Final answer:
The statement is false; by the end of 1862, the Union had not yet gained control of the entire Mississippi River, as the stronghold of Vicksburg remained in Confederate hands until July 1863. The correct answer is option b.
Step-by-step explanation:
The statement that by the end of 1862, northern forces had succeeded in gaining control of the length of the Mississippi River is false. While considerable progress was made by Union forces in the spring and summer of 1862, including the capture of New Orleans and Memphis, they had not yet achieved complete control over the river. The critical stronghold of Vicksburg, Mississippi remained in Confederate hands until it eventually fell to Union forces under General Ulysses S. Grant on July 4, 1863, marking a turning point in the war by granting the Union full control over the Mississippi River and effectively splitting the Confederacy in two.