A plane P passes through the line of intersection of the planes 2

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 Multiple Choice QuestionsMultiple Choice Questions

11.

A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B and C respectively. The locus of the centre of the sphere OABC, O being the origin, is

  • xa + yb + zc = 1

  • ax + by + cz = 1

  • ax + by + cz = 2

  • xa + yb +zc = 2


12.

The equation of the plane passing through the line of intersection of the planes x + y + z = 1, 2x + 3y + 4z = 7 and perpendicular to the piane x - 5y + 3z = 5 is given by :

  • x + 2y + 3z - 6 = 0

  • x + 2y + 3z + 6 = 0

  • x + 4y + 5z - 8 = 0

  • 3x + 4y + 5z + 8 = 0


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13.

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). What are the direction ratios of the line of intersection of the.given planes ?

  • (2, - 5, - 3)

  • (1, - 5, - 3)

  • (2, 5, 3)

  • (1, 3, 5)


A.

(2, - 5, - 3)

To find, intersection point first put z = 0 2x  y = 2   x + y = 1        3x = 3              x = 1     x + y = 1At x - 1 = 0  x, y, z = 1, 0, 0            z = 32           y = 1 + z = 1 + 32x, y, z = 0, 52, 32Point of intersectionx1, y1, z1  1, 0, 0x2, y2, z2  0, 52, 32Hence direction ratios of the line of intersection of given plane lt 2, - 5, - 3


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14.

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). What is the equation of the plane P ?

  • 2x + 5y - 2 = 0

  • 5x + 2y - 5 = 0

  • x + y - 2 = 0

  • 2x - y - 2z = 0


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15.

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). If the plane P touches the sphere x2 + y + z2 = r, then what is r equal to ?

  • 229

  • 429

  • 529

  • 1


16.

Let Q be the image of the point P 2, 1, 5) in the plan3x 2y + 2+ 1 = 0.

Consider the following :

  1. The coordinates of Q are (4, - 3, - 1).
  2. PQ is of length more than 8 units.
  3. The point (1, - 1, - 3) is the mid-point of the line segment PQ and lies on the given plane. Which of the above statements are correct ?

  • 1 and 2 only

  • 2 and 3 only

  • 1 and 3 only

  • 1, 2 and 3


17.

Let Q be the image of the point P 2, 1, 5) in the plan3x 2y + 2+ 1 = 0.

Consider the following :

  1. The direction ratios of the line segment PQ ar< 3, 2, 2 >.
  2. The sum of the squares of direction cosines of thline segment PQ is unity.

Which of the above statements is/are correct ?

  • 1 only

  • 2 only

  • Both 1 and 2

  • Neither 1 nor 2


18.

A line L passes through the point P(5, - 6, 7) and is parallel to the planes x + y + z = 1 and 2x - y - 2z = 3.

What are the direction ratios of the line of intersectioof the given plane?

  • < 1, 4, 3 >

  • < - 1, - 4, 3 >

  • < 1, - 4, 3 >

  • < 1, - 4, - 3 >


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19.

A line L passes through the point P(5, - 6, 7) and is parallel to the planes x + y + z = 1 and 2x - y - 2z = 3.

What is the equation of the line ?

  • x - 5 - 1 = y + 64 = z - 7 - 3

  • x + 5 - 1 = y - 64 - z + 7 - 3

  • x - 5 - 1 = y + 64 - z - 7 - 3

  • x - 5 - 1 = y + 6 - 4 - z - 7 - 3


20.

A straight line passes through the point (1, 1, 1) makes an angle 60o with the positive direction of z-axis, and the cosine of the angles made by it with the positive directions of the y-axis and the x-axis are in the ration 3 : 1. What is the acute angle between the two possible positions of the line ?

  • 90°

  • 60°

  • 45°

  • 30°


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