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CBSE Class 12 Mathematics Solved Question Paper 2016

Short Answer Type

11.
Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4)
are coplanar.

Given four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4).
therefore,

$AB with rightwards arrow on top space equals space left parenthesis negative 4 i with hat on top minus 6 j with hat on top minus 2 k with hat on top right parenthesis AC with rightwards arrow on top space equals space left parenthesis negative i with hat on top plus 4 j with hat on top plus 3 k with hat on top right parenthesis AD with rightwards arrow on top space equals space left parenthesis negative 8 i with hat on top plus j with hat on top plus 3 k with hat on top right parenthesis therefore space space vertical line AB with rightwards arrow on top space space AC with rightwards arrow on top space space AD with rightwards arrow on top vertical line space equals space open vertical bar table row cell negative 4 end cell cell negative 6 end cell cell negative 2 end cell row cell negative 1 end cell 4 3 row cell negative 8 end cell 1 3 end table close vertical bar equals space minus 4 left parenthesis 12 space plus space 3 right parenthesis space plus space 6 left parenthesis negative 3 space plus space 24 right parenthesis space minus space 2 left parenthesis 1 space plus space 32 right parenthesis equals space minus 60 space plus space 126 space minus space 66 equals 0 Hence comma space Four space points space straight A comma space straight B comma space straight C comma straight D space are space coplanar.$

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Long Answer Type

12.

Let A= R × R and * be a binary operation on A defined by

(a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for *

on A. Also find the inverse of every element (a, b) ∈ A.

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13.

Prove space that space straight y equals fraction numerator 4 space sin space straight theta over denominator 2 space plus cos space straight theta end fraction space minus straight theta space is space an space increasing space function space of space straight theta space on space open square brackets 0 comma straight pi over 2 close square brackets

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14.

Show that semi-vertical angle of a cone of maximum volume and given slant height is cos^-1 $fraction numerator 1 over denominator square root of 3 end fraction$ .

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15.

Using properties of determinants, prove that

$open vertical bar table row cell left parenthesis straight x plus straight y right parenthesis squared end cell zx zy row zx cell left parenthesis straight z plus straight y right parenthesis squared end cell xy row zy xy cell left parenthesis straight z plus straight x right parenthesis squared end cell end table close vertical bar space space equals space 2 xyz left parenthesis straight x plus straight y plus straight z right parenthesis cubed$

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16.

If space straight A space equals space open square brackets table row 1 0 2 row 0 2 1 row 2 0 3 end table close square brackets space and space straight A cubed minus 6 straight A squared plus 7 straight A space plus kI subscript 3 space equals space 0 space find space straight k.

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Short Answer Type

17.

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