JEE MAIN - Mathematics (2023 - 25th January Morning Shift - No. 9)
Let $$y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})$$. Then $$y' - y''$$ at $$x = - 1$$ is equal to
496
976
464
944
Explanation
$$
\begin{aligned}
& y=\frac{1-x^{32}}{1-x}=1+x+x^2+x^3+\ldots+x^{31} \\\\
& y^{\prime}=1+2 x+3 x^2+\ldots+31 x^{30} \\\\
& y^{\prime}(-1)=1-2+3-4+\ldots+31=16 \\\\
& y^{\prime \prime}(x)=2+6 x+12 x^2+\ldots+31.30 x^{29} \\\\
& y^{\prime \prime}(-1)=2-6+12 \ldots 31.30=480 \\\\
& y^{\prime \prime}(-1)-y^{\prime}(-1)=-496
\end{aligned}
$$
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