Show that lines:Also, find the equation of the plane containing

Subject

Mathematics

Class

CBSE Class 12

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 Multiple Choice QuestionsShort Answer Type

21.

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.

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22. If space straight y space equals space straight e to the power of ax. cosbx comma space then space prove space that
space space space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y space equals space 0
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23. If space straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals space straight a to the power of straight b comma space space then space find space dy over dx.
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24.

If straight x equals straight a space sin space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis space and space straight y space equals straight b space cos space 2 straight t left parenthesis 1 minus cos space 2 straight t right parenthesis comma space then space find dy over dx space at space straight t space equals straight pi over 4.

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25.

Evaluate: integral fraction numerator left parenthesis straight x plus 3 right parenthesis straight e to the power of straight x over denominator left parenthesis straight x plus 5 right parenthesis cubed end fraction dx

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26.

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-held fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:
School/Article                     School X                   School Y                School Z
Hand-held fans                         30                         40                         35
      Mats                                  12                         15                         20
       Toys                                 70                         55                         75
Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation. 

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 Multiple Choice QuestionsLong Answer Type

27.

Sketch the region bounded by the curves straight y equals square root of 5 minus straight x squared end root space and space straight y space equals open vertical bar straight x minus 1 close vertical bar and find its area using integration. 

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28.

Find the particulars solution of the differential equation straight x squared dy space equals space left parenthesis 2 xy plus straight y squared right parenthesis dx comma space given that y =1 when x = 1.

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29.

Find the particulars solution of the differential equation left parenthesis 1 plus straight x squared right parenthesis dy over dx equals open parentheses straight e to the power of straight m space tan to the power of negative 1 straight x end exponent end exponent minus straight y close parentheses comma space given space that space straight y space equals space 1 space when space straight x space equals space 0.

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30.

Show that lines:
straight r with rightwards arrow on top space equals space straight i with hat on top space plus straight j with hat on top space plus straight k with hat on top space plus space straight lambda open parentheses straight i with hat on top minus straight j with hat on top space plus space straight k with hat on top close parentheses
straight r with rightwards arrow on top space equals space 4 straight j with hat on top space plus space 2 straight k with hat on top space plus space straight mu open parentheses 2 straight j with hat on top minus straight j with hat on top plus 3 straight k with hat on top close parentheses space are space coplanar.
Also, find the equation of the plane containing these lines. 


straight r with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda left parenthesis straight i with hat on top minus straight j with hat on top plus straight k with hat on top right parenthesis space... left parenthesis straight i right parenthesis
Converting into cartesian form,
fraction numerator straight x minus 1 over denominator 1 end fraction space equals fraction numerator straight y minus 1 over denominator negative 1 end fraction equals fraction numerator straight z minus 1 over denominator 1 end fraction
left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis space equals space left parenthesis 1 comma space 1 comma space 1 right parenthesis
straight a subscript 1 space equals space 1 comma space space straight b subscript 1 space equals negative 1 comma space space straight c subscript 1 space equals 1
straight r with rightwards arrow on top equals 4 straight j with hat on top plus 2 straight k with hat on top plus straight mu left parenthesis 2 straight i with hat on top minus straight j with hat on top plus 3 straight k with hat on top right parenthesis space space space... left parenthesis ii right parenthesis 
Converting into cartesian form,
fraction numerator straight x minus 0 over denominator 2 end fraction equals fraction numerator straight y minus 4 over denominator negative 1 end fraction equals fraction numerator straight z minus 2 over denominator 3 end fraction
left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis space equals space left parenthesis 0 comma space 4 comma space 2 right parenthesis
straight a subscript 2 equals space 2 comma space space straight b subscript 2 equals negative 1 comma space space straight c subscript 2 equals 3
Condition for the lines to be coplanar is
open vertical bar table row cell 0 minus 1 end cell cell space 4 minus 1 end cell cell space 2 minus 1 end cell row 1 cell negative 1 end cell 1 row 2 cell negative 1 end cell 3 end table close vertical bar space equals space open vertical bar table row cell negative 1 end cell cell space space 3 end cell cell space space 1 end cell row 1 cell negative 1 end cell cell space space 1 end cell row 2 cell negative 1 end cell cell space space 3 end cell end table close vertical bar equals 0
therefore space the space lines space are space coplanar.
Intersection space of space the space two space lines space is
Let space the space equation space be space straight a left parenthesis straight x minus straight x subscript 1 right parenthesis plus straight b left parenthesis straight y minus straight y subscript 1 right parenthesis plus straight c left parenthesis straight z minus straight z subscript 1 right parenthesis space equals space 0 space... left parenthesis iii right parenthesis
Direction space ratios space of space the space plane space is
straight a minus straight b plus straight c equals 0
2 straight a minus straight b plus 3 straight c space equals 0
Solving space by space cross minus multiplication comma
fraction numerator straight a over denominator negative 3 plus 1 end fraction equals fraction numerator straight b over denominator 2 minus 3 end fraction equals fraction numerator straight c over denominator negative 1 plus 2 end fraction
straight a equals negative 2 straight lambda comma space space straight b equals negative straight lambda comma space space straight c equals straight lambda
Since the plane passes through (0, 4, 2) from line (ii)
straight a left parenthesis straight x minus 0 right parenthesis plus straight b left parenthesis straight y minus 4 right parenthesis plus straight c left parenthesis straight z minus 2 right parenthesis space equals 0
rightwards double arrow space minus 2 λx minus straight lambda left parenthesis straight y minus 4 right parenthesis plus straight lambda left parenthesis straight z minus 2 right parenthesis space equals 0
rightwards double arrow space minus 2 straight x minus straight y plus 4 plus straight z minus 2 equals 0
rightwards double arrow negative 2 straight x minus straight y plus straight z equals negative 2
rightwards double arrow 2 straight x plus straight y minus straight z equals 2

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