If the angle between the line x = and the plane x + 2y + 3z = 4

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 Multiple Choice QuestionsMultiple Choice Questions

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11.

If the angle between the line x =fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator straight lambda end fraction and the plane x + 2y + 3z = 4 is cos-1 square root of 5 over 14 end root then λ equal

  • 2/3

  • 3/2

  • 2/5

  • 2/5


A.

2/3

fraction numerator straight x minus 0 over denominator 1 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator straight lambda end fraction space... space left parenthesis 1 right parenthesis
x + 2y + 3z = 4 ....... (2)

Angle between the line and plane is
cos space left parenthesis 90 minus space straight theta right parenthesis space equals space fraction numerator straight a subscript 1 straight a subscript 2 space plus straight b subscript 1 straight b subscript 2 space plus space straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 superscript 2 plus straight b subscript 1 superscript 2 plus straight c subscript 1 superscript 2 end root space square root of straight a subscript 2 superscript 2 plus straight b subscript 2 superscript 2 plus straight c subscript 2 superscript 2 end root end fraction
rightwards double arrow space sin space straight theta space equals space fraction numerator 1 space plus space 4 space plus space 3 straight lambda over denominator square root of 14 space straight x space square root of 5 space plus space straight lambda squared end root end fraction space equals space fraction numerator 5 space plus space 3 straight lambda over denominator square root of 14 space straight x space end root square root of 5 space plus space straight lambda squared end root end fraction space... space left parenthesis 3 right parenthesis

But given that angle between line and plane is

rightwards double arrow space sin space straight theta space equals space fraction numerator 3 over denominator square root of 14 end fraction
therefore space from space left parenthesis 3 right parenthesis
fraction numerator 3 over denominator square root of 14 end fraction space equals space fraction numerator space 5 space plus space 3 straight lambda over denominator square root of 14 space end root space straight x square root of space 5 space plus straight lambda squared end root end fraction
rightwards double arrow space 9 space left parenthesis 5 space plus space straight lambda squared right parenthesis space equals space 25 space plus space 9 straight lambda squared space plus space 30 space straight lambda
30 space straight lambda space equals space 20
straight lambda space equals space 2 divided by 3
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12.

Statement-1 : The point A(1, 0, 7) is the mirror image of the point B(1, 6, 3) in the line: straight x over 1 space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 3 end fraction
Statement-2: The line:straight x over 1 space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 3 end fraction  bisects the line segment joining A(1, 0, 7) and B(1, 6, 3).

  • Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. 

  • Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. 

  • Statement-1 is true, Statement-2 is false.

  • Statement-1 is true, Statement-2 is false.

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13.

A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ

  • 30°

  • 45°

  • 60°

  • 60°

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14.

The line L given by straight x over 5 space plus straight y over straight b space equals space 1 passes through the point (13, 32). The line K is parallel to L and has the equation straight x over straight c space plus straight y over 3 space equals space 1 . Then the distance
between L and K is

  • fraction numerator 23 over denominator square root of 15 end fraction
  • square root of 17
  • fraction numerator 17 over denominator square root of 15 end fraction
  • fraction numerator 17 over denominator square root of 15 end fraction
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15.

Let k be an integer such that triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point 

  • open parentheses 2 comma 1 half close parentheses
  • open parentheses 2 comma negative 1 half close parentheses
  • open parentheses 1 comma negative 3 over 2 close parentheses
  • open parentheses 1 comma negative 3 over 2 close parentheses
857 Views

16.

The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point

  • open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 third close parentheses
  • open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 half close parentheses
  • open parentheses fraction numerator 1 over denominator 2 comma end fraction 1 half close parentheses
205 Views

17.

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to line, x/1 = y/4= z/5 is Q, then PQ is equal to

  • 6 square root of 7
  • 3 square root of 5
  • 2 square root of 42
  • 2 square root of 42
245 Views

18.

The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal perpendicular to both the linesfraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y plus 2 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 4 over denominator 3 end fraction space and space fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z plus 7 over denominator negative 1 end fraction is

  • fraction numerator 10 over denominator square root of 74 end fraction
  • fraction numerator 20 over denominator square root of 74 end fraction
  • fraction numerator 10 over denominator square root of 83 end fraction
  • fraction numerator 10 over denominator square root of 83 end fraction
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19.

The lines p(p2+ 1) x – y + q = 0 and (p2+ 1)2x + (p2+ 1) y + 2q = 0 are
perpendicular to a common line for

  • no value of p

  • exactly one value of p

  • exactly two values of p

  • exactly two values of p

115 Views

20.

Let the line fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator negative 5 end fraction space equals space fraction numerator straight z plus 2 over denominator 2 end fraction lie in the plane x + 3y – αz + β = 0. Then (α, β) equals

  • (6, – 17)

  • (–6, 7)

  • (5, –15)

  • (5, –15)

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