Algebra - Study Mode

[#176] The value of $$frac{{48.3 imes left[ {{{left( {4.95}
ight)}^2} + 4.95 imes 13.25}
ight]}}{{left[ {{{left( {12.55}
ight)}^2} - {{left( {5.65}
ight)}^2}}
ight] imes 19.8}} = ?$$
Correct Answer

(C) 1.75

Explanation

Solution: $$eqalign{
& frac{{48.3 imes left[ {{{left( {4.95}
ight)}^2} + 4.95 imes 13.25}
ight]}}{{left[ {{{left( {12.55}
ight)}^2} - {{left( {5.65}
ight)}^2}}
ight] imes 19.8}} cr
& = frac{{48.3 imes 4.95left[ {4.95 + 13.25}
ight]}}{{left( {12.55 + 5.65}
ight)left( {12.55 - 5.65}
ight) imes 19.8}} cr
& = frac{{48.3 imes 4.95 imes 18.2}}{{18.2 imes 6.9 imes 19.8}} cr
& = frac{7}{4} cr
& = 1.75 cr} $$

[#177] If $${a^2} + frac{2}{{{a^2}}} = 16,$$ xa0 then find the value of $$frac{{72{a^2}}}{{{a^4} + 2 + 8{a^2}}}.$$
Correct Answer

(A) 3

Explanation

Solution: $$eqalign{
& {a^2} + frac{2}{{{a^2}}} = 16 cr
& frac{{72{a^2}}}{{{a^4} + 2 + 8{a^2}}} cr
& = frac{{72}}{{{a^2} + frac{2}{{{a^2}}} + 8}} cr
& = frac{{72}}{{16 + 8}} cr
& = 3 cr} $$

[#178] If a 2 + b 2 = 4b + 6a - 13, then what is the value of a + b?
Correct Answer

(C) 5

Explanation

Solution: a 2 + b 2 = 4b + 6a - 13 a 2 + 9 - 3a + b 2 + 4 - 4b = 0 (a - 3) 2 + (b - 2) 2 = 0 a = 3, b = 2 a + b = 5

[#179] If a + b + c = 0, then the value of $$frac{{{a^2}}}{{bc}} + frac{{{b^2}}}{{ca}} + frac{{{c^2}}}{{ab}}$$ xa0 is:
Correct Answer

(B) 3

Explanation

Solution: $$eqalign{
& { ext{If }}a + b + c = 0, cr
& { ext{then }} Rightarrow {a^3} + {b^3} + {c^3} = 3abc cr
& = frac{{{a^2}}}{{bc}} + frac{{{b^2}}}{{ca}} + frac{{{c^2}}}{{ab}} cr
& = frac{{{a^3} + {b^3} + {c^3}}}{{abc}} cr
& = frac{{3abc}}{{abc}} cr
& = 3 cr} $$

[#180] If 2x + 3y - 5z = 18, 3x + 2y + z = 29 and x + y + 3z = 17, then what is the value of xy + yz + zx ?
Correct Answer

(B) 52

Explanation

Solution: $$eqalign{
& 2x + 3y - 5z = 18,......left( 1
ight) cr
& 3x + 2y + z = 29,......left( 2
ight) cr
& x + y + 3z = 17,......left( 3
ight) cr
& { ext{Equation}}left( 2
ight){ ext{ and Equation}}left( 1
ight),{ ext{we get}} cr
& 3x + 2y + z = 29 cr
& underline {2x + 3y - 5z = 18} o left( {{ ext{Subtracting}}}
ight) cr
& x - y + 6z = 11,......left( 4
ight) cr
& { ext{Adding Equation}}left( 4
ight){ ext{and Equation}}left( 3
ight), cr
& 2x + 9z = 28,......left( 5
ight) cr
& { ext{And Equation}}left( 2
ight){ ext{and}},2 imes { ext{Equation}}left( 3
ight), cr
& 3x + 2y + z = 29 cr
& underline {2x + 2y + 6z = 34} o left( {{ ext{Subtracting}}}
ight) cr
& x - 5z = - 5,......left( 6
ight) cr
& { ext{Equation}}left( 5
ight){ ext{and}},2 imes { ext{Equation}}left( 6
ight), cr
& 2x + 9z = 28 cr
& underline {2x - 10z = - 10} o left( {{ ext{Subtracting}}}
ight) cr
& 19z = 38 cr
& herefore z = 2 cr
& { ext{Now, from equation}}left( { ext{6}}
ight), cr
& x - 5 imes 2 = - 5 cr
& x = 10 - 5 cr
& x = 5 cr
& { ext{And from equation}}left( { ext{3}}
ight), cr
& 5 + y + 3 imes 2 = 17 cr
& y = 17 - 11 cr
& y = 6 cr
& herefore xy + yz + zx cr
& = 5 imes 6 + 6 imes 2 + 5 imes 2 cr
& = 30 + 12 + 10 cr
& = 52 cr} $$