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Question

The curve y=ax3+bx2+cx+5 touches the x−axis at P(−2,0) and cuts the y-axis at the point Q where its gradient is 3. Find the equation of the curve completely.

A
A=1/2,B=3/2,C=3
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B
A=1/2,B=3/2,C=8
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C
A=1/2,B=3/4,C=3
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D
A=1/2,B=3/2,C=3
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Solution

The correct option is C A=1/2,B=3/4,C=3
A] Equation of curve y=ax3+bx2+cx+5 __ (1)
It meets y axis at o where x = 0
substitute x = 0 in (1)
y=0+0+0+5,y=5p=(0,5)
From (1),
dydx=3ax2+2bx+c __ (2)
at p, y=0+0+c=c
gradient is 3 c=3
(1) touches x axis at (-2,0)
slope of tangent at (-2,0) is 0
at (-2,0)y=12a4b+3 __ (4)
12a4b+3=0
(-2,0) lies on (1)
8a+4b2c+5=0 __ (3)
subtracting (3)-(4) 4a+2=0 a=1/2
4b=3b=3/2 & c=3

1177618_1201599_ans_926399251b394b9089cae49dfe5986f2.jpg

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