A line passing through the point A with position vector  is pa

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 Multiple Choice QuestionsLong Answer Type

301. Show that the lines:
fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 7 over denominator 4 end fraction space equals fraction numerator straight z plus 3 over denominator negative 5 end fraction comma space space fraction numerator straight x minus 8 over denominator 7 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 3 end fraction are co-planar, find their common point and the equation of the plane in which way they lie. 
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 Multiple Choice QuestionsShort Answer Type

302. Find the angle between each of the following pairs of lines:
A line with direction ratios 2 : 2 : 1.
A line joining (3, 1, 4) to (7, 2, 12)
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303. Find the angle between each of the following pairs of lines:
5, –12, 13;    – 3, 4, 5
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304. Find the angle between each of the following pairs of lines:
1, 1, 2;   square root of 3 minus 1 comma space space minus square root of 3 minus 1 comma space 4
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305.

Write the sum of intercepts cut off by the planestraight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses minus 5 space equals 0 on the three axes.

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306.

Find the coordinates of the foot of perpendicular drawn from the point A
(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

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 Multiple Choice QuestionsLong Answer Type

307.

Find the equation of the plane which contains the line of intersection of the planes
straight r with rightwards arrow on top. left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top right parenthesis minus 4 space equals space 0
and
straight r with rightwards arrow on top. left parenthesis negative 2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis plus 5 space space equals space 0

and whose intercept on the x-axis is equal to that of on y-axis.

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 Multiple Choice QuestionsShort Answer Type

308.

Find the distance of a point (2, 5, −3) from the plane straight r with rightwards arrow on top. open parentheses 6 straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top close parentheses space equals 4.

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309.

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

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310.

A line passing through the point A with position vector straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top is parallel to the vector straight b with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top. Find the length of the perpendicular drawn on this line from a point p with vector stack straight r subscript 1 with rightwards arrow on top space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top.


Let the equation of the line be straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda straight b with rightwards arrow on top
 Here comma space straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top
straight b with rightwards arrow on top equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top
therefore space Equation space of space the space line space is space straight r with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top plus bold lambda left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis
Let space straight L space be space the space foot space of space the space perpendicular space and space straight P space be space the space required space point space from
which space we space have space to space find space the space length space of space the space perpendicular
straight P open parentheses straight alpha with rightwards arrow on top close parentheses space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top
PL with rightwards arrow on top space equals space Position space of space vector space of space straight L space minus space Position space vector space of space straight P
space space space space space space equals 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses minus open parentheses straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top close parentheses
space space space space space space equals 3 straight i with hat on top minus straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses space.... left parenthesis straight i right parenthesis
Now 
PL with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 left square bracket Since space PL with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top right square bracket
space space open square brackets 3 straight i with hat on top minus straight k with hat on top plus straight lambda open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses close square brackets. space open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses equals 0
space rightwards double arrow open square brackets straight i with hat on top left parenthesis 3 plus 2 straight lambda right parenthesis plus straight j with hat on top left parenthesis 3 straight lambda right parenthesis plus space straight k with hat on top left parenthesis negative 1 plus 6 straight lambda right parenthesis close square brackets. space open parentheses 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top close parentheses space equals space 0
space rightwards double arrow left parenthesis 3 plus 2 straight lambda right parenthesis 2 plus left parenthesis 3 straight lambda right parenthesis 3 plus left parenthesis negative 1 plus 6 straight lambda right parenthesis 6 space equals space 0
space space rightwards double arrow 6 plus 4 straight lambda plus 9 straight lambda minus 6 plus 36 straight lambda space equals space 0
space space space rightwards double arrow 49 straight lambda space equals space 0
space space space space space therefore space space straight lambda space equals space 0
space space PL with rightwards arrow on top equals 3 straight i with hat on top minus straight k with hat on top space left square bracket from space left parenthesis ii right parenthesis right square bracket
space space open vertical bar PL with rightwards arrow on top close vertical bar space equals open vertical bar square root of 3 squared plus left parenthesis negative 1 right parenthesis squared end root close vertical bar
space space therefore space space open vertical bar PL with rightwards arrow on top close vertical bar space equals space square root of 10


space space space space space
Length of the perpendicular drawn on the line from P = square root of 10

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