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RD Sharma solutions for Class 12 Mathematics chapter 29 - The Plane

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) Chapter 29: The Plane

Ex. 29.1Ex. 29.2Ex. 29.3Ex. 29.4Ex. 29.5Ex. 29.6Ex. 29.7Ex. 29.8Ex. 29.9Ex. 29.10Ex. 29.11Ex. 29.12Ex. 29.13Ex. 29.14Ex. 29.15Very Short AnswersMCQ

Chapter 29: The Plane Exercise 29.1 solutions [Pages 4 - 5]

Ex. 29.1 | Q 1.1 | Page 4

Find the equation of the plane passing through the following points.

(2, 1, 0), (3, −2, −2) and (3, 1, 7)

Ex. 29.1 | Q 1.2 | Page 4

Find the equation of the plane passing through the following points.

(−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)

Ex. 29.1 | Q 1.3 | Page 4

Find the equation of the plane passing through the following point

(1, 1, 1), (1, −1, 2) and (−2, −2, 2)

Ex. 29.1 | Q 1.4 | Page 4

Find the equation of the plane passing through the following points.

(2, 3, 4), (−3, 5, 1) and (4, −1, 2)

Ex. 29.1 | Q 1.5 | Page 4

Find the equation of the plane passing through the following point

(0, −1, 0), (3, 3, 0) and (1, 1, 1)

Ex. 29.1 | Q 2 | Page 5

Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.

Ex. 29.1 | Q 3.1 | Page 5

Show that the following points are coplanar.
(0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0)

Ex. 29.1 | Q 3.2 | Page 5

Show that the following points are coplanar.

(0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)

Ex. 29.1 | Q 4 | Page 5

Find the coordinates of the point where the line through (3, -4 , -5 ) and B (2, -3 , 1) crosses the plane passing through three points L(2,2,1), M(3,0,1) and N(4, -1,0 ) . Also, find the ratio in which diveides the line segment AB.

Chapter 29: The Plane Exercise 29.2 solutions [Page 7]

Ex. 29.2 | Q 1 | Page 7

Write the equation of the plane whose intercepts on the coordinate axes are 2, −3 and 4.

Ex. 29.2 | Q 2.1 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
4x + 3y − 6z − 12 = 0

Ex. 29.2 | Q 2.2 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

2x + 3y − z = 6

Ex. 29.2 | Q 2.3 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

2x − y + z = 5

Ex. 29.2 | Q 3 | Page 7

Find the equation of a plane which meets the axes at AB and C, given that the centroid of the triangle ABC is the point (α, β, γ).

Ex. 29.2 | Q 4 | Page 7

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

Ex. 29.2 | Q 5 | Page 7

A plane meets the coordinate axes at AB and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

Chapter 29: The Plane Exercise 29.3 solutions [Pages 13 - 14]

Ex. 29.3 | Q 1 | Page 13

Find the vector equation of a plane passing through a point with position vector $2 \hat{i} - \hat{j} + \hat{k}$ and perpendicular to the vector  $4 \hat{i} + 2 \hat{j} - 3 \hat{k} .$

Ex. 29.3 | Q 2.1 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

$\vec{r} \cdot \left( 12 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + 5 = 0$

Ex. 29.3 | Q 2.2 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

$\vec{r} \cdot \left( - \hat{i} + \hat{j} + 2 \hat{k} \right) = 9$

Ex. 29.3 | Q 3 | Page 13

Find the vector equations of the coordinate planes.

Ex. 29.3 | Q 4.1 | Page 13

Find the vector equation of each one of following planes.

2x − y + 2z = 8

Ex. 29.3 | Q 4.2 | Page 13

Find the vector equation of each one of following planes.

x + y − z = 5

Ex. 29.3 | Q 4.3 | Page 13

Find the vector equation of each one of following planes.

x + y = 3

Ex. 29.3 | Q 5 | Page 13

Find the vector and Cartesian equations of a plane passing through the point (1, −1, 1) and normal to the line joining the points (1, 2, 5) and (−1, 3, 1).

Ex. 29.3 | Q 6 | Page 13

$\vec{n}$ is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and Cartesian forms of the equation of a plane which passes through (2, 1, −1) and is normal to $\vec{n}$ .

Ex. 29.3 | Q 7 | Page 13

The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, −4, 3). Find the equation of the plane.

Ex. 29.3 | Q 8 | Page 13

Find the equation of the plane passing through the point (2, 3, 1), given that the direction ratios of the normal to the plane are proportional to 5, 3, 2.

Ex. 29.3 | Q 9 | Page 13

If the axes are rectangular and P is the point (2, 3, −1), find the equation of the plane through P at right angles to OP.

Ex. 29.3 | Q 10 | Page 13

Find the intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 and also find the direction cosines of the normal to the plane.

Ex. 29.3 | Q 11 | Page 13

A plane passes through the point (1, −2, 5) and is perpendicular to the line joining the origin to the point

$\text{ 3 } \hat{i} + \hat{j} - \hat{k} .$ Find the vector and Cartesian forms of the equation of the plane.

Ex. 29.3 | Q 12 | Page 13

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.

Ex. 29.3 | Q 13.1 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

x − y + z − 2 = 0 and 3x + 2y − z + 4 = 0

Ex. 29.3 | Q 13.2 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 3 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 2 \hat{i} - 2 \hat{j} - 2 \hat{k} \right) = 5$
Ex. 29.3 | Q 14 | Page 13

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.

Ex. 29.3 | Q 15 | Page 14

Find a vector of magnitude 26 units normal to the plane 12x − 3y + 4z = 1.

Ex. 29.3 | Q 16 | Page 14

If the line drawn from (4, −1, 2) meets a plane at right angles at the point (−10, 5, 4), find the equation of the plane.

Ex. 29.3 | Q 17 | Page 14

Find the equation of the plane which bisects the line segment joining the points (−1, 2, 3) and (3, −5, 6) at right angles.

Ex. 29.3 | Q 18 | Page 14

Find the vector and Cartesian equations of the plane that passes through the point (5, 2, −4) and is perpendicular to the line with direction ratios 2, 3, −1.

Ex. 29.3 | Q 19 | Page 14

If O be the origin and the coordinates of P be (1, 2,−3), then find the equation of the plane passing through P and perpendicular to OP.

Ex. 29.3 | Q 20 | Page 14

Find the vector equation of the plane with intercepts 3, –4 and 2 on xy and z-axis respectively.

Ex. 29.3 | Q 21 | Page 14

Find the vector equation of the plane with intercepts 3, –4 and 2 on xy and z-axis respectively.

Chapter 29: The Plane Exercise 29.4 solutions [Page 19]

Ex. 29.4 | Q 1 | Page 19

Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{k}$ as the unit vector normal to it.

Ex. 29.4 | Q 2 | Page 19

Find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector  $\hat{i} - \text{2 } \hat{j} - \text{2 } \hat{k} .$

Ex. 29.4 | Q 3 | Page 19

Reduce the equation 2x − 3y − 6z = 14 to the normal form and, hence, find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.

Ex. 29.4 | Q 4 | Page 19

Reduce the equation $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) + 6 = 0$ to normal form and, hence, find the length of the perpendicular from the origin to the plane.

Ex. 29.4 | Q 5 | Page 19

Write the normal form of the equation of the plane 2x − 3y + 6z + 14 = 0.

Ex. 29.4 | Q 6 | Page 19

The direction ratios of the perpendicular from the origin to a plane are 12, −3, 4 and the length of the perpendicular is 5. Find the equation of the plane.

Ex. 29.4 | Q 7 | Page 19

Find a unit normal vector to the plane x + 2y + 3z − 6 = 0.

Ex. 29.4 | Q 8 | Page 19

Find the equation of a plane which is at a distance of $3\sqrt{3}$  units from the origin and the normal to which is equally inclined to the coordinate axes.

Ex. 29.4 | Q 9 | Page 19

find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Find also the perpendicular distance of the origin from this plane

Ex. 29.4 | Q 10 | Page 19

Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is  $2 \hat{i} - 3 \hat{j} + 4 \hat{k} .$ Also, find its Cartesian form.

Ex. 29.4 | Q 11 | Page 19

Find the distance of the plane 2x − 3y + 4z − 6 = 0 from the origin.

Chapter 29: The Plane Exercise 29.5 solutions [Pages 22 - 23]

Ex. 29.5 | Q 1 | Page 22

Find the vector equation of the plane passing through the points (1, 1, 1), (1, −1, 1) and (−7, −3, −5).

Ex. 29.5 | Q 2 | Page 23

Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).

Ex. 29.5 | Q 3 | Page 23

Find the vector equation of the plane passing through points A (a, 0, 0), B (0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .$

Ex. 29.5 | Q 4 | Page 23

Find the vector equation of the plane passing through the points (1, 1, −1), (6, 4, −5) and (−4, −2, 3).

Ex. 29.5 | Q 5 | Page 23

Find the vector equation of the plane passing through the points $3 \hat{i} + 4 \hat{j} + 2 \hat{k} , 2 \hat{i} - 2 \hat{j} - \hat{k} \text{ and } 7 \hat{i} + 6 \hat{k} .$

Chapter 29: The Plane Exercise 29.6 solutions [Page 29]

Ex. 29.6 | Q 1.1 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( - \hat{i} + \hat{j} \right) = 4$

Ex. 29.6 | Q 1.2 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 6 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + 6 \hat{j} - 2 \hat{k} \right) = 9$

Ex. 29.6 | Q 1.3 | Page 29
Find the angle between the given planes.
$\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) = 9$

Ex. 29.6 | Q 2.1 | Page 29

Find the angle between the planes.

2x − y + z = 4 and x + y + 2z = 3

Ex. 29.6 | Q 2.2 | Page 29

Find the angle between the planes.

x + y − 2z = 3 and 2x − 2y + z = 5

Ex. 29.6 | Q 2.3 | Page 29

Find the angle between the planes.

x − y + z = 5 and x + 2y + z = 9

Ex. 29.6 | Q 2.4 | Page 29

Find the angle between the planes.
2x − 3y + 4z = 1 and − x + y = 4

Ex. 29.6 | Q 2.5 | Page 29

Find the angle between the planes.

2x + y − 2z = 5 and 3x − 6y − 2z = 7

Ex. 29.6 | Q 3.1 | Page 29

Show that the following planes are at right angles.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( - \hat{i} - \hat{j} + \hat{k} \right) = 3$

Ex. 29.6 | Q 3.2 | Page 29

Show that the following planes are at right angles.

x − 2y + 4z = 10 and 18x + 17y + 4z = 49

Ex. 29.6 | Q 4.1 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

$\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 7 \text{ and } \vec{r} \cdot \left( \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \right) = 26$

Ex. 29.6 | Q 4.2 | Page 29

Determine the value of λ for which the following planes are perpendicular to each ot

2x − 4y + 3z = 5 and x + 2y + λz = 5

Ex. 29.6 | Q 4.3 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

3x − 6y − 2z = 7 and 2x + y − λz = 5

Ex. 29.6 | Q 5 | Page 29

Find the equation of a plane passing through the point (−1, −1, 2) and perpendicular to the planes 3x + 2y − 3z = 1 and 5x − 4y + z = 5.

Ex. 29.6 | Q 6 | Page 29

Obtain the equation of the plane passing through the point (1, −3, −2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

Ex. 29.6 | Q 7 | Page 29

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y − z = 1 and 3x − 4y + z = 5.

Ex. 29.6 | Q 8 | Page 29

Find the equation of the plane passing through the points (1, −1, 2) and (2, −2, 2) and which is perpendicular to the plane 6x − 2y + 2z = 9.

Ex. 29.6 | Q 9 | Page 29

Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.

Ex. 29.6 | Q 10 | Page 29

Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Ex. 29.6 | Q 11 | Page 29

Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.

Ex. 29.6 | Q 12 | Page 29

Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x + 3y − 2z = 5 and x + 2y − 3z = 8.

Ex. 29.6 | Q 13 | Page 29

Find the equation of the plane passing through (abc) and parallel to the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Ex. 29.6 | Q 14 | Page 29

Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Ex. 29.6 | Q 15 | Page 29

Find the vector equation of the plane through the points (2, 1, −1) and (−1, 3, 4) and perpendicular to the plane x − 2y + 4z = 10

Chapter 29: The Plane Exercise 29.7 solutions [Page 33]

Ex. 29.7 | Q 1.1 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k} \right)$

Ex. 29.7 | Q 1.2 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 1 + s - t \right) \hat{t} + \left( 2 - s \right) \hat{j} + \left( 3 - 2s + 2t \right) \hat{k}$

Ex. 29.7 | Q 1.3 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( \hat{i} + 2 \hat{j} - \hat{k} \right) + \mu\left( - \hat{i} + \hat{j} - 2 \hat{k} \right)$

Ex. 29.7 | Q 1.4 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( 4 \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$

Ex. 29.7 | Q 2.1 | Page 33

Find the Cartesian forms of the equations of the following planes. $\vec{r} = \left( \hat{i} - \hat{j} \right) + s\left( - \hat{i} + \hat{j} + 2 \hat{k} \right) + t\left( \hat{i} + 2 \hat{j} + \hat{k} \right)$

Ex. 29.7 | Q 2.2 | Page 33

Find the Cartesian forms of the equations of the following planes.

$\vec{r} = \left( 1 + s + t \right) \hat{i} + \left( 2 - s + t \right) \hat{i} + \left( 3 - 2s + 2t \right) \hat{k}$

Ex. 29.7 | Q 3.1 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j} + \left( 2\lambda + \mu \right) \hat{k}$

Ex. 29.7 | Q 3.2 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \mu\left( 5 \hat{i} - 2 \hat{j} + 7 \hat{k} \right)$

Chapter 29: The Plane Exercise 29.8 solutions [Pages 39 - 40]

Ex. 29.8 | Q 1 | Page 39

Find the equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2).

Ex. 29.8 | Q 2 | Page 39

Find the equation of the plane through (3, 4, −1) which is parallel to the plane $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + 2 = 0 .$

Ex. 29.8 | Q 3 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3).

Ex. 29.8 | Q 4 | Page 39

Find the equation of the plane through the point $2 \hat{i} + \hat{j} - \hat{k}$ and passing through the line of intersection of the planes $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} - \hat{k} \right) = 0 \text{ and } \vec{r} \cdot \left( \hat{j} + 2 \hat{k} \right) = 0 .$

Ex. 29.8 | Q 5 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − y = 0 and 3z − y = 0 and perpendicular to the plane 4x + 5y − 3z = 8

Ex. 29.8 | Q 6 | Page 39

Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0 and which is perpendicular to the plane 5x + 3y − 6z+ 8 = 0.

Ex. 29.8 | Q 7 | Page 39

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x − y + z + 3 = 0 and passing through the origin.

Ex. 29.8 | Q 8 | Page 39

Find the vector equation (in scalar product form) of the plane containing the line of intersection of the planes x − 3y + 2z − 5 = 0 and 2x − y + 3z − 1 = 0 and passing through (1, −2, 3).

Ex. 29.8 | Q 9 | Page 39

Find the equation of the plane that is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z − 4 = 0, 2x + y − z + 5 = 0.

Ex. 29.8 | Q 10 | Page 39

Find the equation of the plane through the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} \right) + 6 = 0 \text{ and } \vec{r} \cdot \left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) = 0,$ which is at a unit distance from the origin.

Ex. 29.8 | Q 11 | Page 39

Find the equation of the plane passing through the intersection of the planes 2x + 3y − z+ 1 = 0 and x + y − 2z + 3 = 0 and perpendicular to the plane 3x − y − 2z − 4 = 0.

Ex. 29.8 | Q 12 | Page 39

Find the equation of the plane that contains the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) - 4 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + \hat{j} - \hat{k} \right) + 5 = 0$ and which is perpendicular  to the plane $\vec{r} \cdot \left( 5 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) + 8 = 0 .$

Ex. 29.8 | Q 13 | Page 39

Find the equation of the plane passing through (abc) and parallel to the plane $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Ex. 29.8 | Q 14 | Page 39

Find the equation of the plane passing through the intersection of the planes  $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) = 7, \vec{r} \cdot \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) = 9$ and the point (2, 1, 3).

Ex. 29.8 | Q 15 | Page 40

Find the equation of the plane through the intersection of the planes 3x − y + 2z = 4 and x + y + z = 2 and the point (2, 2, 1).

Ex. 29.8 | Q 16 | Page 40

Find the vector equation of the plane through the line of intersection of the planes x + yz = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0.

Ex. 29.8 | Q 17 | Page 40

Find the vector equation of the plane passing through the intersection of the planes

$\vec{r} \cdot \left( \hat{ i } + \hat{ j }+ \hat{ k }\right) = \text{ 6 and }\vec{r} \cdot \left( \text{ 2 } \hat{ i} +\text{ 3 } \hat{ j } + \text{ 4 } \hat{ k } \right) = - 5$ and the point (1, 1, 1).

Ex. 29.8 | Q 18 | Page 40

Find the equation of the plane which contains the line of intersection of the planes x $+$  2y $+$  3 $z -$  4 $=$  0 and 2 $x + y - z$ $+$ 5  $=$ 0 and whose x-intercept is twice its z-intercept. Hence, write the equation of the plane passing through the point (2, 3,  $-$ 1) and parallel to the plane obtained above.

Ex. 29.8 | Q 19 | Page 40

Find the equation of the plane through the line of intersection of the planes $x + y + z =$1 and 2x $+$ 3 $+$ y $+$ 4$z =$ 5 and twice of its $y$ -intercept is equal to three times its $z$-intercept

Chapter 29: The Plane Exercise 29.9 solutions [Page 49]

Ex. 29.9 | Q 1 | Page 49

Find the distance of the point  $2 \hat{i} - \hat{j} - 4 \hat{k}$  from the plane  $\vec{r} \cdot \left( 3 \hat{i} - 4 \hat{j} + 12 \hat{k} \right) - 9 = 0 .$

Ex. 29.9 | Q 2 | Page 49

Show that the points $\hat{i} - \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 3 \hat{j} + 3 \hat{k}$ are equidistant from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7 \hat{k} \right) + 9 = 0 .$

Ex. 29.9 | Q 3 | Page 49

Find the distance of the point (2, 3, −5) from the plane x + 2y − 2z − 9 = 0.

Ex. 29.9 | Q 4 | Page 49

Find the equations of the planes parallel to the plane x + 2y − 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).

Ex. 29.9 | Q 5 | Page 49

Show that the points (1, 1, 1) and (−3, 0, 1) are equidistant from the plane 3x + 4y − 12z + 13 = 0.

Ex. 29.9 | Q 6 | Page 49

Find the equations of the planes parallel to the plane x − 2y + 2z − 3 = 0 and which are at a unit distance from the point (1, 1, 1).

Ex. 29.9 | Q 7 | Page 49

Find the distance of the point (2, 3, 5) from the xy - plane.

Ex. 29.9 | Q 8 | Page 49

Find the distance of the point (3, 3, 3) from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7k \right) + 9 = 0$

Ex. 29.9 | Q 9 | Page 49

If the product of the distances of the point (1, 1, 1) from the origin and the plane x − y + z+ λ = 0 be 5, find the value of λ.

Ex. 29.9 | Q 10 | Page 49

Find an equation for the set of all points that are equidistant from the planes 3x − 4y + 12z = 6 and 4x + 3z = 7.

Ex. 29.9 | Q 11 | Page 49

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C (5, 3, −3).

Ex. 29.9 | Q 12 | Page 49

A plane makes intercepts −6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

Ex. 29.9 | Q 13 | Page 49

Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes  $x - y +$ 2  z = 3 and 2 x - 2  y +z + 12 = 0.

Chapter 29: The Plane Exercise 29.10 solutions [Page 51]

Ex. 29.10 | Q 1 | Page 51

Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.

Ex. 29.10 | Q 2 | Page 51

Find the equation of the plane which passes through the point (3, 4, −1) and is parallel to the plane 2x − 3y + 5z + 7 = 0. Also, find the distance between the two planes.

Ex. 29.10 | Q 3 | Page 51

Find the equation of the plane mid-parallel to the planes 2x − 2y + z + 3 = 0 and 2x − 2y + z + 9 = 0.

Ex. 29.10 | Q 4 | Page 51

Find the distance between the planes $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + 4 \hat{j} + 6 \hat{k} \right) + 7 = 0 .$

Chapter 29: The Plane Exercise 29.11 solutions [Pages 61 - 62]

Ex. 29.11 | Q 1 | Page 61

Find the angle between the line $\vec{r} = \left( 2 \hat{i}+ 3 \hat {j} + 9 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  and the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 5 .$

Ex. 29.11 | Q 2 | Page 61

Find the angle between the line $\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}$  and the plane 2x + y − z = 4.

Ex. 29.11 | Q 3 | Page 61

Find the angle between the line joining the points (3, −4, −2) and (12, 2, 0) and the plane 3x − y + z = 1.

Ex. 29.11 | Q 4 | Page 61

The line  $\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 4 .$ Find m

Ex. 29.11 | Q 5 | Page 61

Show that the line whose vector equation is $\vec{r} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$ is parallel to the plane whose vector  $\vec{r} \cdot \left( \hat{i} + \hat{j} - \hat{k} \right) = 7 .$  Also, find the distance between them.

Ex. 29.11 | Q 6 | Page 61

Find the vector equation of the line through the origin which is perpendicular to the plane  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 3 .$

Ex. 29.11 | Q 7 | Page 61

Find the equation of the plane through (2, 3, −4) and (1, −1, 3) and parallel to x-axis.

Ex. 29.11 | Q 8 | Page 61

Find the equation of a plane passing through the points (0, 0, 0) and (3, −1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7} .$

Ex. 29.11 | Q 9 | Page 61

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) = 6$

Ex. 29.11 | Q 10 | Page 61

Prove that the line of section of the planes 5x + 2y − 4z + 2 = 0 and 2x + 8y + 2z − 1 = 0 is parallel to the plane 4x − 2y − 5z − 2 = 0.

Ex. 29.11 | Q 11 | Page 61

Find the vector equation of the line passing through the point (1, −1, 2) and perpendicular to the plane 2x − y + 3z − 5 = 0.

Ex. 29.11 | Q 12 | Page 61

Find the equation of the plane through the points (2, 2, −1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.

Ex. 29.11 | Q 13 | Page 61

Find the angle between the line $\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}$ and the plane

3x + 4y + z + 5 = 0.

Ex. 29.11 | Q 14 | Page 61

Find the equation of the plane passing through the intersection of the planes x − 2y + z = 1 and 2x + y + z = 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from this plane

Ex. 29.11 | Q 15 | Page 61

State when the line $\vec{r} = \vec{a} + \lambda \vec{b}$  is parallel to the plane  $\vec{r} \cdot \vec{n} = d .$Show that the line  $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 3 \hat{i} - \hat{j} + 2 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .$   Also, find the distance between the line and the plane.

Ex. 29.11 | Q 16 | Page 61

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 1$ and the line whose vector equation is  $\vec{r} = \left( - \hat{i} + \hat{j} + \hat{k} \right) + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right)$   are parallel. Also, find the distance between them.

Ex. 29.11 | Q 17 | Page 61

Find the equation of the plane through the intersection of the planes 3x − 4y + 5z = 10 and 2x + 2y − 3z = 4 and parallel to the line x = 2y = 3z.

Ex. 29.11 | Q 18 | Page 62

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines $\vec{r} = \left( \hat{i} + 2 \hat{j} - 4 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)$ and $\vec{r} = \left( \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)$ Also, find the distance of the point (9, −8, −10) from the plane thus obtained.

Ex. 29.11 | Q 19 | Page 62

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line

$\frac{x + 3}{2} = \frac{y - 3}{7} = \frac{z - 2}{5} .$

Ex. 29.11 | Q 20 | Page 62

Find the coordinates of the point where the line  $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$   intersects the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Ex. 29.11 | Q 21 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - 5 \hat{k} \right) + 9 = 0 .$

Ex. 29.11 | Q 22 | Page 62

Find the angle between the line

$\frac{x + 1}{2} = \frac{y}{3} = \frac{z - 3}{6}$  and the plane 10x + 2y − 11z = 3.

Ex. 29.11 | Q 23 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes  $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .$

Ex. 29.11 | Q 24 | Page 62

Find the value of λ such that the line $\frac{x - 2}{6} = \frac{y - 1}{\lambda} = \frac{z + 5}{- 4}$  is perpendicular to the plane 3x − y − 2z = 7.

Ex. 29.11 | Q 25 | Page 62

Find the equation of the plane passing through the points (−1, 2, 0), (2, 2, −1) and parallel to the line $\frac{x - 1}{1} = \frac{2y + 1}{2} = \frac{z + 1}{- 1}$

Chapter 29: The Plane Exercise 29.12 solutions [Page 65]

Ex. 29.12 | Q 1.1 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the   yz - plane .

Ex. 29.12 | Q 1.2 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the  zx - plane .

Ex. 29.12 | Q 2 | Page 65

Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.

Ex. 29.12 | Q 3 | Page 65

Find the distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i}+ 4 \hat{j} + 2 \hat{k} \right)$ and the plane  $\vec{r} . \left( \hat{i} - \hat{j} + \hat{k} \right) = 5 .$

Ex. 29.12 | Q 4 | Page 65

Find the distance of the point (2, 12, 5) from the point of intersection of the line $\vec{r} = 2 \hat{i} - 4 \hat{j}+ 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j} + 2 \hat{k} \right)$ and $\vec{r} . \left( \hat{i} - 2 \hat{j} + \hat{k} \right) = 0$

Ex. 29.12 | Q 5 | Page 65

Find the distance of the point P(−1, −5, −10) from the point of intersection of the line joining the points A(2, −1, 2) and B(5, 3, 4) with the plane  $x - y + z = 5$ .

Ex. 29.12 | Q 6 | Page 65

Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, −4, −5) and B(2, −3, 1) intersects the plane 2x + y + z = 7.

Ex. 29.12 | Q 7 | Page 65

Find the distance of the point (1, -5, 9) from the plane

$x - y + z =$ 5  measured along the line $x = y = z$  .

Chapter 29: The Plane Exercise 29.13 solutions [Pages 73 - 74]

Ex. 29.13 | Q 1 | Page 73

Show that the lines $\vec{r} = \left( 2 \hat{j} - 3 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  are coplanar. Also, find the equation of the plane containing them.

Ex. 29.13 | Q 2 | Page 74

Show that the lines $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} \text{ and }\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$  are coplanar. Also, find the equation of the plane containing them.

Ex. 29.13 | Q 3 | Page 74

Find the equation of the plane containing the line $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}$  and the point (0, 7, −7) and show that the line  $\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$ also lies in the same plane.

Ex. 29.13 | Q 4 | Page 74

Find the equation of the plane which contains two parallel lines$\frac{x - 4}{1} = \frac{y - 3}{- 4} = \frac{z - 2}{5}\text{ and }\frac{x - 3}{1} = \frac{y + 2}{- 4} = \frac{z}{5} .$

Ex. 29.13 | Q 5 | Page 74

Show that the lines  $\frac{x + 4}{3} = \frac{y + 6}{5} = \frac{z - 1}{- 2}$ and 3x − 2y + z + 5 = 0 = 2x + 3y + 4z − 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

Ex. 29.13 | Q 6 | Page 74

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 3$ contains the line whose vector equation is $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right) .$

Ex. 29.13 | Q 7 | Page 74

Find the equation of the plane determined by the intersection of the lines $\frac{x + 3}{3} = \frac{y}{- 2} = \frac{z - 7}{6} \text{ and }\frac{x + 6}{1} = \frac{y + 5}{- 3} = \frac{z - 1}{2}$

Ex. 29.13 | Q 8 | Page 74

Find the vector equation of the plane passing through the points (3, 4, 2) and (7, 0, 6) and perpendicular to the plane 2x − 5y − 15 = 0. Also, show that the plane thus obtained contains the line $\vec{r} = \hat{i} + 3 \hat{j} - 2 \hat{k} + \lambda\left( \hat{i} - \hat{j} + \hat{k} \right) .$

Ex. 29.13 | Q 9 | Page 74

If the lines  $\frac{x - 1}{- 3} = \frac{y - 2}{- 2k} = \frac{z - 3}{2} \text{ and }\frac{x - 1}{k} = \frac{y - 2}{1} = \frac{z - 3}{5}$ are perpendicular, find the value of and, hence, find the equation of the plane containing these lines.

Ex. 29.13 | Q 10 | Page 74

Find the coordinates of the point where the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$ intersect the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Ex. 29.13 | Q 11 | Page 74

Find the vector equation of the plane passing through three points with position vectors  $\hat{i} + \hat{j} - 2 \hat{k} , 2 \hat{i} - \hat{j} + \hat{k} \text{ and } \hat{i} + 2 \hat{j} + \hat{k} .$  Also, find the coordinates of the point of intersection of this plane and the line  $\vec{r} = 3 \hat{i} - \hat{j} - \hat{k} + \lambda\left( 2 \hat{i} - 2 \hat{j} + \hat{k} \right) .$

Ex. 29.13 | Q 12 | Page 74

Show that the lines  $\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}$ and  $\frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}$ are coplanar.

Ex. 29.13 | Q 13 | Page 74

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line  $\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}$ .

Ex. 29.13 | Q 14 | Page 74

Show that the lines  $\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}$ and  $\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}$  are coplanar. Hence, find the equation of the plane containing these lines.

Ex. 29.13 | Q 15 | Page 74

If the line $\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}$  lies in the plane  $lx + my - z =$   then find the value of  $l^2 + m^2$ .

Ex. 29.13 | Q 16 | Page 74

Find the values of  $\lambda$ for which the lines

$\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{\lambda^2}$and  $\frac{x - 3}{1} = \frac{y - 2}{\lambda^2} = \frac{z - 1}{2}$  are coplanar .
Ex. 29.13 | Q 17 | Page 74

If the lines  $x =$  5 ,  $\frac{y}{3 - \alpha} = \frac{z}{- 2}$ and   $x = \alpha$ $\frac{y}{- 1} = \frac{z}{2 - \alpha}$ are coplanar, find the values of  $\alpha$.

Ex. 29.13 | Q 18 | Page 74

If the straight lines  $\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}$ and $\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}$ are coplanar, find the equations of the planes containing them.

Chapter 29: The Plane Exercise 29.14 solutions [Page 77]

Ex. 29.14 | Q 1 | Page 77

Find the shortest distance between the lines

$\frac{x - 2}{- 1} = \frac{y - 5}{2} = \frac{z - 0}{3} \text{ and } \frac{x - 0}{2} = \frac{y + 1}{- 1} = \frac{z - 1}{2} .$

Ex. 29.14 | Q 2 | Page 77

Find the shortest distance between the lines

$\frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1} \text{ and } \frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1} .$

Ex. 29.14 | Q 3 | Page 77

Find the shortest distance between the lines

$\frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z + 2}{1}$ and
$3x - y - 2z + 4 = 0 = 2x + y + z + 1$

Chapter 29: The Plane Exercise 29.15 solutions [Pages 81 - 82]

Ex. 29.15 | Q 1 | Page 81

Find the image of the point (0, 0, 0) in the plane 3x + 4y − 6z + 1 = 0.

Ex. 29.15 | Q 2 | Page 81

Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

Ex. 29.15 | Q 3 | Page 81

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .$

Hence, or otherwise, deduce the length of the perpendicular.

Ex. 29.15 | Q 4 | Page 81

Find the image of the point with position vector $3 \hat{i} + \hat{j} + 2 \hat{k}$  in the plane  $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 4 .$  Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through $3 \hat{i} + \hat{j} + 2 \hat{k} .$

Ex. 29.15 | Q 5 | Page 81

Find the coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x − 2y + 4z + 5 = 0. Also, find the length of the perpendicular.

Ex. 29.15 | Q 6 | Page 82

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured along a line parallel to  $\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6} .$

Ex. 29.15 | Q 7 | Page 82

Find the coordinates of the foot of the perpendicular from the point (2, 3, 7) to the plane 3x − y − z = 7. Also, find the length of the perpendicular.

Ex. 29.15 | Q 8 | Page 82

Find the image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0.

Ex. 29.15 | Q 9 | Page 82

Find the distance of the point with position vector

$- \hat{i} - 5 \hat{j} - 10 \hat{k}$  from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i} + 4 \hat{j} + 12 \hat{k} \right)$  with the plane $\vec{r} \cdot \left( \hat{i} - \hat{j}+ \hat{k} \right) = 5 .$

Ex. 29.15 | Q 10 | Page 82

Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 4 \hat{k} \right) + 5 = 0 .$

Ex. 29.15 | Q 11 | Page 82

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the  point P (3, 2, 1) from the plane 2x − y + z + 1 = 0. Also, find the image of the point in the plane.

Ex. 29.15 | Q 12 | Page 82

Find the direction cosines of the unit vector perpendicular to the plane  $\vec{r} \cdot \left( 6 \hat{i} - 3 \hat{j} - 2 \hat{k} \right) + 1 = 0$ passing through the origin.

Ex. 29.15 | Q 13 | Page 82

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x − 3y + 4z − 6 = 0.

Ex. 29.15 | Q 14 | Page 82

Find the length and the foot of perpendicular from the point $\left( 1, \frac{3}{2}, 2 \right)$  to the plane $2x - 2y + 4z + 5 = 0$ .

Ex. 29.15 | Q 15 | Page 82

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ to the plane  $\vec{r} . \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) - 26 = 0$ Also find image of P in the plane.

Chapter 29: The Plane Exercise Very Short Answers solutions [Pages 83 - 84]

Very Short Answers | Q 1 | Page 83

Write the equation of the plane parallel to XOY- plane and passing through the point (2, −3, 5).

Very Short Answers | Q 2 | Page 83

Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).

Very Short Answers | Q 3 | Page 83

Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).

Very Short Answers | Q 4 | Page 83

Write the general equation of a plane parallel to X-axis.

Very Short Answers | Q 5 | Page 83

Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.

Very Short Answers | Q 6 | Page 83

Write the intercepts made by the plane 2x − 3y + 4z = 12 on the coordinate axes.

Very Short Answers | Q 7 | Page 83

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

Very Short Answers | Q 8 | Page 83

Write the distance between the parallel planes 2x − y + 3z = 4 and 2x − y + 3z = 18.

Very Short Answers | Q 9 | Page 83

Write the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 14$  in normal form.

Very Short Answers | Q 10 | Page 83

Write the distance of the plane  $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12$ from the origin.

Very Short Answers | Q 11 | Page 83

Write the equation of the plane  $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$   in scalar product form.

Very Short Answers | Q 12 | Page 83

Write a vector normal to the plane  $\vec{r} = l \vec{b} + m \vec{c} .$

Very Short Answers | Q 13 | Page 83

Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y −z = 7.

Very Short Answers | Q 14 | Page 83

Write the equation of the plane containing the lines $\vec{r} = \vec{a} + \lambda \vec{b} \text{ and } \vec{r} = \vec{a} + \mu \vec{c} .$

Very Short Answers | Q 15 | Page 83

Write the position vector of the point where the line $\vec{r} = \vec{a} + \lambda \vec{b}$ meets the plane  $\vec{r} . \vec{n} = 0 .$

Very Short Answers | Q 16 | Page 83

Write the value of k for which the line $\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{k}$  is perpendicular to the normal to the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right) = 4 .$

Very Short Answers | Q 17 | Page 84

Write the angle between the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}$  and the plane x + y + 4 = 0.

Very Short Answers | Q 18 | Page 84

Write the intercept cut off by the plane 2x + y − z = 5 on x-axis.

Very Short Answers | Q 19 | Page 84

Find the length of the perpendicular drawn from the origin to the plane 2x − 3y + 6z + 21 = 0.

Very Short Answers | Q 20 | Page 84

Write the vector equation of the line passing through the point (1, −2, −3) and normal to the plane $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right) = 5 .$

Very Short Answers | Q 21 | Page 84

Find the vector equation of the plane, passing through the point (abc) and parallel to the plane $\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2$

Very Short Answers | Q 22 | Page 84

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ .

Very Short Answers | Q 23 | Page 84

Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes.

Chapter 29: The Plane Exercise MCQ solutions [Pages 84 - 86]

MCQ | Q 1 | Page 84

The plane 2x − (1 + λ) y + 3λz = 0 passes through the intersection of the planes

• 2x − y = 0 and y − 3z = 0

• 2x + 3z = 0 and y = 0

• 2x − y + 3z = 0 and y − 3z = 0

• None of these

MCQ | Q 2 | Page 84

The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is

•  45°

• 60°

•  30°

•  75°

MCQ | Q 3 | Page 84

The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

• 7x − 2y + 3z + 81 = 0

• 23x + 14y − 9z + 48 = 0

•  51x − 15y − 50z + 173 = 0

•  None of these

MCQ | Q 4 | Page 84

The distance between the planes 2x + 2y − z + 2 = 0 and 4x + 4y − 2z + 5 = 0 is

• $\frac{1}{2}$

• $\frac{1}{4}$

•  $\frac{1}{6}$

• None of these

MCQ | Q 5 | Page 84

The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is

•  (3, 5, 2)

•  (−3, 5, 2)

•  (3, 5, −2)

• (3, −5, 2)

MCQ | Q 6 | Page 85

The equation of the plane containing the two lines

$\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 0}{3} \text{ and }\frac{x}{- 2} = \frac{y - 2}{- 3} = \frac{z + 1}{- 1}$

•  8x + y − 5z − 7 = 0

•  8x + y + 5z − 7 = 0

• 8x − y − 5z − 7 = 0

•  None of these

MCQ | Q 7 | Page 85

The equation of the plane $\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$  in scalar product form is

•   $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} - 3 \hat{k} \right) = 7$

•  $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 7$

•  $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} + 3 \hat{k} \right) = 7$

•  None of these

MCQ | Q 8 | Page 85

The distance of the line $\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k} \right)$  from the plane $\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5$ is

• $\frac{5}{3\sqrt{3}}$

• $\frac{10}{3\sqrt{3}}$

• $\frac{25}{3\sqrt{3}}$

•  None of these

MCQ | Q 9 | Page 85

The equation of the plane through the line x + y + z + 3 = 0 = 2x − y + 3z + 1 and parallel to the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is

•  x − 5y + 3z = 7

• x − 5y + 3z = −7

•  x + 5y + 3z = 7

•  x + 5y + 3z = −7

MCQ | Q 10 | Page 85

The vector equation of the plane containing the line $\vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + \lambda\left( 3 \hat{i} - 2 \hat{j} - \hat{k} \right)$ and the point  $\hat{i} + 2 \hat{j} + 3 \hat{k}$  is

• $\vec{r} \cdot \left( \hat{i} + 3 \hat{k} \right) = 10$

•  $\vec{r} \cdot \left( \hat{i} - 3 \hat{k} \right) = 10$

•  $\vec{r} \cdot \left( 3\hat{i} - \hat{k} \right) = 10$

• None of these

MCQ | Q 11 | Page 85

A plane meets the coordinate axes at AB and C such that the centroid of ∆ABC is the point (abc). If the equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = k,$ then k =

•  1

•  2

•  3

•  None of these

MCQ | Q 12 | Page 85
The distance between the point (3, 4, 5) and the point where the line $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$ meets the plane x + y + z = 17 is
•  1

• 2

•  3

• None of these

MCQ | Q 13 | Page 85

A vector parallel to the line of intersection of the planes$\vec{r} \cdot \left( 3 \hat{i} - \hat{j} + \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( \hat{i} + 4 \hat{j} - 2 \hat{k} \right) = 2$ is

•  $- 2 \hat{i} + 7 \hat{j}+ 13 \hat{k}$

•   $2 \hat{i} + 7 \hat{j} - 13 \hat{k}$

•  $-2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

•  $2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

MCQ | Q 14 | Page 85

If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}$ then its perpendicular distance from the origin is

• 3/4

•  4/3

•  7/5

•  1

MCQ | Q 15 | Page 85

The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is

•  x − 4y + 2z + 4 = 0

• x + 4y + 2z + 4 = 0

•  x − 4y + 2z − 4 = 0

• None of these

MCQ | Q 16 | Page 86

The distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = 2 \hat{i}- \hat{j} + 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j}+ 12 \hat{k} \right)$   and the plane $\vec{r} \cdot \left( \hat{i} - \hat{j} + \hat{k} \right) = 5$ is

• 9

•  13

•  17

•  None of these

MCQ | Q 17 | Page 86

The equation of the plane through the intersection of the planes ax + by + cz + d = 0 andlx + my + nz + p = 0 and parallel to the line y=0, z=0

• (bl − amy + (cl − anz + dl − ap = 0

•  (am − blx + (mc − bnz + md − bp = 0

•  (na − clx + (bn − cmy + nd − cp = 0

• None of these

MCQ | Q 18 | Page 86

The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is

•  x + y + z = 1

•  x + y + z = 0

• x + y − z = 1

•  x + y + z = 2

Chapter 29: The Plane

Ex. 29.1Ex. 29.2Ex. 29.3Ex. 29.4Ex. 29.5Ex. 29.6Ex. 29.7Ex. 29.8Ex. 29.9Ex. 29.10Ex. 29.11Ex. 29.12Ex. 29.13Ex. 29.14Ex. 29.15Very Short AnswersMCQ

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) RD Sharma solutions for Class 12 Mathematics chapter 29 - The Plane

RD Sharma solutions for Class 12 Maths chapter 29 (The Plane) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 29 The Plane are Three - Dimensional Geometry Examples and Solutions, Introduction of Three Dimensional Geometry, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Angle Between Two Lines, Vector and Cartesian Equation of a Plane, Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Plane Passing Through the Intersection of Two Given Planes, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line.

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