Verified Solution • 2025–26 Session moderate

QJEEM23Ma5502: JEE Main 2023 Mathematics - Class 12 JEEM23Ma5502 Solution: Let . Then at is equal to

Let y(x)=(1+x)(1+x2)(1+x4)(1+x8)(1+x16)y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}}) . Then y′−y′′y' - y'' at x=−1x = - 1 is equal to

Multiple Choice Options

  • A 496
  • B 976
  • C 464
  • D 944

Step-by-Step Verified Solution

amp;y=1−x321−x=1+x+x2+x3+…+x31amp;y′=1+2x+3x2+…+31x30amp;y′(−1)=1−2+3−4+…+31=16amp;y′′(x)=2+6x+12x2+…+31.30x29amp;y′′(−1)=2−6+12…31.30=480amp;y′′(−1)−y′(−1)=−496\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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