GATE 2021 CH – Question 12

Engineering Mathematics · Calculus: Limits, continuity, differentiability, Taylor series and mean value theorem · 1 mark · Multiple choice

The function $\cos(x)$ is approximated using Taylor series around $x=0$ as $\cos(x)\approx1+ax+bx^2+cx^3+dx^4$. The values of $a,b,c$ and $d$ are

$a=1,\ b=-0.5,\ c=-1,\ d=-0.25$
$a=0,\ b=-0.5,\ c=0,\ d=0.042$
$a=0,\ b=0.5,\ c=0,\ d=0.042$
$a=-0.5,\ b=0,\ c=0.042,\ d=0$

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Correct answer: (B) $a=0,\ b=-0.5,\ c=0,\ d=0.042$

Explanation

The Maclaurin series is $\cos x=1-x^2/2!+x^4/4!-\cdots$. Cosine is even, so its odd-power coefficients vanish: $a=c=0$.

The remaining coefficients are $b=-1/2=-0.5$ and $d=1/24=0.0416667$, approximately 0.042. These four values match **B**.