hope u can identify my hand writing
Here's is answer key for whole maths Leaked board paper 2018 =>
LET the four consecutive terms of AP be (a-3d), ( a-d) , (a+d) , ( a+3d).
a-3d + a-d + a + d + a + 3d = 32
4a = 32
a = 32/4 = 8 .
(a-3d) × (a+3d) / (a-d) × (a+d) = 7/15
a² - 3d² / a²- d². [ (a+b) × (a-b) = a2 - b²]
PUTTING THE VALUE OF A IN,
a²-3d² / a² - d² = 64-3d²/64 - d² = 7/15
SOLVE THIS U WILL GET THE VALUE OF D...
AND PUT THE VALUE OF A AND D IN ALL FOUR CONSECUTIVE TERMS OF THE AP...THEN U WILL GET ALL THE NUMBERS
HOPE U WILL UNDERSTAND THIS
assume w.lo.g x> y. then the statement becomes xp−yp≤(x−y)p. since y> 0, divide through out by yp. so we need to show (xy)p−1≤(xy−1)p whenever x> y> 0 and 0< p< 1. let t=xy. so we need to show tp−1≤(t−1)p whenever t> 1 and 0< p< 1.
this is a calculus problem.
let f(t)=(t−1)p−(tp−1) where 0< p< 1.
show that the function f(t) is increasing for t≥1 and when 0< p< 1.
so f(t)≥f(1) and f(1)=0. so, we get the desired result.
edit: to show f(t) is increasing for t≥1 and when 0< p< 1.
we need to show 0< f′(t)=p(t−1)p−1−ptp−1 and since p> 0, all we need to show is that (t−1)p−1> tp−1, ∀t> 1 and 0< p< 1.
since 0< p< 1, we need to show t1−p> (t−1)1−p which is true since p< 1 and t> t−1> 0.
so? what is the qstn? tell us what you anytime we will be your side