) and (x, 2x, 2x), and the lines are perpendicular, find x.

A) 1 B) -1 C) 0 D) 2

20. What are the direction cosines of a line that makes equal angles with all three coordinate axes?

A) (1/sqrt(3), 1/sqrt(3), 1/sqrt(3)) and its variations with negative signs B) (1, 0, 0) C) (1/2, 1/2, 1/2) D) (1/sqrt(2), 1/sqrt(2), 0)

21. If a line has direction ratios (a, b, c), what are the direction ratios of the line parallel to it in the opposite direction?

A) (-a, -b, -c) B) (a, b, c) C) (1/a, 1/b, 1/c) D) (-1/a, -1/b, -1/c)

22. A line has direction ratios (0, 5, 0). What is its direction?

A) Parallel to the x-axis B) Parallel to the y-axis C) Parallel to the z-axis D) Along the line y=x in the xy-plane

23. If l, m, n are direction cosines, which of the following is NOT possible?

A) l=1, m=0, n=0 B) l=1/2, m=1/2, n=1/sqrt(2) C) l=1/3, m=2/3, n=2/3 D) l=1, m=1, n=1

24. What are the direction ratios of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2)?

A) (x2-x1, y2-y1, z2-z1) B) (x1-x2, y1-y2, z1-z2) C) ((x2-x1)/PQ, (y2-y1)/PQ, (z2-z1)/PQ) D) (x1+x2, y1+y2, z1+z2)

25. If a line has direction cosines (2/3, -2/3, 1/3), what are its direction ratios?

A) (2, -2, 1) B) (4/9, 4/9, 1/9) C) (2/9, -2/9, 1/9) D) (2/3, -2/3, 1/3)

26. What is the relationship between the direction ratios (a,b,c) and the direction cosines (l,m,n) of a line?

A) l=a, m=b, n=c B) l=a/sqrt(a^2+b^2+c^2), m=b/sqrt(a^2+b^2+c^2), n=c/sqrt(a^2+b^2+c^2) C) l=sqrt(a^2+b^2+c^2)/a, m=sqrt(a^2+b^2+c^2)/b, n=sqrt(a^2+b^2+c^2)/c D) l=1/a, m=1/b, n=1/c

27. If a line makes angles alpha, beta, gamma with the positive x, y, z axes, then cos^2(alpha) + cos^2(beta) + cos^2(gamma) is equal to:

A) 0 B) 1 C) 2 D) Depends on the angles

28. What are the direction ratios of the z-axis?

A) (1, 0, 0) B) (0, 1, 0) C) (0, 0, 1) D) (1, 1, 0)

29. A line has direction ratios (2, -1, 2). What is the cosine of the angle it makes with the positive y-axis?

A) -1/3 B) 2/3 C) -1/sqrt(5) D) 2/sqrt(5)

30. If two lines have direction cosines (l1, m1, n1) and (l2, m2, n2) and are perpendicular, then:

A) l1*l2 + m1*m2 + n1*n2 = 0 B) l1*l2 + m1*m2 + n1*n2 = 1 C) l1*l2 = m1*m2 = n1*n2 D) l1/l2 = m1/m2 = n1/n2

31. If a line has direction ratios (a, b, c), what is the value of the constant k such that (ka, kb, kc) are its direction cosines?

A) k = 1/sqrt(a^2+b^2+c^2) B) k = sqrt(a^2+b^2+c^2) C) k = 1 D) k = a^2+b^2+c^2

32. What are the direction cosines of a line parallel to the vector i - j + k?

A) (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)) B) (1, -1, 1) C) (1/3, -1/3, 1/3) D) (sqrt(3), -sqrt(3), sqrt(3))

33. If a line has direction ratios (3, 4, 0), what is its direction?

A) Parallel to the x-axis B) Parallel to the y-axis C) Parallel to the z-axis D) Lies in the xy-plane perpendicular to the x-axis

34. The direction ratios of two lines are (1, 2, 3) and (2, 4, 6). What can be said about these lines?

A) They are perpendicular B) They are parallel C) They intersect at the origin D) They are skew

35. If a line makes angles of 45 degrees and 60 degrees with the positive x and y axes respectively, what is the angle it makes with the positive z-axis?

A) 45 degrees B) 60 degrees C) 90 degrees D) 120 degrees

36. What are the direction cosines of a line passing through the origin and the point (-3, 4, -12)?

A) (-3/13, 4/13, -12/13) B) (3/13, -4/13, 12/13) C) (-3/sqrt(253), 4/sqrt(253), -12/sqrt(253)) D) (3/sqrt(253), -4/sqrt(253), 12/sqrt(253))

37. If a line has direction ratios (2, 3, -6), what is the cosine of the angle it makes with the positive z-axis?

A) -6/7 B) 6/7 C) 2/7 D) 3/7

38. Which of the following sets of numbers cannot be direction cosines of a line?

A) (1/sqrt(2), 0, 1/sqrt(2)) B) (1/3, 2/3, 2/3) C) (0.6, 0.8, 0) D) (1, 1, 1)

39. If the direction ratios of a line are (k, 2k, 3k) for k != 0, what are its direction cosines?

A) (1/sqrt(14), 2/sqrt(14), 3/sqrt(14)) B) (k/sqrt(14k^2), 2k/sqrt(14k^2), 3k/sqrt(14k^2)) C) (1/sqrt(6), 2/sqrt(6), 3/sqrt(6)) D) (k, 2k, 3k)

40. What is the sum of the squares of the direction cosines of a line?

A) 0 B) 1 C) 2 D) Varies

41. A line has direction ratios (a, b, c). If b=0 and c=0, what is the direction of the line?

A) Parallel to the y-axis B) Parallel to the z-axis C) Parallel to the x-axis D) It is not a valid line

42. If the direction cosines of two lines are (l1, m1, n1) and (l2, m2, n2) and the angle between them is theta, then cos(theta) is given by:

A) l1*l2 + m1*m2 + n1*n2 B) l1*l2 - m1*m2 - n1*n2 C) (l1+l2)/2 + (m1+m2)/2 + (n1+n2)/2 D) sqrt((l1-l2)^2 + (m1-m2)^2 + (n1-n2)^2)

43. What are the direction ratios of a line making angles 30, 60, and 90 degrees with the positive x, y, and z axes respectively?

A) (sqrt(3)/2, 1/2, 0) B) (sqrt(3), 1, 0) C) (1, 1, 0) D) (30, 60, 90)

44. If a line has direction ratios (5, -12, 0), what is the cosine of the angle it makes with the positive x-axis?

A) 5/13 B) -12/13 C) 0 D) 5/sqrt(169)

45. If l, m, n are direction cosines, what is the value of l*cos(alpha) + m*cos(beta) + n*cos(gamma) where alpha, beta, gamma are angles with axes?

A) 0 B) 1 C) l+m+n D) cos(alpha)+cos(beta)+cos(gamma)

46. What are the direction cosines of a line that passes through the origin and is equally inclined to the coordinate axes?

A) (1/sqrt(3), 1/sqrt(3), 1/sqrt(3)) B) (1, 0, 0) C) (1/2, 1/2, 1/2) D) (1/sqrt(2), 1/sqrt(2), 0)

47. If a line has direction ratios (a, b, c), and a line has direction ratios (b, c, a), what is the condition for them to be perpendicular?

A) ab + bc + ca = 0 B) a+b+c = 0 C) a^2+b^2+c^2 = 0 D) a=b=c

48. If a line makes an angle of 135 degrees with the positive x-axis, what is its direction cosine along the x-axis?

A) -1/sqrt(2) B) 1/sqrt(2) C) -1/2 D) 1/2

49. A line has direction ratios (1, 1, -2). What is the cosine of the angle it makes with the positive y-axis?

A) 1/sqrt(6) B) -1/sqrt(6) C) 1/sqrt(2) D) -1/sqrt(2)

50. If a line has direction ratios (a, b, c), then the direction ratios of the line bisecting the angle between the positive x-axis and the line are proportional to:

A) (1 + a/sqrt(a^2+b^2+c^2), b/sqrt(a^2+b^2+c^2), c/sqrt(a^2+b^2+c^2)) B) (1, b, c) C) (a, b, c) D) (1, 0, 0)

Direction ratios and direction cosines - Online Test

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