Simplification - Study Mode
[#141] If $$frac{p}{a} + frac{q}{b} + frac{r}{c} = 1$$ xa0 xa0 and $$frac{a}{p} + frac{b}{q} + frac{c}{r} = 0$$ xa0 xa0 where a, b, c, p, q, r are non-zero real numbers, then $$frac{{{p^2}}}{{{a^2}}} + frac{{{q^2}}}{{{b^2}}} + frac{{{r^2}}}{{{c^2}}}$$ xa0xa0 is equal to = ?
Correct Answer
(B) 1
Explanation
Solution: $$eqalign{
& frac{a}{p} + frac{b}{q} + frac{c}{r} = 0,,,, cr
& Rightarrow aqr + bpr + cpq = 0....({ ext{i}}) cr
& frac{p}{a} + frac{q}{b} + frac{r}{c} = 1,, cr
& Rightarrow {left( {frac{p}{a} + frac{q}{b} + frac{r}{c}}
ight)^2} = 1 cr
& Rightarrow { ext{ }}frac{{{p^2}}}{{{a^2}}} + frac{{{q^2}}}{{{b^2}}} + frac{{{r^2}}}{{{c^2}}}, + 2left( {frac{{pq}}{{ab}} + frac{{pr}}{{ac}} + frac{{qr}}{{bc}}}
ight) = 1 cr
& Rightarrow frac{{{p^2}}}{{{a^2}}} + frac{{{q^2}}}{{{b^2}}} + frac{{{r^2}}}{{{c^2}}} + frac{{2left( {pqc + prb + qra}
ight)}}{{abc}} = 1 cr
& Rightarrow frac{{{p^2}}}{{{a^2}}} + frac{{{q^2}}}{{{b^2}}} + frac{{{r^2}}}{{{c^2}}} = 1....left[ {{ ext{using (i)}}}
ight] cr} $$
[#142] If $$x = sqrt 3 { ext{ + }}sqrt 2 { ext{,}}$$ xa0xa0then the value of $${x^3} - frac{1}{{{x^3}}}$$ xa0 is?
Correct Answer
(C) $$22sqrt 2 $$
Explanation
Solution: $$eqalign{
& x = sqrt 3 + sqrt 2 cr
& frac{1}{x} = frac{1}{{sqrt 3 + sqrt 2 }} imes frac{{sqrt 3 - sqrt 2 }}{{sqrt 3 - sqrt 2 }} cr
& frac{1}{x} = sqrt 3 - sqrt 2 cr
& {x^3} - frac{1}{{{x^3}}} cr
& = {left[ {x - frac{1}{x}}
ight]^3} + 3 imes x imes frac{1}{x}left( {x - frac{1}{x}}
ight) cr} $$ $$ = {left( {sqrt 3 + sqrt 2 - sqrt 3 + sqrt 2 }
ight)^3} + $$ xa0xa0 xa0 $$3left( {sqrt 3 + sqrt 2 - sqrt 3 + sqrt 2 }
ight)$$ $$eqalign{
& = {left( {2sqrt 2 }
ight)^3} + 3left( {2sqrt 2 }
ight) cr
& = 16sqrt 2 + 6sqrt 2 cr
& = 22sqrt 2 cr} $$
[#143] The value (1001) 3 is = ?
Correct Answer
(A) 1003003001
Explanation
Solution: (1001) 3 =1001 × 1001 × 1001 =1002001 × 1001 =1003003001
[#144] The Value of ($$sqrt {6} $$ + $$sqrt {10} $$ - $$sqrt {21} $$ - $$sqrt {35} $$) × ($$sqrt {6} $$ - $$sqrt {10} $$ + $$sqrt {21}$$ - $$sqrt {35} $$) = ?
Correct Answer
(D) 10
Explanation
Solution: ($$sqrt {6} $$xa0 + $$sqrt {10} $$xa0 - $$sqrt {21} $$xa0 - $$sqrt {35} $$) × ($$sqrt {6} $$xa0 - $$sqrt {10} $$xa0 + $$sqrt {21} $$xa0 - $$sqrt {35} $$) = {($$sqrt {6} $$xa0 - $$sqrt {35} $$) + ($$sqrt {10} $$xa0 - $$sqrt {21} $$)} × {($$sqrt {6} $$xa0 - $$sqrt {35} $$) - ($$sqrt {10} $$xa0 - $$sqrt {21} $$)} = ($$sqrt {6} $$xa0 - $$sqrt {35} $$) 2 - ($$sqrt {10} $$xa0 - $$sqrt {21} $$) 2 = 6 + 35 - 2$$sqrt {210} $$xa0 - 10 - 21 + 2$$sqrt {210} $$ = 41 - 31 = 10
[#145] The expression $$frac{1}{{x - 1}} - $$ xa0$$frac{1}{{x + 1}} - $$xa0 $$frac{2}{{{x^2} + 1}} - $$ xa0$$frac{4}{{{x^4} + 1}}$$ xa0is equal to = ?
Correct Answer
(B) $$frac{8}{{{x^8} - 1}}$$
Explanation
Solution: $$eqalign{
& { ext{Given expression,}} cr
& left( {frac{1}{{x - 1}} - frac{1}{{x + 1}}}
ight) - frac{2}{{{x^2} + 1}} - frac{4}{{{x^4} + 1}} cr
& = left[ {frac{{left( {x + 1}
ight) - left( {x - 1}
ight)}}{{left( {x - 1}
ight)left( {x + 1}
ight)}}}
ight] - frac{2}{{{x^2} + 1}} - frac{4}{{{x^4} + 1}} cr
& = left( {frac{2}{{{x^2} - 1}} - frac{2}{{{x^2} + 1}}}
ight) - frac{4}{{{x^4} + 1}} cr
& = left[ {frac{{2left( {{x^2} + 1}
ight) - 2left( {{x^2} - 1}
ight)}}{{left( {{x^2} - 1}
ight)left( {{x^2} + 1}
ight)}}}
ight] - frac{4}{{{x^4} + 1}} cr
& = frac{4}{{{x^4} - 1}} - frac{4}{{{x^4} + 1}} cr
& = frac{{4left( {{x^4} + 1}
ight) - 4left( {{x^4} - 1}
ight)}}{{left( {{x^4} - 1}
ight)left( {{x^4} + 1}
ight)}} cr
& = frac{8}{{{x^8} - 1}} cr} $$