Class 11

Math

Algebra

Sequences and Series

Assertion :STATEMENT -1 : If $x,y,z$ are the sides of a triangle such that $x+y+z=1$, then $[32x−1+2y−1+2z−1 ]≥((2x−1)(2y−1)(2z−1))_{1/3}$

Reason :STATEMENT-2 : For positive numbers their A.M., G.M. and H.M. satisfy the relation $A.M.>G.M.>H.M.$

- Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
- Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
- Statement -1 is True, Statement -2 is False
- Statement -1 is False, Statement -2 is True

Similarly, $1>2x$ and $1>2y$

$∴2x−1<0,2y−1<0,2z−1<0$

We know that for positive numbers A.M $≥$G.M, then

$3(1−2x)+(1−2y)+(1−2z) ≥{(1−2x)(1−2y)(1−2z) }_{1/3}$

Now, multiplying both sides by $−1$, we get

$3(1−2x)+(1−2y)+(1−2z) ≤{(2x−1)(2y−1)(2z−1) }_{1/3}$

$∴$ statement-1 is false $∴$ statement-2 is true

Hence, option 'D' is correct.