The marks obtained in the improvement exam will be considered for college admissions, as the new score reflects your latest performance. However, it is important to note that:

1. The improvement exam score of 51/100 in mathematics is lower than your previous score of 58/100.

2. Colleges generally consider the best scores from multiple attempts of the same exam for admissions purposes. But in your case, the improvement exam score is lower.

3. If the exam allows only one attempt, which is the case for many state-level entrance tests, the single score obtained will be used for admissions.

4. Some colleges may consider both the original and improvement exam scores and take the higher of the two for admission purposes. But this practice is not universal.

It is advisable to check the specific policies of the colleges you are applying to regarding the consideration of improvement exam scores. In general, the lower score of 51/100 in the improvement exam will likely be used for your college applications, unless the college has a policy of considering the higher of the two scores.