CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 07

Question. Distance of the point \( (\alpha, \beta, \gamma) \) from y-axis is
(a) \( \beta \)
(b) \( |\beta| \)
(c) \( |\beta| + |\gamma| \)
(d) \( \sqrt{\alpha^2 + \gamma^2} \)
Answer: (d) \( \sqrt{\alpha^2 + \gamma^2} \)

Question. If the direction cosines of a line are \( k, k, k \) then
(a) \( k > 0 \)
(b) \( 0 < k < 1 \)
(c) \( k = 1 \)
(d) \( k = \frac{1}{\sqrt{3}} \text{ or } -\frac{1}{\sqrt{3}} \)
Answer: (d) \( k = \frac{1}{\sqrt{3}} \text{ or } -\frac{1}{\sqrt{3}} \)
 

Question. The distance of the plane \( \vec{r} \cdot \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 1 \) from the origin is
(a) 1
(b) 7
(c) \( \frac{1}{7} \)
(d) None of the options
Answer: (a) 1

Question. The sine of the angle between the straight line \( \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5} \) and the plane \( 2x - 2y + z = 5 \) is
(a) \( \frac{10}{6\sqrt{5}} \)
(b) \( \frac{4}{5\sqrt{2}} \)
(c) \( \frac{2\sqrt{3}}{5} \)
(d) \( \frac{\sqrt{2}}{10} \)
Answer: (d) \( \frac{\sqrt{2}}{10} \)

Question. The reflection of the point \( (\alpha, \beta, \gamma) \) in the xy-plane is
(a) \( (\alpha, \beta, 0) \)
(b) \( (0, 0, \gamma) \)
(c) \( (-\alpha, -\beta, \gamma) \)
(d) \( (\alpha, \beta, -\gamma) \)
Answer: (d) \( (\alpha, \beta, -\gamma) \)
 

Question. The coordinates of the foot of the perpendicular drawn from the point \( (2, 5, 7) \) on the x-axis are given by
(a) \( (2, 0, 0) \)
(b) \( (0, 5, 0) \)
(c) \( (0, 0, 7) \)
(d) \( (0, 5, 7) \)
Answer: (a) \( (2, 0, 0) \)

Question. P is a point on the line segment joining the points \( (3, 2, -1) \) and \( (6, 2, -2) \). If x co-ordinate of P is 5, then its y co-ordinate is
(a) 2
(b) 1
(c) -1
(d) -2
Answer: (a) 2

Question. If \( \alpha, \beta, \gamma \) 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) \( \cos \alpha, \cos \beta, \cos \gamma \)

Question. 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) \( \sqrt{b^2 + c^2} \)

Question. The equations of x-axis in space are
(a) \( x = 0, y = 0 \)
(b) \( x = 0, z = 0 \)
(c) \( x = 0 \)
(d) \( y = 0, z = 0 \)
Answer: (d) \( y = 0, z = 0 \)

Question. A line makes equal angles with co-ordinate axis. Direction cosines of this line are
(a) \( \pm (1, 1, 1) \)
(b) \( \pm \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \)
(c) \( \pm \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \)
(d) \( \pm \left(\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\right) \)
Answer: (b) \( \pm \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \)

Question. P is the point on the line segment joining the points \( (3, 2, -1) \) and \( (6, 2, -2) \). If x co-ordinate of P is 5, then its y co-ordinate is
(a) 2
(b) 1
(c) -1
(d) -2
Answer: (a) 2

Question. The sine of the angle between the straight line \( \frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5} \) and the plane \( 2x - 2y + z = 5 \) is
(a) \( \frac{10}{6\sqrt{5}} \)
(b) \( \frac{4}{5\sqrt{2}} \)
(c) \( \frac{2\sqrt{3}}{5} \)
(d) \( \frac{\sqrt{2}}{10} \)
Answer: (d) \( \frac{\sqrt{2}}{10} \)

Question. 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) 9 sq units

Question. The locus represented by \( xy + yz = 0 \) is
(a) a pair of perpendicular lines
(b) a pair of parallel lines
(c) a pair of perpendicular planes
(d) a pair of parallel planes
Answer: (c) a pair of perpendicular planes

Question. In the intercepts made by the plane \( 2x - 3y + 5z + 4 = 0 \) on the coordinate axes are
(a) \( -2, \frac{4}{3} \text{ and } -\frac{4}{5} \)
(b) \( -2, -\frac{4}{3} \text{ and } \frac{4}{5} \)
(c) \( \frac{4}{3}, -\frac{4}{3} \text{ and } \frac{7}{3} \)
(d) \( -2, -\frac{4}{3} \text{ and } -\frac{4}{5} \)
Answer: (a) \( -2, \frac{4}{3} \text{ and } -\frac{4}{5} \)

Question. 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) 14 units

Question. 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) (1, 0, 7)

Question. The coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, -1, 0) are
(a) (0, -2, 7)
(b) (3, -2, 5)
(c) (1, -2, -7)
(d) (1, -2, 7)
Answer: (d) (1, -2, 7)

Question. 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) \( \left( \frac{-7}{3}, \frac{2}{3}, \frac{11}{3} \right) \)
(b) \( \left( \frac{-5}{3}, \frac{2}{3}, \frac{19}{3} \right) \)
(c) \( \left( \frac{4}{3}, \frac{2}{3}, \frac{11}{3} \right) \)
(d) None of the options
Answer: (b) \( \left( \frac{-5}{3}, \frac{2}{3}, \frac{19}{3} \right) \)

Assertion-Reason Questions 

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : Direction ratios of the normal to the plane \( 2x - 3y + 5z + 1 = 0 \) are 2, -3, 5.
Reason (R) : Equation of a plane is \( ax + by + cz + d = 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Clearly, both Assertion (A) and Reason (R) are true. But Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : The angle between the lines whose direction cosines are \( \frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2} ; \frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2} \) is 120°.
Reason (R) : The angle between two lines whose direction ratios are \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) is given by \( \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
We have, \( \cos \theta = \frac{-\sqrt{3}}{4} \times \frac{-\sqrt{3}}{4} + \frac{1}{4} \times \frac{1}{4} + \left( \frac{-\sqrt{3}}{2} \right) \times \frac{\sqrt{3}}{2} \)
\( = \frac{3}{16} + \frac{1}{16} - \frac{3}{4} = \frac{3 + 1 - 12}{16} \)
\( = \frac{-8}{16} = -\frac{1}{2} \)
\( \implies \) \( \theta = 120^\circ \).
Clearly, both Assertion (A) and Reason (R) are true Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : The intercept cut off by the plane \( 2x + y - z = 5 \) on x-axis is \( \frac{5}{2} \).
Reason (R) : Equation of the plane in intercept form is \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
We have, \( 2x + y - z = 5 \)
\( \implies \) \( \frac{x}{\frac{5}{2}} + \frac{y}{5} + \frac{z}{-5} = 1 \)
\( \therefore \) x-intercept is \( \frac{5}{2} \).
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : Direction cosines of z-axis are 0, 0, 1.
Reason (R) : If l, m, n be the direction cosines of a line then \( l^2 + m^2 + n^2 = 1 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : The equation of the plane \( \vec{r} \cdot (3\hat{i} - 2\hat{j} + \hat{k}) = 2 \) in Cartesian form is \( 3x - 2y + z = 2 \).
Reason (R) : General equation of the plane parallel to z-axis is \( ax + by + d = 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.