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Determinants

Short Answer Type

141.

Find values of k if area of triangle is 4 sq. units and vertices are (k, 0), (4, 0), (0, 2).

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142.

Find values of k if area of triangle is 4 sq. units and vertices are (– 2, 0), (0, 4), (0, k)

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143. Find the equation of the line joining A(l, 3) and B(0, 0) using determinants and find k if D(A, 0) is a point such that area of triangle ABD is 3 sq. units.

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144.

Find equation of line joining (1, 2) and (3, 6) using determinants.

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145.

Find equation of line joining (3, 1) and (9, 3) using determinants.

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Long Answer Type

146. A triangle has its three sides equal to a, b and c. If the coordinates of its vertices are A(x₁ y₁), B(x₂, y₂) and C(x₃ y₃), show that

space open vertical bar table row cell straight x subscript 1 end cell cell space space straight y subscript 1 end cell cell space 2 end cell row cell straight x subscript 2 end cell cell space straight y subscript 2 end cell cell space 2 end cell row cell straight x subscript 3 end cell cell space straight y subscript 3 end cell cell space 2 end cell end table close vertical bar squared space equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b minus straight c right parenthesis.

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Multiple Choice Questions

147. If area of triangle is 35 sq. units with vertices (2, – 6), (5, 4) and (k, 4). Then k is

12
-2
-12, -2
-12, -2

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Short Answer Type

148.

Find the minor of element 6 in the determinant $increment space equals space open vertical bar table row 1 cell space 2 end cell cell space 3 end cell row 4 cell space 5 end cell cell space 6 end cell row 7 cell space 8 end cell cell space 9 end cell end table close vertical bar$

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149.
Find the minors and co-factors of the elements a₁₁, a₂₁ in the determinant.

increment space equals space open vertical bar table row cell straight a subscript 11 end cell cell space straight a subscript 12 end cell cell space straight a subscript 13 end cell row cell straight a subscript 21 end cell cell space straight a subscript 22 end cell cell straight a subscript 23 end cell row cell straight a subscript 31 end cell cell space straight a subscript 32 end cell cell straight a subscript 33 end cell end table close vertical bar

increment space equals space open vertical bar table row cell straight a subscript 11 end cell cell space straight a subscript 12 end cell cell space straight a subscript 13 end cell row cell straight a subscript 21 end cell cell space straight a subscript 22 end cell cell space straight a subscript 23 end cell row cell straight a subscript 31 end cell cell space straight a subscript 32 end cell cell space straight a subscript 33 end cell end table close vertical bar

Minor of

straight a subscript 11 space equals space straight M subscript 11 space equals space open vertical bar table row cell straight a subscript 22 end cell cell space space space space straight a subscript 23 end cell row cell straight a subscript 32 end cell cell space space space straight a subscript 33 end cell end table close vertical bar space equals space straight a subscript 22 straight a subscript 33 minus straight a subscript 23 straight a subscript 32

Co-factor of

straight a subscript 11 space equals space straight A subscript 11 space equals space left parenthesis negative 1 right parenthesis to the power of 1 plus 1 end exponent straight M subscript 11 space equals space straight M subscript 11 space equals space straight a subscript 22 straight a subscript 33 minus straight a subscript 23 straight a subscript 32

Minor of

straight a subscript 21

equals straight M subscript 21 space equals space open vertical bar table row cell straight a subscript 12 end cell cell space space straight a subscript 13 end cell row cell straight a subscript 13 end cell cell space space straight a subscript 33 end cell end table close vertical bar space equals space straight a subscript 12 straight a subscript 33 space minus space straight a subscript 13 straight a subscript 32

Co-factor of

straight a subscript 21 space equals space straight A subscript 21 space equals space left parenthesis negative 1 right parenthesis to the power of 2 plus 1 end exponent space space straight M subscript 21 space equals space minus straight M subscript 21 space equals space minus straight a subscript 12 straight a subscript 33 plus straight a subscript 13 straight a subscript 32

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150.

Find minors and cofactors of all the elements of the determinant:
$open vertical bar table row 1 cell space space minus 2 end cell row 4 cell space space space 3 end cell end table close vertical bar$ .