Question

If a function $f( x )= ax ^3+ bx ^2+ cx + d$ where $a , b , c$ and $d$ are integers and $a >0$ is such that $f\left(\sin \frac{\pi}{18}\right)=0$. Then the smallest possible value of $f(1)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

$ \sin \frac{\pi}{18}=\sin 10^{\circ}, \sin 30^{\circ}=\frac{1}{2} $
$\text { also } \sin 30^{\circ}=3 \sin 10^{\circ}-4 \sin ^3 10^{\circ}$
$ \frac{1}{2}=3 \sin 10^{\circ}-4 \sin ^3 10^{\circ} $
$8 \sin ^3 10^{\circ}+0 \sin ^2 10^{\circ}-6 \sin 10^{\circ}+1=0 $ .....(1)
$ f\left(\sin ^3 0^{\circ}\right)=0$
$\sin ^3 10^{\circ}+ b \sin ^2 10^{\circ}+ c \sin 10^{\circ}+ d =0 $ ....(2)
$\text { comparing }(1) \text { and }(2)$
$a =8, b =0, c =-6, d =1 $
$\text { hence } f (1)= a + b + c + d $
$ f (1)=3 $