Three Dimensional Geometry

More Topic from Mathematics

Question 1

A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance

  • 2H/A-B

  • H/A+B

  • H/2(A+B)

  • H/A-B

Solution

B.

H/A+B

(A + B) = d = H
d = (H/A+B)

Question 2

A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ

  • 30°

  • 45°

  • 60°

  • 75°

Solution

C.

60°

cos2α +cos2β + cos2γ = 1
α = 45°,β = 120°, γ = θ
rightwards double arrow 1 half space plus space 1 fourth space equals space 1 minus cos squared space straight theta
rightwards double arrow space 3 over 4 space equals space sin squared space straight theta
rightwards double arrow space sin squared space straight theta space equals space open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared space equals space sin squared space 60 to the power of straight o
straight theta space equals space 60 to the power of straight o

Question 3

A line makes the same angle θ, with each of the x and z-axis. If the angle β, which it makes with y-axis, is such that sin2β = 3sin2θ , then cos2θ equals 

  • 2/3

  • 1/5

  • 3/5

  • 2/3

Solution

C.

3/5

A line makes angle θ with x-axis and z-axis and β with y-axis.
∴ l = cosθ, m = cosβ,n = cosθ
We know that, l2+ m2+ n2= 1

cos2θ + cos2β +cos2θ =1
2cos2θ = 1- cos2β
2cos2θ = sin2β
But sin2β = 3 sin2θ
therefore from equation (i) and (ii)
3sin2θ = 2cos2θ
3(1-cos2θ) = 2cos2θ
3-3cos2θ = 2cos2θ
3 = 5cos2θ

Question 4

A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by

  • (3a, 3a, 3a), (a, a, a)

  • (3a, 2a, 3a), (a, a, a)

  • (3a, 2a, 3a), (a, a, 2a)

  • (2a, 3a, 3a), (2a, a, a)

Solution

B.

(3a, 2a, 3a), (a, a, a)

Any point on the line straight x over 1 space equals fraction numerator straight y plus straight a over denominator 1 end fraction space equals straight z over 1 space equals space straight t subscript 1 space left parenthesis say right parenthesis space is space left parenthesis straight t subscript 1 comma space straight t subscript 1 minus straight a comma space straight t subscript 1 right parenthesis and any point on the line fraction numerator straight x plus straight a over denominator 2 end fraction space equals space straight y over 1 space equals straight z over 1 space equals space straight t subscript 2 space left parenthesis say right parenthesis space is space left parenthesis 2 straight t subscript 2 minus straight a comma space straight t subscript 2 comma space straight t subscript 2 right parenthesis
Now direction cosine of the lines intersecting the above lines is proportional to (2t2 – a – t1, t2 – t1 + a, t2 – t1).
Hence 2t2 – a – t1 = 2k , t2 – t1 + a = k and t2 – t1 = 2k
On solving these, we get t1 = 3a , t2 = a. Hence points are (3a, 2a, 3a) and (a, a, a).

Question 5

Angle between the tangents to the curve y = x2 − 5x + 6 at the points (2, 0) and (3, 0) is

  • π/2

  • π/4

  • π/6

  • π

Solution

A.

π/2