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Which of the following options are correct?A. `[(.^(n)C_(0)+.^(n)C_(3)+.^(n)C_(6)+…)-1/2 (.^(n)C_(1)+.^(n)C_(2)+.^(n)C_(4)+.^(n)C_(5)+.)]^(2)+3/4(.^(n)C_(1)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(5)+…….)^(2)=1`B. If `a` and `b` are two positive numbers such that `a^(5)b^(2)=4` then the maximum value of `log_(2^(1//5))(a^(2)).log_(2^(1//2))(b^(2))` is equal to 4C. Constant term in `((((((x-2)^(2)-2)^(2)-2)^(2)-2)^(2)-2)^(2)………2)^(2)` is equal to 2D. The coeffocient of `x^(24)` in `((.^(25)C_(1))/(.^(25)C_(0))-x)(x-2^(2)(.^(25)C_(2))/(.^(25)C_(1)))(x-3^(2)(.^(5)C_(3))/(.^(2)C_(2)))(x-4^(2) (.^(25)C_(4))/(.^(25)C_(3)))………..(x-25^(2)(.^(25)C_(25))/(.^(25)C_(24)))` is equal to 2925 |
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Answer» Correct Answer - A::B::D (A) `(1+x)^(n)=.^(n)C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+.^(n)C_(3)x^(3)+……..+.^(n)C_(n)x^(n)` Put `x=omega` where `omega=-1/2+(isqrt(3))/2` `(.^(n)C_(0)+.^(n)C_(3)+.^(n)C_(6)+……)+….+(.^(n)C_(1)+.^(n)C_(4)+.^(n)C_(7)+……..)omega +(.^(n)C_(2)+.^(n)C_(5)+.^(n)C_(8)+.....)omega^(2)=(1+omega)^(n)` `(.^(n)C_(0)+.^(n)C_(3)+.^(n)C_(6)+.........)+(.^(n)C_(1)+.^(n)C_(4)+.^(n)C_(7)+......)(-1/2+(isqrt(3))/2)+(.^(n)C_(2)+.^(n)C_(5)+.^(n)C_(8)+.)(-1/2-(isqrt(3))/2)=(-omega^(2))^(n)` `=(.^(n)C_(0)+.^(n)C_(3)+.^(n)C_(6)+...)-1/2(.^(n)C_(1)+.^(n)C_(2)+.^(n)C_(4)+.^(n)C_(5)+...)+ (isqrt(3))/2(.^(n)C_(1)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(5)+...........)=(-omega^(2))^(n)` Take modulus of both sides (B) `a^(5)b^(2)=4implies5log2(a)+2log_(2)(b)=2` `y=10log_(2)a.4 log_(2)b=40log_(2)a.log_(2)b` A.M. `ge` G.Mgt `implies(5log_(2)(a)+2log_(2)(b))/2 ge sqrt(10 log_(2)a.log_(2)b)` `log_(2)a.log_(2)b le 1/10 impliesyle4, =4` when `5 log_(2)a=2 log_(2)bimpliesa^(5)=b^(2)=2` (C) For constant term put `x=0` and get constant term `=4` (D) Coefficient of `x^(24)=1.25+2.24+3.23+.+25.1=2925` |
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