Conjugated double bond is present in:
(This question has multiple correct options)
(A) $1,2-butadiene$
(B) $1,3-butadiene$
(C) $1,3-pentadiene$
(D) $\beta -butylene$

Hint: Conjugated dienes are more stable than any other dienes because it involves resonance of electrons. Conjugations take place between a single bond and double bond present. Electrons are delocalized during conjugations.

Complete step by step solution:
Dienes can be divided into three classes, depending on the relative location of the double bonds:
1.Cumulated dienes have the double bonds sharing a common atom.
2.Conjugated dienes have conjugated double bonds separated by one single bond. Conjugated diene is more stable than other dienes because of resonance.
3.Unconjugated dienes have the double bonds separated by two or more single bonds. They are usually less stable than isomeric conjugated dienes. This can be also known as an isolated diene.
Compounds that contain more than two double bonds are called polyenes. Polyenes and dienes share many properties.
Now looking in each of the options given above to see whether it is conjugated double bond or not.
Considering option (A), 1,2-butadiene. The structure of 1,2-butadiene is as follows:
\[C{{H}_{3}}-CH=C=C{{H}_{2}}\]
Since it has double bonds sharing a common atom, then it is not conjugated diene. It is an allene.
Considering option (B), 1,3-butadiene. The structure of 1,3-butadiene is as follows:
\[C{{H}_{2}}=CH-CH=C{{H}_{2}}\]
It has two double bonds separated by one single bond. Therefore, it is a conjugated diene.
Considering option (C), 1,3-pentadiene. The structure of 1,3-pentadiene is as follows:
\[C{{H}_{2}}=CH-CH=CH-C{{H}_{3}}\]
It has two double bonds separated by one single bond. Therefore, it is a conjugated diene.
Considering option (D)$\beta -butylene$. The structure of $\beta -butylene$ is as follows:
\[C{{H}_{3}}-CH=CH-C{{H}_{3}}\]
It has only one double bond. Therefore it is not a diene.

Therefore, the correct answers are the B and D option.

Note: Many molecules contain two double bonds, called dienes. Cumulative dienes are generally known as Allene. Compounds that contain more than two double bonds are called polyenes. Polyenes and dienes share many properties.