If ,  then is true that  Justify your answer. from Mathemat

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 Multiple Choice QuestionsShort Answer Type

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101.

If straight a with rightwards arrow on top space equals straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top,  then is true that open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top close vertical bar space plus space open vertical bar straight c with rightwards arrow on top close vertical bar space ? Justify your answer.


When straight a with rightwards arrow on top space equals space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top, then straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top form a triangle ABC where
                        AB with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space BC with rightwards arrow on top space equals space straight c with rightwards arrow on top comma space space AC with rightwards arrow on top space equals space straight a with rightwards arrow on top
 
Since sum of two sides of a triangle is always greater than the third side
              therefore space space space space open vertical bar straight b with rightwards arrow on top close vertical bar space plus space open vertical bar straight c with rightwards arrow on top close vertical bar space greater than space open vertical bar straight a with rightwards arrow on top close vertical bar
or space space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space less than space open vertical bar straight b with rightwards arrow on top close vertical bar space plus space open vertical bar straight c with rightwards arrow on top close vertical bar
therefore space space space straight a with rightwards arrow on top space equals straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space does space not space imply space in space general space that space open vertical bar straight a with rightwards arrow on top close vertical bar space equals open vertical bar straight b with rightwards arrow on top close vertical bar space plus space open vertical bar straight c with rightwards arrow on top close vertical bar.


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 Multiple Choice QuestionsMultiple Choice Questions

102. In triangle ABC, which of the following is not true

  • AB with rightwards arrow on top space plus space BC with rightwards arrow on top space plus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top
  • AB with rightwards arrow on top space plus space BC with rightwards arrow on top space minus space AC with rightwards arrow on top space equals space 0 with rightwards arrow on top
  • AB with rightwards arrow on top space plus space BC with rightwards arrow on top space minus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top
  • AB with rightwards arrow on top space plus space BC with rightwards arrow on top space minus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top
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103.

If straight a with rightwards arrow on top and straight b with rightwards arrow on top are two collinear vectors, then which of the following are incorrect:

  • straight b with rightwards arrow on top equals space straight lambda space straight a with rightwards arrow on top comma space space for space some space scalar space straight lambda
  • straight a with rightwards arrow on top space equals plus-or-minus space straight b with rightwards arrow on top
  • the respective components of straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are proportional

  • the respective components of straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are proportional

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104.

If straight a with rightwards arrow on top is a non-zero vector of magnitude ‘a’ and λ a non-zero scalar, then straight lambda straight a with rightwards arrow on top is a unit vector if

  • straight lambda space equals space 1
  • straight lambda equals negative 1
  • straight a space equals space open vertical bar straight lambda close vertical bar
  • straight a space equals space open vertical bar straight lambda close vertical bar
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 Multiple Choice QuestionsShort Answer Type

105. Find the position vector of a point R which divides the line joining straight P left parenthesis 3 straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top right parenthesis space and space straight Q left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis in the ratio 2 : 1 
(i) internally and (ii) externally.
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106. Find the position vector of a point R which divides the line joining two points straight P left parenthesis 2 straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space and space straight Q left parenthesis straight a with rightwards arrow on top space minus space 3 space straight b with rightwards arrow on top right parenthesis externally in the ratio 1 :2. Also, show that P is the middle point of the line segment RQ.
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 Multiple Choice QuestionsLong Answer Type

107. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top and negative straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top respectively, in the ratio 2 : 1
(i) internally                  (ii) externally
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 Multiple Choice QuestionsShort Answer Type

108.

If straight a with rightwards arrow on top an straight b with rightwards arrow on top are position vectors of points A and B respectively, then find the position vector of points of trisection of AB.

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109. Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, – 2).
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110. Find the vector from the origin O to the centroid of the triangle whose vertices are (1, – 1, 2), (2, 1, 3) and (–1, 2, –1).
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