Iltutmish was a prominent ruler of the Delhi Sultanate in medieval India. He is known for his military campaigns and administrative reforms during his reign from 1211 to 1236. Among the given options, all of them are correct.

1. Iltutmish gave the Turkish kingdom legal status:
- Iltutmish was a Turkish slave who rose to power and established the Delhi Sultanate. By giving the Turkish kingdom legal status, he solidified the position of the Turks as the ruling elite in the Delhi Sultanate.

2. Iltutmish recaptured Bayna and Thangir:
- Bayna and Thangir were territories that had been previously lost by the Delhi Sultanate. Iltutmish successfully recaptured these territories, demonstrating his military prowess and expanding the empire's territories.

3. Iltutmish reconquered Nagaur in Jodhpur:
- Nagaur was a strategic city in the region of Jodhpur, and it had been previously captured by the Rajputs. Iltutmish led a successful campaign to reconquer Nagaur, further expanding the influence of the Delhi Sultanate in Rajasthan.

4. Iltutmish failed to recapture Varanasi:
- Varanasi, also known as Kashi, was a significant city in the eastern part of the Delhi Sultanate. Iltutmish attempted to recapture Varanasi but was unsuccessful. This failure highlights the limitations of his military campaigns and the resilience of the local powers in the region.

Therefore, all of the given options are correct. Iltutmish's reign was marked by territorial expansion, military achievements, and administrative reforms. He played a crucial role in shaping the Delhi Sultanate and consolidating Turkish rule in India.

Iltutmish, also known as Shams-ud-din Iltutmish, was the third ruler of the Delhi Sultanate in medieval India. He ascended to the throne in 1211 and ruled until his death in 1236. During his reign, Iltutmish made significant contributions to the consolidation and expansion of the Delhi Sultanate.

Iltutmish's Achievements:

1. Granting Legal Status to the Turkish Kingdom:
Iltutmish granted legal status to the Turkish kingdom, which refers to his efforts to establish a stable and organized administration for the Delhi Sultanate. He implemented various administrative and legal reforms, including the introduction of the Iqta system, which involved granting land revenue assignments to military commanders in exchange for their services. This system helped in strengthening the central authority and maintaining control over the vast territories of the Sultanate.

2. Recapture of Bayna and Thangir:
Iltutmish successfully recaptured the regions of Bayna and Thangir. These areas had previously been under the control of local Rajput rulers, but Iltutmish's military campaigns led to their recapture by the Delhi Sultanate. This expansion of territory further solidified the Sultanate's dominance in the region.

3. Reconquest of Nagaur in Jodhpur:
Iltutmish also reconquered Nagaur, a strategic city located in the Jodhpur region of present-day Rajasthan. The city had been captured by the Rajput ruler Prithviraj Chauhan in the past, but Iltutmish's forces successfully reclaimed it for the Delhi Sultanate. This victory further enhanced the Sultanate's control over northern India.

4. Failure to Recapture Varanasi:
However, it is important to note that Iltutmish was unsuccessful in his attempts to recapture Varanasi. Varanasi, also known as Kashi, was a significant cultural and religious center in northern India. Despite his efforts, Iltutmish was unable to regain control over the city, which remained outside the dominion of the Delhi Sultanate.

Conclusion:
In conclusion, all the statements (a, b, c, and d) are correct. Iltutmish granted legal status to the Turkish kingdom, recaptured Bayna and Thangir, reconquered Nagaur in Jodhpur, but failed to recapture Varanasi. These achievements and setbacks played a crucial role in shaping the territorial expansion and administration of the Delhi Sultanate under Iltutmish's rule.