Chapter 11 3D by nehalwan

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                        Three –Dimensional Geometry (1 mark questions)
1. Find the vector equation of the line through the point (5,2,-4) and which is parallel to the vector 3î + 2ĵ
   - 8k.

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

3. Find the angle between two planes:
           3x - 6y + 2z = 7 and 2x + 2y – 2z = 5

4. A straight line makes angles 60˚ and 45˚ with the positive direction of x – axis and y – axis
   respectively. What angle does it make with the z – axis ?

5. Find the equation of the plane passing through the point (-1,0,7) and parallel to the plane
   3x - 5y + 4z = 11.

6. Find the distance from the point (2,5,-3) to plane 6x – 3y +2z = 4.

7. Show that the normals to the planes r . (î – ĵ + k) = 4 and r .( 3î + 2ĵ - k) -11 = 0 are perpendicular to
   each other.

8. Write the equation of the plane whose intercepts on the co- ordinate axis are -4, 2 and 3.

9. If the line r = (î - 2ĵ + k)+λ(2î + ĵ + 2k) is parallel to the plane r .( 3î - 2ĵ + mk) = 14, find the value
   of m.

10. If 4x + 4y – λz =0 is the equation of the plane through the origin that contains the line
                      x–1 = y+1 = z ,
                       2        3        4

   find the value of ‘λ’.



                                         SECTION - B(4 marks)

1. Find the shortest distance between the lines l1 and l2 whose vector equations are :-
    r = î + ĵ + λ(2î – ĵ + k)
    r = 2î + ĵ – k + μ(3î - 5ĵ + 2k)

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

3. Find the vector equation of the line passing through (1,2,3) and parallel to the planes r. (î – ĵ + 2k) = 5
    and r .( 3î + ĵ + k ) = 6.

4. Find the vector equation of the plane passing through the points A(1,-2,5) , B(0,-5,-1) and C(-3,5,0).

5. Find the image of the point (3,-2,1) in the plane 3x - y + 4z = 2.
                                                          2
6. Find the image of the point (1,6,3) in the line x = y – 1 = z – 2 .
                                                       1       2        3
7. If l1, m1, n1 ; l2 , m2 , n2 are the direction cosines of two mutually perpendicular lines. Show that the
    direction cosines of the line perpendicular to both of them are m1n2 – m2n1 , n1l2 – n2l1 , l1m2 – l2m1.

8. Verify that l1+l2+l3 , m1+m2+m3 , n1+n2+n3 can be taken as the direction cosines of a line equally
                √3           √3         √3
   inclined to three mutually perpendicular lines with direction cosines l1,m1,n1;l2,m2,n2 and l3,m3,n3.


                                        SECTION – C(6 Marks)
1. A line makes angles α, β, γ and δ with the diagonals of a cube, prove that cos2α + cos2β + cos2γ +
   cos2δ = 4
           3

2. Show that the lines r1 = î + ĵ – k + λ(3î - ĵ)   and       r2 = 4î – k + μ(2î + 3k) intersect.
   Find their point of intersection.

3. Find the equation of the plane through the intersection of the planes r .(î + 3ĵ)+6 = 0 and
   r .(3î - ĵ - 4k) = 0 which is at a unit distance from the origin.

4. Show that the lines :
                           x - 1 = y – 3 = z and x – 4 = y – 1 = z – 1
                            2         4       -1         3         -2          1
     are coplanar. Also find the equation of the plane containing these lines.

5. If a variable line in two adjacent positions has direction cosines l, m, n and l+δl , m+δm , n+δn ,
   show that the small angle δθ between two positions is given by:-
                        (δθ)2 = (δl)2 + ( δm)2 + ( δn)2

6. Find the equations of the line of shortest distance between the lines :
               x – 8 = y +9 = z-10 and x-15 = y-29 = 5-z
                 3      -16       7            3        8     5
   also find the shortest distance between the lines.

7. A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A
   , B , C. Show that the locus of the centroid of ΔABC is x -2 + y -2 +z -2 = p-2 .

8. Find the equation of the perpendicular from the point (3,-1,11) to the line x = y-2 = z-3 .
                                                                               2    3     4
   Also find the foot of perpendicular and the length of perpendicular.

9.    Find the distance of the point (1,-2,3) from the plane x - y + z = 5 measured parallel to the line
      x+1 = y+3 = z+1 .
        2      3       -6
                                                       3


                                                    Answers

                                      Three Dimensional Geometry

                                                1 MARKS

  1) r = (5 î + 2ĵ-4 k ) + λ(3 î + 2ĵ- 8k )                    2) 6
                                                                 √29

  3) cos-1 5√3                         4) 60º                  5) 3x - 5y + 4z – 25 = 0
            21

  6) 13 units                          8) -3x + 6y + 4z = 12    9) m = -2
      7

  10) λ= 5
                                                4 MARKS

1) 10           2) x – z + 2 = 0
                                              √59

 3) r = î + 2ĵ + 3k + λ(-3î + 5ĵ + 4 k )                       4) r .( 3î + ĵ – k ) = - 4

 5) (0,-1,-3)                          6) (1,0,7)

                                                6 MARKS
  2) (4,0,-1)

 3) r .( -2î + 4ĵ + 4k ) + 6 = 0 or                            4) 2x – 5y -16z +13 = 0
     r . (4î + 2ĵ -4 k ) + 6 = 0

 6) x-5 = y – 7 = z-3 , 14 units                               8)(2,5,7) , √53 units
     2       3      6                                              x-3 = y + 1 = z-11
                                                                   -1       6     -4
 9) 1 unit

								
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