Algebra - Study Mode
[#346] If 2s = a + b + c, then the value of s(s - c) + (s - a) (s - b) is?
Correct Answer
(A) ab
Explanation
Solution: $$eqalign{
& { ext{According to the question,}} cr
& { ext{If ,}}2s = a + b + c cr
& Leftrightarrow s = frac{{a + b + c}}{2} cr
& { ext{Let, }} cr
& a = 10 cr
& b = 10 cr
& c = 10 cr
& x08ecause s = frac{{a + b + c}}{2} cr
& Rightarrow s = frac{{10 + 10 + 10}}{2} cr
& Rightarrow s = frac{{30}}{2} cr
& Rightarrow s = 15 cr
& herefore sleft( {s - c}
ight) + left( {s - a}
ight)left( {s - b}
ight) cr
& = 15left( {15 - 10}
ight) + left( {15 - 10}
ight)left( {15 - 10}
ight) cr
& = 75 + 25 cr
& = 100 cr
& { ext{Now check from option, }} cr
& { ext{Option 'A' }}ab = 10 imes 10 cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 100left( {{ ext{Satisfied}}}
ight) cr} $$
[#347] When x m is multiplied by x n , product is 1. The relation between m and n is?
Correct Answer
(D) m = -n
Explanation
Solution: $$eqalign{
& {x^m} imes {x^n} = 1 cr
& {x^{m + n}} = {x^0}left[ {{x^0} = 1}
ight] cr
& m + n = 0 cr
& m = - n cr} $$
[#348] If $${64^{x + 1}} = frac{{64}}{{{4^x}}}{ ext{,}}$$ xa0 then the value of x is?
Correct Answer
(B) 0
Explanation
Solution: $$eqalign{
& {64^{x + 1}} = frac{{64}}{{{4^x}}} cr
& Rightarrow {left( {{4^3}}
ight)^{x + 1}} - frac{{{4^3}}}{{{4^x}}} cr
& Rightarrow {4^{3x + 3}} = {4^{3 - x}} cr
& Rightarrow 3x + 3 = 3 - x cr
& Rightarrow 4x = 0 cr
& Rightarrow x = 0 cr} $$
[#349] If ax 2 + bx + c = a(x - p) 2 , then the relation among a, b, c would be?
Correct Answer
(C) b 2 = 4ac
Explanation
Solution: $$eqalign{
& a{x^2} + bx + c = a{left( {x - p}
ight)^2} cr
& Rightarrow a{x^2} + bx + c = a{left( {{x^2} + {p^2} - 2px}
ight)^2} cr
& Rightarrow a{x^2} + bx + c = a{x^2} + a{p^2} - 2apx cr
& { ext{Comparing confficients of }}{x^2}{ ext{and }}x cr
& Rightarrow b = - 2ap cr
& Rightarrow p = - frac{b}{{2a}},.......(1) cr
& and{ ext{ }}c = a{p^2} cr
& Rightarrow c = a imes frac{{{b^2}}}{{4{a^2}}}left[ {{ ext{From (i)}}}
ight] cr
& Rightarrow 4ac = {b^2} cr} $$
[#350] If a 2 + b 2 + c 2 + 3 = 2(a + b + c) then the value of (a + b + c) is?
Correct Answer
(B) 3
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} + {c^2} + 3 = 2left( {a + b + c}
ight) cr
& Rightarrow {a^2} + {b^2} + {c^2} + 3 = 2a + 2b + 2c cr
& Rightarrow {a^2} - 2a + 1 + {b^2} - 2b + 1 + {c^2} - 2c + 1 = 0 cr
& Rightarrow {left( {a - 1}
ight)^2} + {left( {b - 1}
ight)^2} + {left( {c - 1}
ight)^2} = 0 cr
& a = 1 cr
& b = 1 cr
& c = 1 cr
& herefore left( {a + b + c}
ight) cr
& = 1 + 1 + 1 cr
& = 3 cr} $$