Which of the following functions has (have) y-symmetry or origin symmetry? (i) ` f(x) = x^(2) sin x" "(ii) f(x) = log ((1-x)/(1+x))` (iii)` f(x) = x/(

Let `K = x^(1/log y+1/log z)xxy^(1/log z+1/log x)xxz^(1/log x+1/log y)` ` :. log K = log x[1/log y+1/log z]+log y [1/log z+1/log x]+log z [1/log x+1/log y]` Putting ` log...

(i)`f_(1)(x) = 2 log_(10)x" is defined for "x gt 0` ` f_(2)(x) = log_(10)x^(2)" is defined for "x^(2) gt 0 or x in R - {0}` Therefore ` f_(1)(x) and...

`log_(3)4log_(4)5log_(5)6log_(6)7log_(7)8log_(8)9` `=(log4)/(log3)xx(log)5/(log4)xx(log6)/(log5)xx(log7)/(log6)xx(log8)/(log7)xx(log9)/(log8)` `=(log9)/(log3)=log_(3)9=2`

Correct Answer - A::B::C (1) ` log_(10)(10/2) * log_(10)(10xx2)+(log_(10)2)^(2)` ` = (1-log_(10)2)(1+log_(10)2)+(log_(10)2)^(2)=1` (2) `(log2^(2)xx3)/(log(48//4)) = 1` (3) ` -log_(5)log_(3)9^(1//10)=- log_(5)log_(3)3^(1//5)` ` = - log_(5)(1//5)=1` (4) ` 1/4 log_(sqrt3/2)(64/27) = 1/6 log_(sqrt3/2)(sqrt3/2)^(-6)=-...

Correct Answer - 4 `log_(6) 54+log_(x)16=log_(sqrt2)x-log_(36). 4/9` ` or 1+2 log_(6) 3+log_(x) 16 = 2 log_(2) x- log_(6). 2/3` ` or 1+2 log_(6) 3+log_(x) 16=2 log_(2)x-log_(6) 2+log_(6) 3` `or 1+ log_(x)...

Correct Answer - 2 ` N = (log_(5)250)/(log_(50)5)-(log_(5)10)/(log_(1250)5)` ` = (log_(5)250)(log_(5)50)-(log_(5)10)(log_(5)1250)` ` = (3+log_(5)2)(2+log_(5)2)-(1+log_(5)2)xx(4+log_(5)2)` ` = (log_(5)2)^(2)+5 log_(5)2+6-[(log_(5)2)^(2)+5 log_(5)2 + 4]` = 2

(i) ` f(x) = (-x)^(2) sin (-x) =- x^(2) sin x =- f(x)`, hence function has origin symmetry (ii) `f(-x) = log (-x+sqrt(1+(-x)^(2)))` `= log(((-x+sqrt(1+x^(2)))(x+sqrt(1+x^(2))))/((x+sqrt(1+x^(2)))))` `= log(1/(x+sqrt(1+x^(2))))` ` = -...

(i) Since `1 in [-pi//2, pi//2], sin^(-1) (sin 1) = 1` (ii) Since `2 in [-pi//2, pi//2], sin^(-1) (sin 2) != 2` `:. Sin^(-1) (sin 2) = sin^(-1) (sin (pi)...

Correct Answer - B Let `x = sin theta and sqrtx = sin phi, " where " x in [0, 1]` `rArr theta, phi in [0, pi//2]` `rArr theta - phi...

Correct Answer - C `I=-e^(-x)log(e^(x)+1)+int(e^(-x)e^(x))/(e^(x)+1)dx ` `=-e^(-x)log(e^(x)+1)+int(e^(-x))/(e^(-x)+1)dx` `=-e^(-x)log(e^(x)+1)-log(e^(-x)+1)+C` `=-e^(-x)log(e^(x)+1)-log(1+e^(x))+x+C` `=-(e^(-x)+1)log(e^(x)+1)+x+C`