Algebra - Study Mode
[#306] If a 2 = b + c, b 2 = a + c, c 2 = b + a, then what will be the value of $$frac{1}{{a + 1}}$$ xa0+ $$frac{1}{{b + 1}}$$ xa0+ $$frac{1}{{c + 1}}$$ ?
Correct Answer
(C) 1
Explanation
Solution: $$eqalign{
& {a^2} = b + c,{ ext{ }}{b^2} = a + c,{ ext{ }}{c^2} = b + a cr
& { ext{Taking }}a = 2,{ ext{ }}b = 2{ ext{ and }}c = 2 cr
& { ext{So,}}{left( 2
ight)^2} = 2 + 2 cr
& x08oxed{4 = 4} cr
& { ext{Now,}}frac{1}{{a + 1}} + { ext{ }}frac{1}{{b + 1}} + { ext{ }}frac{1}{{c + 1}} cr
& { ext{Put }}a = 2,{ ext{ }}b = 2{ ext{ and }}c = 2 cr
& = frac{1}{{2 + 1}} + { ext{ }}frac{1}{{2 + 1}} + { ext{ }}frac{1}{{2 + 1}} cr
& = frac{1}{3} + frac{1}{3} + frac{1}{3} cr
& = 1 cr} $$
[#307] If $$a + b = 2c,$$ xa0 find $$frac{a}{{a - c}}$$ xa0+ $$frac{c}{{b - c}}$$ xa0 = ?
Correct Answer
(B) 1
Explanation
Solution: $$eqalign{
& a + b = 2c cr
& { ext{Taking }}a = 2,{ ext{ }}b = 4{ ext{ and }}c = 3 cr
& { ext{So,}}2 + 4 = 2 imes 3 cr
& x08oxed{6 = 6} cr
& { ext{Now,}}frac{a}{{a - c}} + { ext{ }}frac{c}{{b - c}} cr
& = frac{2}{{2 - 3}} + { ext{ }}frac{3}{{4 - 3}} cr
& = frac{2}{{ - 1}} + frac{3}{1} cr
& = 1 cr} $$
[#308] If $$frac{a}{b} = frac{1}{2},$$ xa0 find the value of the expression $$frac{{left( {2a - 5b}
ight)}}{{left( {5a + 3b}
ight)}}$$ xa0 = ?
Correct Answer
(C) $$frac{{ - 8}}{{11}}$$
Explanation
Solution: $$eqalign{
& frac{a}{b} = frac{1}{2} cr
& { ext{Let }}a = x,{ ext{ }}b = 2x cr
& { ext{Then,}}frac{{left( {2a - 5b}
ight)}}{{left( {5a + 3b}
ight)}} cr
& = frac{{2x - 10x}}{{5x + 6x}} cr
& { ext{ = }}frac{{ - 8x}}{{11x}} cr
& = frac{{ - 8}}{{11}} cr} $$
[#309] If for a non - zero x, 3x 2 + 5x + 3 = 0, then the value of $${x^3} + frac{1}{{{x^3}}}$$ xa0 is?
Correct Answer
(A) $$frac{{10}}{{27}}$$
Explanation
Solution: $$eqalign{
& 3{x^2} + 5x + 3 = 0 cr
& 3{x^2} + 3 = - 5x cr
& { ext{Divide by }}3x{ ext{ both sides}} cr
& x + frac{1}{x} = frac{{ - 5}}{3} cr
& { ext{Then, }}{x^3} + frac{1}{{{x^3}}} cr
& = {left( {frac{{ - 5}}{3}}
ight)^3} - 3 imes left( {frac{{ - 5}}{3}}
ight) cr
& = frac{{ - 125}}{{27}} + 5 cr
& = frac{{10}}{{27}} cr} $$
[#310] The factors of x 4 + x 2 + 25 are:
Correct Answer
(B) (x 2 + 3x + 5)(x 2 - 3x + 5)
Explanation
Solution: x 4 + x 2 + 25 Check by the option, (x 2 - 3x + 5)(x 2 - 3x + 5) = x 4 - 3x 3 + 5x 2 + 3x 3 - 9x 2 + 15x + 5x 2 - 15x + 25 = x 4 + x 2 + 25 Other Method Let the value of x = 1 then, x 4 + x 2 + 25 = 27 Now check options, and there is only one option that is equal to 27. (1 + 3 + 5)(1 - 3 + 5) = 9 × 3 = 27