let for a is not equal to a1 is not equal to a is not equal to 0 f(x)=ax2+bx+c g(x)=a1x2+b1x+c1 and p(x)=f(x)-g(x). if p(x)=0 only for x=-1 and p(-2)=2, then the value of p(2)is:

Asked by abhilipsa satpathy | 16th Jun, 2013, 03:37: PM

Expert Answer:

f(x)=ax2+bx+c
g(x)=a1x2+b1x+c1
 
and p(x)=f(x)-g(x) = (ax2+bx+c) - (a1x2+b1x+c1)
p(x) = (a-a1)x2 + (b-b1)x + c - c1
 
Now, p(-1) = 0 
i.e. a-a1 -(b-b1) + c-c1 = 0
Hence, (b-b1) = a-a1 + c-c1
 
Also, p(-2) = 2
(a-a1)(-2)^2 + (b-b1)*(-2) + c-c1 = 2
4(a-a1) -2(b-b1) + c-c1 = 2
4(a-a1) - 2(a-a1 + c-c1) + c-c1 = 2
2(a-a1) - (c-c1) = 2
 
Now, p(2) = (a-a1)(2)^2 + (b-b1)*(2) + c-c1 
= 4(a-a1) + 2(a-a1 + c-c1) + c-c1
= 6(a-a1) + 3(c-c1)
= 3(2(a-a1) + (c-c1))
= 3(2+c-c1 + c-c1)
= 3(2+ 2(c-c1))
= 6(1+ c-c1)
Now, since, c-c1 is not equal to zero, so, 1+ c-c1 > 1 and p(2) is  a multiple of 6, so, 
from the options you can see that only the 4th option i.e. p(2) = 18 is possible. 
Hence answer is option 4. 
 

Answered by  | 17th Jun, 2013, 04:21: AM

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