Multiple choice question of 3 dimensional geometry

Multiple choice (3 Dimensional Geometry)

Choose and write the correct option in the following question:(Multiple choice question of 3 dimensional geometry)

1.) The co-ordinates of the foot of the perpendicular drawn from the point (2, -3, 4) on the y-axis is

(a) (2, 3, 4)                 (b) (-2, -3, -4)

(c) (0, -3, 0)                (d) (2, 0, 4)

Answer (c)

2.) If a lines makes angles α, β and γ with the axis respectively, then \cos 2\alpha +\cos 2\beta +\cos 2\gamma=

(a) -2                             (b) -1

(c) 1                                (d) 2

Answer (b)

3.) Distance of the point (α, β, γ) from y-axis is

(a) β                               (b) |β|

(c) |β|+|γ|                    (d) √(α²+γ²)

Answer (d)

4.) If the direction cosines of a line are k, k, k then

(a) k > 0                        (b) 0 < k < 0

(c) k = 1                          (d) k=\frac{1}{\sqrt{3}} or -\frac{1}{\sqrt{3}}

Answer (d)

5.) The angle between the two diagonals of a cube is

(a) 30°                          (b) 45°

(c) \cos^{-1}(\frac{1}{\sqrt{3}}   (d) \cos^{-1}\frac{1}{3}

Answer (d)

6.) If a line makes angles α, β, γ, δ with four diagonals of a cube, then

\cos^2\alpha +\cos^2\beta+\cos^2\gamma +\cos^2\delta is equal to

(a) 1/3                         (b) 2/3

(c) 4/3                         (d) 8/3

Answer (c)

7.) If a line makes angle π/3 and π/4 with x-axis and y-axis respectively,then the angle made by the line with z-axis is

(a) π/2                         (b) π/3

(c) π/4                          (d) 15π/12

Answer (b)

8.) P is a point on the line segment joining the points (3, 2, -1) and (6, 2, -2). If x-coordinate of P is 5, then its y co-ordinate is

(a) 2                          (b) 1

(c) -1                          (d)  -2

Answer (a)

9.) If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively. then the direction cosines of the line are

(a) \sin \alpha,\sin\beta, \sin\gamma

(b) \cos\alpha, \cos\beta, \cos\gamma

(c) \tan\alpha, \tan\beta, \tan\gamma

(d) \cos^2\alpha,\cos^2\beta, \cos^2\gamma

Answer (b)

10.) The distance of a point P(a, b, c) from x-axis is

(a) \sqrt{a^2+c^2}       (b) \sqrt{a^2+b^2}

(c) \sqrt{b^2+c^2}       (d) b^2+c^2

Answer (c)

11.) The equation of x-axis in space are

(a) x = 0, y = 0                  (b) x = 0, y = 0

(c) x = 0                               (d) y = 0, z = 0

Answer (d)

12.) A line makes equal angles with co-ordinate axis. Direction cosine of this line are

(a) ±(1, 1, 1)                        (b) ±(1/√3, 1/√3, 1/√3)

(c) ±(1/3, 1/3, 1/3)           (d)  ±(1/√3, -1/√3, -1/√3)

Answer (b)

13.) The distance of the point (1, 6, 3) to the line \vec{r} =(\hat{j}+2\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k}) is

(a) √13                                (b) 13

(c) 2√13                               (d) None of these

Answer (a)

14.) The two lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ will be perpendicular, if and only if 

(a) aa’ + cc’ + 1 = 0          (b) aa’ + bb’ + cc’ + 1 = 0

(c) aa’ + bb’ + cc’ =0       (d) (a + a’)(b + b’)(c + c’) = 0

Answer (a)

15.) The area of the quadrilateral ABCD where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2) is equal to

(a) 9 sq.units                    (b) 18 sq.units

(c) 27 sq.units                   (d) 81 sq.units

Answer (a)

16.) The straight line \frac{x-3}{3}=\frac{y-2}{1}=\frac{z-1}{0} is

(a) Parallel to x-axis         (b) Parallel to y-axis

(c) Parallel to z-axis           (d) Perpendicular to z-axis

Answer (d)

17.) The shortest distance between the lines given by \vec{r} = (8+3\lambda)\hat{i}-(9+16\lambda)\hat{j}+(10+7\lambda)\hat{k} and \vec{r} = 15\hat{i} +29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k}) is

(a) 7 units                            (b) 2 units

(c) 14 units                           (d) 3 units

Answer (c)

18.) The image of the point (1, 6, 3) in the line \frac{x}{1}= \frac{y-1}{2}=\frac{z-2}{2} is

(a) (2, 0, 5)                         (b) (1, 3, 4)

(c) (1, 0, 7)                          (d) (-3, -2, 0)

Answer (c)

19.) The shortest distance between the line \frac{x-3}{3}=\frac{y}{0}= \frac{z}{-4} and the y -axis is

(a) 1/5                                   (b) 1

(c) 0                                        (d) 12/5

Answer (d)

20.) The co-ordinates of the foot of perpendicular drawn from point A (1, 8, 4) to the line joining the points B(0, -1, 3) and C (2, -3, -1) are

(a) (-7/3, 2/3, 11/3)             (b) (-5/3, 2/3, 19/3)

(c) (4/3, 2/3, 11/3)               (d) None of these

Answer (b)


 

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