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CBSE Class 12 Mathematics Solved Question Paper 2015

Short 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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Long 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.

straight x squared dy space equals space left parenthesis 2 xy plus straight y squared right parenthesis dx space space space rightwards double arrow dy over dx space equals space fraction numerator 2 xy plus straight y squared over denominator straight x squared end fraction space.... left parenthesis straight i right parenthesis space space space Let space straight y space equals space vx comma space space space space dy over dx equals straight v plus straight x dv over dx space space space Substituting space in space left parenthesis straight i right parenthesis comma space we space straight g space space space space space straight v plus straight x dv over dx space equals space fraction numerator 2 vx squared plus straight v squared straight x squared over denominator straight x squared end fraction space space rightwards double arrow space space straight v plus straight x dv over dx space equals space 2 straight v plus straight v squared space space space rightwards double arrow space straight x dv over dx equals space straight v squared plus straight v space space space rightwards double arrow space fraction numerator dv over denominator straight v squared plus straight v end fraction equals dx over straight x

Integrating both sides,

rightwards double arrow space integral fraction numerator dv over denominator straight v squared plus straight v end fraction equals integral dx over straight x rightwards double arrow integral fraction numerator straight v plus 1 minus straight v over denominator straight v left parenthesis straight v plus 1 right parenthesis end fraction. dv space equals space integral dx over straight x rightwards double arrow integral dv over straight v minus integral fraction numerator dv over denominator straight v plus 1 end fraction equals integral dx over straight x rightwards double arrow space logv minus log open vertical bar straight v plus 1 close vertical bar equals logx space plus space logC rightwards double arrow log open vertical bar fraction numerator straight v over denominator straight v plus 1 end fraction close vertical bar space equals space log open vertical bar Cx close vertical bar rightwards double arrow log open vertical bar fraction numerator begin display style bevelled straight y over straight x end style over denominator begin display style bevelled straight y over straight x end style plus 1 end fraction close vertical bar equals log open vertical bar Cx close vertical bar

rightwards double arrow space fraction numerator straight y over denominator straight y plus straight x end fraction equals Cx space left square bracket Removing space logarithm space in space both space sides right square bracket therefore space straight y equals Cxy plus Cx squared comma space which space is space the space general space solution. space Putting space straight y space equals space 1 space space and space straight x space equals space 1 comma 1 space equals space straight C space plus space straight C space space rightwards double arrow space space 2 straight C space equals space 1 space space space rightwards double arrow space straight C equals 1 half space space space straight y space equals space xy over 2 plus straight x squared over 2 space therefore 2 straight y space equals space xy plus straight y squared comma space which space is space the space particular space solution.

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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.