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Question

f(x)=∣ ∣x+c1x+ax+ax+bx+c2x+ax+bx+bx+c3∣ ∣ and g(x)=(c1x)(c2x)(c3x)
Which of the following is not a constant term in f(x)?

A
bg(a)ag(b)ba
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B
bg(a)af(b)ba
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C
bf(a)ag(b)ba
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D
none of these
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Solution

The correct option is A bg(a)ag(b)ba
Given,g(x)=(c1x)(c2x)(c3x).
g(a)=(c1a)(c2a)(c3a) ....(i)
g(b)=(c1b)(c2b)(c3b) ....(ii)
f(x)=∣ ∣x+c1x+ax+ax+bx+c2x+ax+bx+bx+c3∣ ∣
Clearly,f(x) is linear in x.
Let f(x)=αx+β
f(a)=aα+β
and f(b)=bα+β
β=bf(a)af(b)ba ....(iii)
Now, f(a)=∣ ∣c1a00bac2a0babac3a∣ ∣
f(a)=(c1a)(c2a)(c3a)
f(a)=g(a) (using (i)) .....(iv)
f(b)=∣ ∣c1babab0c2bab00c3a∣ ∣
f(b)=(c1b)(c2b)(c3b)
f(b)=g(b) .....(v)
Put this value in (iii), we get
β=bg(a)ag(b)ba

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