The root locus plot of the roots of the characteristics equation of a closed loop system having the open loop transfer function $$\frac{{{\text{K}}\left( {{\text{s}} + 1} \right)}}{{2\left( {2{\text{s}} + 1} \right)\left( {3{\text{s}} + 1} \right)}}$$    will have a definite number of loci for variation of K from 0 to $$\infty $$. The number of loci is

We are given the open-loop transfer function:

𝐺
(
𝑠
)
𝐻
(
𝑠
)
=
𝐾
(
𝑠
+
1
)
2
(
2
𝑠
+
1
)
(
3
𝑠
+
1
)
G(s)H(s)=
2(2s+1)(3s+1)
K(s+1)
​

Let’s analyze it step by step:

Step 1: Simplify the Transfer Function
𝐺
(
𝑠
)
𝐻
(
𝑠
)
=
𝐾
(
𝑠
+
1
)
6
(
𝑠
+
0.5
)
(
𝑠
+
1
3
)
G(s)H(s)=
6(s+0.5)(s+
3
1
​
)
K(s+1)
​

Step 2: Determine Poles and Zeros
Poles: From the denominator:

𝑠
=
−
0.5
s=−0.5

𝑠
=
−
1
3
s=−
3
1
​


Zero: From the numerator:

𝑠
=
−
1
s=−1

So:

Number of poles = 2

Number of zeros = 1

Step 3: Number of Root Locus Branches (Loci)
The number of loci (root locus branches) is always equal to the number of poles of the open-loop transfer function.

Thus, the number of loci = 2

✅ Final Answer: D. 2

solution plz