CBSE Class 12th Mathematics Chapter 11 - Three - dimensional Geometry Important Questions with Answers

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If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines. (Delhi 2019)

Question 2.

Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector 2i? + 2j? – 3k?. (Delhi 2019)

Question 3.

What are the direction cosines of a line which makes equal angles with the coordinate axes? (All India 2019, 2009, 2008C; Foreign 2011)

Question 4.

A line passes through the point with position vector 2i? – j? + 4k? and is in the direction of the vector i? + j? – 2k?. Find the equation of the line in cartesian form. (All India 2019)

Question 5.

Find the value of ?, so that the lines

\frac{1 - x}{3} = \frac{7 y - 14}{λ} = \frac{z - 3}{2}

and

\frac{7 - 7 x}{3 λ} = \frac{y - 5}{1} = \frac{6 - z}{5}

are at right angles. Also, find whether the lines are intersecting or not. (Delhi 2019)

Question 6.

If the lines

\frac{x - 1}{- 3} = \frac{y - 2}{2 λ} = \frac{z - 3}{2}

and

\frac{x - 1}{3 λ} = \frac{y - 1}{2} = \frac{z - 6}{- 5}

are perpendicular, find the value of ?. Hence find whether the lines are intersecting or not. (All India 2019)

Question 7.

Find the cartesian and vector equations of the plane passing through the points A(2, 5, – 3), B(- 2, – 3, 5) and C(5, 3, – 3). (All India 2019)

Question 8.

Find the vector and cartesian equations of the plane passing through the points (2, 2, – 1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above. (Delhi 2019; Foreign 20114)

Question 9.

Find the vector equation of the plane that contains the lines

\vec{r}

= (i? + j?) + ? (i? + 2j? – k?) and the points (- 1, 3, – 4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane, thus obtained. (Delhi 2019; All India 2012C)

Question 10.

Find the vector and cartesian equations of the plane passing through the points having position vectors i? + j? – 2k?, 2i? – j? + k? and i? + 2j? + k?. Write the equation of a plane passing through a point (2, 3, 7) and parallel to the plane obtained above. Hence, find the distance between the two parallel planes. (All India 2019)

Question 1.

Find the equation of the line passing through (2, – 1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5) and (3, 2, 4). Also, find their point of intersection. (ll India 2019; Foreign 2011)

Question 2.

Find the shortest distance between the lines

\vec{r}

= (4i? – j?) + ?(i? + 2j? – 3k?) and

\vec{r}

= (i? – j? + 2k?) + ?(2i? + 4j? – 5k?). (CBSE 2018)

Question 3.

Find the shortest distance between the lines

\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}

and

\frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}

. (CBSE 2018C)

Question 4.

Find the distance of the point P(-1, – 5, – 10) from the point of intersection of the line

\vec{r}

= 2i? – j? + 2k? + ? (3i? + 4j? + 2k?) and the plane

\vec{r}

? (i? – j? + k?) = 5.
(CBSE 2018; Delhi 2014; All India 2014C, 2011)

Question 5.

Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes

\vec{r}

? (i? – j? + 2k?) = 5 and

\vec{r}

? (3i? + j? + 2k?) = 6. Also, find the point of intersection of the line, thus obtained with the plane

\vec{r}

? (2i? + j? + k?) = 4 (CBSE 2018C; All India 2013)

Question 6.

If a line makes angles 90° and 60°, respectively with the positive directions of X and Y-axes, find the angle which it makes with the positive direction of Z-axis. (Delhi 2017)