Subject

Mathematics

Class

CBSE Class 12

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Sample Papers

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 Multiple Choice QuestionsLong Answer Type

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31.

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.


stack 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 0

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 equals 0

straight r with rightwards arrow on top space 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 plus straight lambda open curly brackets 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 close curly brackets minus 4 plus 5 straight lambda space equals 0

rightwards double arrow space straight r with rightwards arrow on top left square bracket left parenthesis 1 minus 2 straight lambda right parenthesis straight i with hat on top plus left parenthesis negative 2 plus straight lambda right parenthesis straight j with hat on top space plus left parenthesis 3 plus straight lambda right parenthesis straight k with hat on top right square bracket minus 4 plus 5 straight lambda equals 0

left parenthesis 1 minus 2 straight lambda right parenthesis space straight x space plus left parenthesis negative 2 plus straight lambda right parenthesis straight y plus left parenthesis 3 plus straight lambda right parenthesis straight z space equals negative 5 straight lambda plus 4

rightwards double arrow space fraction numerator straight x over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator 1 minus 2 straight lambda end fraction end style end fraction plus fraction numerator straight y over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator negative 2 plus straight lambda end fraction end style end fraction plus fraction numerator straight z over denominator begin display style fraction numerator negative 5 straight lambda plus 4 over denominator 3 plus straight lambda end fraction end style end fraction equals 1

therefore space fraction numerator negative 5 straight lambda plus 4 over denominator 1 minus 2 straight lambda end fraction space equals fraction numerator negative 5 straight lambda plus 4 over denominator negative 2 plus straight lambda end fraction

rightwards double arrow space 1 minus 2 straight lambda space equals negative 2 plus straight lambda

rightwards double arrow space minus 3 straight lambda space equals space minus 3

rightwards double arrow space straight lambda space equals 1

therefore space equation space of space the space required space plane
minus straight x minus straight y plus 4 straight z equals 1
straight x plus straight y minus 4 straight z minus 1 space equals space 0

vector space equation space of space the space required space plane

equals straight r with rightwards arrow on top. left parenthesis straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis minus 1 space equals 0
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32.

A retired person wants to invest an amount of Rs. 50, 000. His broker recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximise his returns.

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