GATE 2026 IN Q.13: solution and answer

GATE 2026 · INQ.13 · 1 mark · MCQ

Consider a function $f (z) = z^{2} + z + 1$ where $z \in C$ is a complex variable. A simple closed contour $γ$ in $z$ -plane encloses the point $z = 1 + 0 j$ .

The value of integral $\oint_{γ} \frac{f ( z )}{z - 1} d z =$ ________.

(A)
$6 π j$
(B)
$3 π j$
(C)
$12 π j$
(D)
$π j$

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Answer and solution

ANSWER

$\oint_{γ} \frac{f ( z )}{z - 1} d z = 2 π j f (1) = 2 π j \times 3 = 6 π j$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2026 IN (IIT Guwahati)

$f (z) = z^{2} + z + 1$ is a polynomial, so it is analytic everywhere (entire), including on and inside $γ$ .
The integrand $\frac{f ( z )}{z - 1}$ is analytic everywhere except at $z = 1$ . Since $f (1) = 3 \neq = 0$ , that point is a simple pole, and $γ$ encloses it. It is the only singularity inside $γ$ .
Apply the Cauchy integral formula, taking $γ$ in the usual positive (anticlockwise) sense: $\oint_{γ} \frac{f ( z )}{z - z _{0}} d z = 2 π j f (z_{0}), z_{0} = 1$
Evaluate $f$ at the pole: $f (1) = 1^{2} + 1 + 1 = 3$ .
Hence $\oint_{γ} \frac{f ( z )}{z - 1} d z = 2 π j \times 3 = 6 π j$
Check with the residue theorem: the residue at the simple pole $z = 1$ is $lim_{z \to 1} (z - 1) \frac{f ( z )}{z - 1} = f (1) = 3$ , so the integral is $2 π j \times 3 = 6 π j$ . The answer is option (A).

KEY IDEA

Cauchy integral formula: if $f$ is analytic on and inside a simple closed contour $γ$ traversed anticlockwise, and $z_{0}$ lies inside $γ$ , then $\oint_{γ} \frac{f ( z )}{z - z _{0}} d z = 2 π j f (z_{0})$ . Here $f$ is a polynomial, so the integral is simply $2 π j$ times $f (1)$ , whatever shape $γ$ has.

WATCH OUT FOR

Leaving out the factor $2 π$ . Writing $π j f (1) = 3 π j$ gives option (B), and dropping the factor 3 gives $π j$ , option (D). Another slip is evaluating $f$ at $z = 0$ or at the wrong point, or squaring twice, which can lead to $12 π j$ (C). The integral is $2 π j$ times $f$ evaluated at the enclosed pole, $z_{0} = 1$ .

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GATE 2026 IN Q.13: solution and answer

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