Plus Two Maths Chapter Wise Questions and Answers Chapter 11 Three Dimensional Geometry

are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 11 Three Dimensional Geometry

.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Chapter Wise Questions |

Chapter | Chapter 11 |

Chapter Name | Three Dimensional Geometry |

Number of Questions Solved | 41 |

Category | Kerala Plus Two |

## Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 11 Three Dimensional Geometry

**Short Answer Type Questions (Score 3)**

Question 1.

Given the straight line

i. What are its direction ratios? **(1) **

ii. Find the distance from the point (2,4,-1) to the given line. **(1)**

iii. Obtain the equation of the straight line passing through (2, 4, -1) and parallel to the given line. **(1)**

Answer:

i. Direction ratios are 1, 4, -9

ii. 7

Question 2.

i. Find the equation of the plane containing the line of intersection of the plane

x+y+z-6=0 and 2x +3y + 4z + 5 = 0. **(1)**

ii. If the plane passes through the point (1, 1, 1) find its equation. **(2)**

Answer:

Question 3.

Find the equation of the plane through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane

2x + 6y + 6z – 1= 0. **(6)**

Answer:

\quad is perpendicular to the plane.

2x + 6y + 6z -1 = 0 and hence it is parallel to the required plane.

Question 4.

A variable plane is at a constant distance P from origin and meets the axes in

A (p,0,0), B (0, q, 0), C (0, 0, r).

i. Write the equation of the plane passing through A, B and C. **(1)**

ii. Find the centriod of Δ ABC. **(1)**

iii. Show that the locus of P is

**(1)**

Answer:

i. Equation of the plane is perpendicular distance from origin = p

Question 5.

Consider the points (-1,2,4) and (1,0,5). Find the direction cosines of the line joining the two points **(3)**

Answer:

d.r’s are proportional to 1-1, 0-2,5-4 i.e, 2, -2, 1; d.c’s are

Question 6.

Find the angle between the line

Ans:

Question 7.

Consider the cartesian equation of the line

i. Write the co-ordinates of the point, in which the line passes through. **(1)**

ii. Write the direction ratios of the line. **(1)**

iii. Find the vector equation of the above line. **(1)**

Answer:

i. (5, -4, 6)

ii. Direction ratios are 3, 7, 2

iii.

Question 8.

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

2x-y = 0, 3z – y = 0 and perpendicular to the plane 4x + 5y – 3z = 8. **(3)**

Answer:

Let the equation of the plane be (2x – y) + A (3z – y) = 0

i.e., 2x-(1+λ)y + 3λ z = 0———– (1)

By the given condition,

2(4) – (1+ x )5+3 A (-3)= 0

⇒

using in (1), 28x- 17y + 9z = 0

Question 9.

Prove that the points (2, 4, 4), (1, 5, 4), (2, 6, 2) and (0, 6, 4) are coplanar.**(3)**

Answer:

The equation of a plane passing through the points (2, 4, 4), (1, 5, 4) and (2, 6, 2) is

Substituting (0, 6, 4) in this equation 0 + 6 + 4=10 which is true.

The points lie in the same plane.Thus the given points are coplanar.

Question 10.

Find the angle between the lines whose direction ratios are a, b, c and b – c ,c – a, a-b **(3)**

Answer:

**Long Answer Type Questions**

**(Score 4)**

Question 1.

Consider the pair of lines whose equations are

and

i. Write the direction ratios of the lines. **(1)**

ii. Check whether the lines are skew lines. **(2)**

iii. Find the angle between these two lines. **(1)**

Answer:

i. Direction ratios of the lines:

2, 5, -3 and -1, 8, 4

Question 2.

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

x + y + 4z + 5 = 0 and 2x – y + 3z + 6 = 0 and containing the point (1, 0, 0). **(4)**

Answer:

The equation of the plane through the line of intersection of the given planes is

(x+y + 4z + 5)+λ (2x-y+3z + 6)=0…(l)

If (1) passes through (1,0, 0), we have

(1 + 5) + λ (2 + 6) = 0

6 + λ 8 = 0, 3 + 4λ = 0

⇒ λ = –

Putting X = in (1) we obtain the

equation of the required plane as

(x + y + 4z + 5) – (2x – y + 3z + 6) = 0

i.e., 2x – 7y – 7z – 2 = 0

Question 3.

Show that if a plane has the intercepts ; a, b, c and is P units from the origin, then

**(4)**

Answer:

The equation of a plane in the intercept form is

are

Question 4.

If O is 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. **(4)**

Answer:

Direction ratios of OP are 1, 2, -3

OP is perpendicular to the plane

∴ Direction ratios of the normal to the plane are a = 1, b = 2, c = 3

P (1, 2, – 3) is a point on the plane.

Equation of the plane is

a(x-x_{1}) + &(y-y_{1})+c(z-z_{1}) = 0

i.e., 1(x-1)+2(y-2) – 3 (z + 3) = 0

x-1 + 2y-4-3z-9 = 0 ;

x + 2y-3z-14 = 0

The equation of the plane is

x + 2y-3z-14 = 0

Question 5.

Consider the equation of the line 3x + 4 = 4y – 1 = 1 – 4z.

i. Find the direction ratios of this line. **(2)**

ii. What is the vector equation of a line passing through (2, 4, 6) and parallel to the above line ? **(2)**

Answer:

Question 6.

a. Find the equation of the plane through the point (1, 2, 3) perpendicular to the planes

x – y + z = 2 and

2x + y – 3z = 5. **(2)**

b. Find the distance between the parallel planes x-2y+2z-8= 0 and 6y-3x- 6z = 57 **(2)**

Answer:

a. Any plane through (1,2,3) is

a (x -1) + b (y – 2) + c (z – 3) = 0….(1)

(1) is ⊥ r to planes, x – y + z = 2 and 2x + y – 3z = 5,

ie, a – b + c = 0 and 2a + b -3c = 0

a = 2k, b = 5k, c = 3k.

Put this is (1).

2k (x – 1) + 5k (y – 2) + 3k (z – 3) = 0

2(x – 1) + 5 (y – 2) + 3 (z – 3) = 0

2x + 5y + 3z – 21 = 0

b. Give planes are x – 2y + 2z – 8 = 0

and 3x-6y + 6z+57 = 0;

ie, x-2y + 2z +19 = 0

Question 7.

i. A plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point

(p, q, r). Show that the equation of the plane is

**(2)**

ii. A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the three

co-ordinate axes is constant. Show that the plane passes through a fixed point. **(2)**

Answer:

i. Let the equation of the required plane

Question 8.

Consider the vector

i. Find the direction cosines of the given vector. **(3)**

ii. Evaluate the angles at which it is inclined to the positive direction of each of the co-ordinate axes. **(1)**

Answer:

Question 9.

a. Find the angle between the planes 2x-y + z = 6 and x + y + 2z = 7. **(2)**

b. The point in the XY plane which is equidistance from the points (2,0,3),(0,3,2) and (0,0,1) is…… **(2)**

Answer:

b. Let P (x, y, 0) is a point on xy plane A (2, 0,3), B (0, 3, 2) & C (0,0, 1) PA = PB = PC

Question 10.

a. Find the perpendicular distance of the points (6, 5, 8) from γ – axis .**(1)**

b. Find the position vector of the point where the

meets the plane **(3)**

Answer:

**Very Long Answer Type Questions (Score 6)**

Question 1.

Let α , β ,γ be the angles made by a line with the positive directions of the X, Y, Z axes respectively.

i. Write the direction cosines of the line. **(1)**

ii. Show that **(2)**

iii. Show that

**(3)**

Answer:

Question 2.

A straight line is passing through (3, 8, 3) and is parallel to the vector

i. Form its equation. **(2)**

ii. Show that it is not coplanar with the lines **(1)**

iii. Calculate the shortest distance between the lines. **(3)**

Answer:

Question 3.

A variable plane is at a constant distance P from origin and meets the axes in

A (p,0,0), B (0, q, 0), C (0, 0, r).

i. Write the equation of the plane passing through A, B and C **(3)**

ii. Find the centriod of Δ ABC. **(2)**

iii. Show that the locus of P is **(1)**

Answer:

Equation of the plan is

perpendicular distance from orgin =p

The plane cuts the axes at (a,0,0), (0,b,0) and (0,0,c).

Question 4.

i. Find the equation of the plane passing through the points (2, 1, 0), (3, -2, -2) and

(3, 1, 7). **(2)**

ii. Find ‘x’ such that the points A(3,2,1), B(4, x, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar. **(2)**

iii. Find the equation of the plane containing the lines

and

**(2)**

Answer:

Question 5.

Consider the planes 3x – 4y + 5z = 10 and 2x + 2y – 3z = 4

a. Write the equation of the planes through the line of intersection of the above planes. **(2)**

b. Write the direction ratios of the line x = 2y = 3z .**(2)**

c. If the above line is parallel to the obtained plane, show that the plane is

x – 20y + 27z = 14. **(2)**

Answer:

Question 6.

i. Find the direction cosines of the vector **(2)**

ii. Find the distance of the point (2, 3, 4) from the plane **(2)**

iii. Find the shortest distance between the lines

and **(2)**

Answer:

i. d.r’s are 2, 2,-1 ; d.c’s are

Question 7.

Find the vector equation of the plane passiing through the intersection of the planes and the point (1,1,1). **(6)**

Answer:

The vector equation of the planes are

The corresponding cartesian equations are

x + y + z- 6 = 0 and 2x + 3y + 4z +5 = 0

The equation of the plane passing through the intersection of the given plane is

(x + y +z-6)+ λ (2x + 3y + 4z +5) = 0….(1)

Since (1) passes through the point (1,1,1),

we get (1+ 1 + 1 – 6)+ λ (2 + 3 + 4 + 5) = 0

-3+ λ (14) = 0

∴

The equation of the required plane is

14x + 14y +14z – 84+6x+9y + 12z +15 = 0

20x + 23y +26z – 69 = 0

Question 8.

Find the distance between the two planes: 2x+3y + 4z = 4 and 4x + 6y + 8z = 12. **(6)**

Answer:

Question 9.

a. Find the shortest distance between the skew lines and

**(3)**

b. If a line makes angles α , β ,γ with the three axes, show that **(3)**

Answer:

a. shortest distance

Question 10.

Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1) j crosses the plane, passing through the points (2,2,1) (3,0,1) and (4,-1,0). **(6)**

Answer:

**Edumate Questions & Answers**

Question 1.

Choose the correct answer from the bracket.

i. If a line in the space makes angle α , β ,γ with the coordinates axes, then cos^{2} α + cos^{2} β + cos^{2} γ is

equal to

{(a)1, (b)2 (c)0 (d)3}

ii. The direction rations of the line

{(a)(6,-2,-2), (b) (1,2,2)

(c) (6,1,-2), (d) (0,0,0)}

iii. If the vector equation of a line is

iv. If the cartesian equation of a plane ( is x + y + z = 12) ,then the vector equation of of the lane is

Answer:

i.a. 1

ii. b. (1,-2,2)

iii. b.

iv. b.

Question 2.

Cartesian equation of two lines are

i. Write the vector equation of the lines

ii. Find the shortest distance between the lines

Answer:

Question 3.

Consider the lines

i. Write the vector equation of the lines

ii. Find the shortest distance between the lines

Answer:

Question 4.

Let the vector equation of a plane be

1. Write the cartesian equation of the plane.

2. Find the distance from the point (2,1,3) to the plane

Answer:

i. 2x + 4y + 3z = 12

ii. Distance of the point (x_{1}y_{1}z_{1})to the plane Ax+By+Cz=D is.

Question 5.

i. Find the equation of the plane through the point (1,2,3) and perpendicular to the plane

x-y + z = 2 and 2x + y-3z = 5

ii. Find the distance between the planes x-2y+ 2z-8 = 0 and 6y-3x-6z-57 = 0

Answer:

Question 6.

Consider the Cartesian equation of the

i. Find the vector equation of the line.

ii. Find the intersecting point of the line with the plain 5x+2y-6z-7=0.

iii. Find the angle made by the line with the plane 5x+2y-6z-7=0.

Answer:

i. The vector equation is

ii. Any point on the line is

Question 7.

Consider the vector equation of two planes

i. Find the vector equation of any plane through the interesection of the above two planes

ii. Find the vector equation of the plane through the intersection of the above planes and the point (1,2,-1)

Answer:

i. The cartesian equation are

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

Required equation of the plane is

(2x + y + z- 3) + λ(x -y-z- 4) = 0

**NCERT Questions & Answers**

Question 1.

Consider the straight lines

i. Show that these lines are not co planar.

ii. Compute the shortest distance between the lines.

Answer:

Question 2

Find the foot and hence the length of the perpendicular from P(5,7,3) to the line

Find also the equation of the plane in which the perpendicular and the given straight line.

Answer:

Let N be the foot of the perpendicular drawn from P on the given line. Then the co-ordinates of N be

(3λ + 15, 8λ + 29, -5λ + 5)

Then the direction ratios of PN are

3λ + 10, 8λ+ 22,-5λ + 2

Since PN is perpendicular to the given line,

3(3λ + 10)+8(8λ + 22) -5(-5λ + 2) = 0

⇒ λ = -2

Thus the co-ordinates of N are (9, 13, 15)

Let the equation of the plane be a(x-15)+b(y-29)+c(z-5)= 0 — (1)

Where 3a+8b-5c = 0 ————- (2)

If the plane (1) passes through P(5,7,3)

we get 5a +11b + c = 0 —– (3)

Solving (2) & (3)

Substituting the values of a, b, c in (1)

9x-4y-z = 14

Question 3.

Length of the shortest distance between two lines

Answer:

Question 4.

a. A line passes through the point (3,-2,5) and parallel to the vector

1. What is the vector equation of the line?

2. What is the cartesian equation of the line?

3. Find the shortest distance between the skew lines whose vector equations are

b. Find the shortest distance between the skew lines whose vector equations are

Answer:

We hope the given Plus Two Maths Chapter Wise Questions and Answers Chapter 11 Three Dimensional Geometry will help you. If you have any query regarding Plus Two Maths Chapter Wise Questions and Answers Chapter 11 Three Dimensional Geometry, drop a comment below and we will get back to you at the earliest.