Algebra - Study Mode

[#376] If a + b + c = 0, then the value of $$left( {frac{{a + b}}{c} + frac{{b + c}}{a} + frac{{c + a}}{b}}
ight)$$ xa0xa0 $$left( {frac{a}{{b + c}} + frac{b}{{c + a}} + frac{c}{{a + b}}}
ight) = ,?$$
Correct Answer

(C) 9

Explanation

Solution: a + b + c = 0 Have values a = 1 b = 2 c = - 3 $$ Rightarrow left( {frac{{a + b}}{c} + frac{{b + c}}{a} + frac{{c + a}}{b}}
ight)$$ xa0 xa0 $$left( {frac{a}{{b + c}} + frac{b}{{c + a}} + frac{c}{{a + b}}}
ight)$$ $$ Rightarrow left( {frac{{1 + 2}}{{ - 3}} + frac{{2 - 3}}{1} + frac{{ - 3 + 1}}{2}}
ight)$$ xa0 xa0 $$left( {frac{1}{{2 - 3}} + frac{2}{{ - 3 + 1}} + frac{{ - 3}}{{1 + 2}}}
ight)$$ $$eqalign{
& Rightarrow left( { - 1 - 1 - 1}
ight)left( { - 1 - 1 - 1}
ight) cr
& Rightarrow - 3 imes - 3 cr
& Rightarrow 9 cr} $$

[#377] If a, b, c are non - zero $$a + frac{1}{b} = 1$$ xa0 and $$b + frac{1}{c} = 1,$$ xa0 then the value of abc is?
Correct Answer

(A) -1

Explanation

Solution: $$eqalign{
& a + frac{1}{b} = 1,{ ext{ }}b + frac{1}{c} = 1 cr
& { ext{Values of }}a,{ ext{ }}b,{ ext{ }}c{ ext{ assume}} cr
& a = frac{1}{2} cr
& b = 2 cr
& c = - 1 cr
& herefore abc cr
& = frac{1}{2} imes 2 imes - 1 cr
& = - 1 cr} $$

[#378] If 3x + 5y + 7z = 49 and 9x + 8y + 21z = 126, then what is the value of y?
Correct Answer

(C) 3

Explanation

Solution: By eliminating variable 'z' as there are three unknowns & only 2 equations. By putting z = 0 3x + 5y = 49 . . . . . . (i) 9x + 8y = 126 . . . . . . (ii) Multiplying by 3 in equation (i) and subtracting equation (ii) 9x + 15y = 147 9x + 8y = 126 $$overline {,,,,,,,,,,,,7{ ext{y}} = 21,,} $$ y = 3 Alternate solution: 3x + 5y + 7z = 49 9x + 8y + 21z = 126 Assume value x, y, z x = 2, y = 3, z = 4 3(2) + 5(3) + 7(4) = 49 6 + 15 + 28 = 49 49 = 49 value satisfied 9(2) + 8(3) + 21(4) = 126 18 + 24 + 84 = 126 126 = 126 value satisfied y = 3

[#379] If x 2 + 8y 2 + 12y - 4xy + 9 = 0, then the value of (7x + 8y) is:
Correct Answer

(A) -33

Explanation

Solution: x 2 + 8y 2 + 12y - 4xy + 9 = 0 x 2 + 4y 2 - 4xy + 4y 2 + 12y + 9 = 0 (x - 2y) 2 + (2y + 3) 2 = 0 x = 2y 2y = -3 y = $$frac{{ - 3}}{2}$$ x = -3 (7x + 8y) = 7 × (-3) + 8 × $$left( {frac{{ - 3}}{2}}
ight)$$ = -21 - 12 = -33

[#380] If (x + y) 3 + 8(x - y) 3 = (3x + Ay)(3x 2 + Bxy + Cy 2 ), then the value of A + B + C is:
Correct Answer

(A) 0

Explanation

Solution: (x + y) 3 + 8(x - y) 3 = (3x + Ay)(3x 2 + Bxy + Cy 2 ) a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) a = (x + y) b = 2(x - y) (x + y) 3 + 8(x - y) 3 = (3x - y)(x 2 + y 2 + 2xy - 2x 2 + 2y 2 + 4x 2 + 4y 2 - 8xy) (x + y) 3 + 8(x - y) 3 = (3x - y)(3x 2 - 6xy + 7y 2 ) (3x + Ay)(3x 2 + Bxy + Cy 2 ) = (3x - y)(3x 2 - 6xy + 7y 2 ) Compare A = -1 B = -6 C = 7 A + B + C = -1 - 6 + 7 = 0