1
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Question

# (a) If x be real prove that the expression x+22x2+3x+6 takes all values in the interval [−113′13].(b) Show that the value of tanxtan3x or sinxcos3xcosxsin3xWhenever defined never lies between 1/3 and 3.(c) If x is real, the maximum value of 3x2+9x+173x2+9x+7 is

A
41
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B
1
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C
177
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D
14
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Solution

## The correct option is A 41(a) If the given expression be y, then 2x2y+(3y−1)x+(6y−2)=0If y≠0 then Δ≥0 for real x i.e. B2−4AC≥0or −39y2+10y+1≥0or (13y+1)(3y−1)≤0∴ −113≤y≤13If y=0 then x=−2 which is real and this value of y is included in the above range.(b) y=tanxtan3x=t(1−3t2)3t−t3=1−3t23−t3, as t≠0∵ t=0 will make y indeterminate∴ y(3−t2)=1−3t2or t2=3y−1y−3=+ive =(3y−1)(y−3)(y−3)2=3(y−1/3)(y−3)(y−3)2Above will be +ive if y<1/3 or y>3 as denominator is +iveHence we conclude that y cannot lie between 13 and 3.(c) Ans (a).Proceeding as in part (a), (b), yϵ[1,41]∴ Max value of y is 41.

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