Considering the following reaction sequence, the correct statement(s) is (are)

The correct statement(s) are: Compounds P and Q are carboxylic acids and Compounds P and S react with hydroxylamine to give the corresponding oximes.

Let us analyze the step-by-step chemical transformations visible in the reaction sequence image:

Step 1: Formation of Compound P
Benzene reacts with succinic anhydride in the presence of anhydrous ${\mathrm{AlCl}}_3$ (Friedel-Crafts acylation). The succinic anhydride ring opens to form a keto-acid compound:
Compound P: $C_6{H}_5-C(=O)-{\mathrm{CH}}_2-{\mathrm{CH}}_2-\mathrm{COOH}$ (3-benzoylpropanoic acid).
Compound P contains both a carbonyl ketone group ($-C(=O)-$) and a carboxylic acid group ($-\mathrm{COOH}$).

Step 2: Formation of Compound Q
Compound P undergoes Clemmensen reduction using amalgam zinc and concentrated hydrochloric acid ($\mathrm{Zn/Hg,\; HCl}$). The carbonyl group ($>C=O$) is reduced to a methylene group ($>{\mathrm{CH}}_2$), while the carboxylic acid group remains unchanged:
Compound Q: $C_6{H}_5-{\mathrm{CH}}_2-{\mathrm{CH}}_2-{\mathrm{CH}}_2-\mathrm{COOH}$ (4-phenylbutanoic acid).
Hence, both P and Q contain terminal $-\mathrm{COOH}$ groups and are carboxylic acids.

Step 3: Formation of Compound R
Reaction of 4-phenylbutanoic acid (Q) with thionyl chloride (${\mathrm{SOCl}}_2$) converts the carboxylic acid into an acyl chloride:
Compound R: $C_6{H}_5-{\mathrm{CH}}_2-{\mathrm{CH}}_2-{\mathrm{CH}}_2-\mathrm{COCl}$ (4-phenylbutanoyl chloride).

Step 4: Formation of Compound S
Treatment of acyl chloride (R) with anhydrous ${\mathrm{AlCl}}_3$ undergoes intramolecular Friedel-Crafts acylation to form a cyclic ketone:
Compound S: $\alpha$-Tetralone (1-tetralone), which contains a carbonyl keto group ($>C=O$) fused to the benzene ring.

Now let us evaluate each statement option:

1. Compounds P and Q are carboxylic acids:
As derived, Compound P (3-benzoylpropanoic acid) and Compound Q (4-phenylbutanoic acid) both contain carboxylic acid functionality. (Correct)

2. Compounds P and S react with hydroxylamine to give the corresponding oximes:
Both Compound P and Compound S contain carbonyl keto groups ($>C=O$). Carbonyl compounds react readily with hydroxylamine (${\mathrm{NH}}_2\mathrm{OH}$) to form oximes ($>C=N-\mathrm{OH}$). (Correct)

3. Compound S decolorizes bromine water:
Compound S ($\alpha$-tetralone) is a saturated cyclic ketone fused to an aromatic ring. It does not have non-aromatic carbon-carbon double or triple bonds to undergo addition reaction with bromine water. Thus, it does not decolorize bromine water. (Incorrect)

4. Compound R reacts with dialkylcadmium to give the corresponding tertiary alcohol:
Dialkylcadmium ($R_2\mathrm{Cd}$) reacts with acyl chlorides (like Compound R) to give ketones, not tertiary alcohols, because dialkylcadmium is not nucleophilic enough to react further with the resulting ketone. (Incorrect)