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Q.

Solve the following equations:


Suppose x1, x2  and x3 are roots of (11 - x)3 + (13 - x)3 = (24 - 2x)3. What is the sum of  x1 + x2 + x3 ?


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a

-36

b

34

c

-34

d

36 

answer is A.

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Detailed Solution

Given:
The roots of  (11 - x)3 + (13 - x)3 = (24 - 2x)3  are x1, x2 and x3 .
Let us consider a general cubic equation of the form ax3 + bx2 + cx + d = 0, where a, b, c, d is non-zero. So, let us consider p, q, and r are the roots. Then, we can write the relationships as below,
p + q + r = − ba
pq + qr + rp = ba
pqr = − da
For (11-x)3 , the expansion is,
(11-x)3 = 113 – 3 × 112 × x + 3 × 11 × x2 x3   [(a-b)3 = a3 − 3a2b + 3ab2 b3]
(11-x)3 = 1331 − 363x + 33x2 x3  …..…..….. (1)
For (13-x)3 the expansion is,
(13-x)3 = 133 – 3 × 132 × x + 3 × 13 × x2 x3
(13-x)3 = 2197 – 507x + 39x2 x3 …..…..….. (2)
For (24-2x)3, the expansion is,
(24-2x)3 = 243 – 3 × 242 × 2x + 3 × 24 × (2x)2 (2x)3 (24-2x)3 = 13824 − 3456x + 288x2 − 8x3 …..…..….. (3)
The equation is,
 (11 - x)3 + (13 - x)3 = (24 - 2x)3     Now, substitute the values from equation (1), (2), and (3)
⇒ (1331 − 363x + 33x2 x3) + (2197 – 507x + 39x2 x3 ) = (13824 − 3456x + 288x2 − 8x3)
⇒ 3528 − 870x + 72x2 − 2x3 = 13824 − 3456x + 288x2 − 8x3
⇒ (3528 − 870x + 72x2 − 2x3) − (13824 − 3456x + 288x2 − 8x3) = 0
⇒ 6x3 − 216x2 + 2586x – 10296 = 0
So, using the relationships in the cubic equation given to us, we can write sum of products as,
p + q + r = − ba
Substitute the values,
⇒ x1 + x2 + x3  = − ( -216)6
x1 + x2 + x3 = −36
Hence, the value of x1 + x2 + x3 is -36.
So, option (1) is correct.
 
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