If and is an integer,
is the sum of all the possible values of that allows the below statement to be true.
Find the value of .
This problem is a part of <Grade 10 CSAT Mock test> series .
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From x 2 − a 2 x ≥ 0 , we get
x ≤ 0 or x ≥ a 2 .
.
And from x 2 − 4 a x + 4 a 2 − 1 < 0 , we get
2 a − 1 < x < 2 a + 1 .
( i ) 0 < a < 2 1
− 1 < 2 a − 1 < x ≤ 0 or a 2 ≤ x < 2 a + 1 < 2 .
Since 0 ≤ a 2 ≤ 4 1 ,
There are two values for x , 0 and 1 , which doesn't satisfy the problem.
( i i ) a = 2 1
.
4 1 ≤ x < 2 .
There is one value for x , which is 1 .
( i i i ) 2 1 < a < 1
a 2 ≤ x < 2 a + 1 .
Since 4 1 < a < 1 and 2 < 2 a + 1 < 3 ,
There are two values for x , 1 and 2 , which doesn't satisfy the problem.
( i v ) a = 1
.
1 < x < 3 .
There is one value for x , which is 2 .
( v ) 1 < a < 2
a 2 ≤ x < 2 a + 1 .
Since 1 < a 2 < 2 and 3 < 2 a + 1 < 1 + 2 2 < 4 ,
There are two values for x , 2 and 3 , which doesn't satisfy the problem.
From ( i ) through ( v ) , we can say that
All possible values for a are 2 1 and 1 .
.
S = 2 3 ,
6 0 0 0 S = 6 0 0 0 × 2 3 = 9 0 0 0 .