Algebra - Study Mode
[#81] If a - b = 4 and a 3 - b 3 = 88, then find the value of a 2 - b 2 .
Correct Answer
(D) 8√6
Explanation
Solution: a - b = 4, a 3 - b 3 = 88 then a 2 - b 2 = ? (a - b) 3 = a 3 - b 3 - 3ab(a - b) 64 = 88 - 12(ab) ab = 2 (a + b) 2 - (a - b) 2 = 4ab (a + b) 2 - 16 = 4 × 2 (a + b) 2 = 24 (a + b) = 2√6 a 2 - b 2 = (a + b) × (a - b) = 2√6 × 4 = 8√6
[#82] If x 3 + y 3 + z 3 = 3(1 + xyz), P = y + z - x, Q = z + x - y and R = x + y - z, then what is the value of P 3 + Q 3 + R 3 - 3PQR?
Correct Answer
(C) 12
Explanation
Solution: x 3 + y 3 + z 3 = 3(1 + xyz) P = y + z - x, Q = z + x - y, R = x + y - z Put y and z = 0 x 3 = 3, P = -x, Q = x, R = x P 3 + Q 3 + R 3 - 3PQR = (-x) 3 + (x) 3 + (x) 3 - 3(-x)(x)(x) = x 3 + 3x 3 = 4x 3 = 4 × 3 = 12
[#83] If ab(a + b) = 1, then what is the value of $$frac{1}{{{a^3}{b^3}}} - {a^3} - {b^3}?$$
Correct Answer
(C) 3
Explanation
Solution: $$eqalign{
& x08ecause ,ableft( {a + b}
ight) = 1 cr
& Rightarrow a + b = frac{1}{{ab}},.......,left( { ext{i}}
ight) cr
& { ext{On cubing both sides}} cr
& Rightarrow {a^3} + {b^3} + 3ableft( {a + b}
ight) = frac{1}{{{a^3}{b^3}}} cr
& Rightarrow {a^3} + {b^3} + 3ab imes frac{1}{{ab}} = frac{1}{{{a^3}{b^3}}}left( {{ ext{from equation }}left( { ext{i}}
ight)}
ight) cr
& Rightarrow frac{1}{{{a^3}{b^3}}} - {a^3} - {b^3} = 3 cr} $$
[#84] ab(a - b) + bc(b - c) + ca(c - a) is equal to:
Correct Answer
(C) (b - a)(b - c)(c - a)
Explanation
Solution: ab(a - b) + bc(b - c) + ca(c - a) Let c = 0 ab(a - b) Now from option C (b - a)(b - c)(c - a) = -(a-b)b(-a) = ab(a - b)
[#85] If $$frac{{a + b}}{c} = frac{6}{5}$$ xa0 and $$frac{{b + c}}{a} = frac{9}{2},$$ xa0 then what is the value of $$frac{{a + c}}{b}?$$
Correct Answer
(D) $$frac{7}{4}$$
Explanation
Solution: $$eqalign{
& frac{{left( {a + b}
ight)}}{c} = frac{6}{5} cr
& frac{{left( {b + c}
ight)}}{a} = frac{9}{2} cr
& frac{{left( {a + c}
ight)}}{b} = ? cr
& c = 5,,frac{{left( {b + c}
ight)}}{a} = frac{9}{2} cr
& Rightarrow frac{{b + 5}}{2} = frac{9}{2} cr
& b = 4,,a = 2 cr
& Rightarrow frac{{left( {a + c}
ight)}}{b} = frac{{2 + 5}}{4} cr
& Rightarrow frac{{left( {a + c}
ight)}}{b} = frac{7}{4} cr} $$