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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.

open square brackets table row 30 12 70 row 40 15 55 row 35 20 75 end table close square brackets space open square brackets table row 25 row 100 row 50 end table close square brackets space space space space equals space open square brackets table row cell 30 cross times 25 plus 12 cross times 100 plus 70 cross times 50 end cell row cell 40 cross times 25 plus 15 cross times 100 plus 55 cross times 50 end cell row cell 35 cross times 25 plus 20 cross times 100 plus 75 cross times 50 end cell end table close square brackets space space space space equals open square brackets table row 5450 row 5250 row 6625 end table close square brackets space equals space open square brackets table row straight X row straight Y row straight Z end table close square brackets

The funds collected by X = Rs.5450, Y= Rs. 5250, Z = Rs. 6625
Total funds collected = Rs. 17325
Value generated: team work.

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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.

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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.