(i) The Doctrine of Lapse was introduced by Lord Dalhousie.
(ii) According to this, if the ruler of a protected or dependent state died without a natural heir, his state would automatically lapse.
(iii) The victims of this doctrine were Satara (1848), Jaitpur, Sambalpur (1850), Udaipur (1852), Nagpur (1853)and Jhansi (1854

In the late 1830s, the East India Company became worried about Russia. It imagined that Russia might expand across Asia and this fear, the British now wanted to secure their control over the North-West. Sind was taken over in 1843. After Maharaja Ranjit Singh's death in 1839, two prolonged wars were fought with the Sikh kingdom. Ultimately, in 1849, Punjab was annexed.

Aurangzeb's Farman had granted the Company only the right to trade duty-free. But the officials of the Company misused this facility. They were carrying on private trade and stopped paying duty for their personal profit. This caused a huge loss of revenue in Bengal.

The EIC used various political, economic and diplomatic methods to expand its rule in India:
(i) Direct Annexation: At times, British fought direct wars. e.g. State of Punjab w~s annexed after the death of Punjab king Maharaja Ranjit Singh.
(ii) Appointment of Residents : EIC residents began interfering in internal affairs deciding as to which person has to be appointed at political or administrative posts.
(iii) Doctrine of Lapse : It was a policy made by lord Dalhousie. The doctrine declared that if an Indian ruler died without a male heir, his kingdom would 'lapse', and would become a part of Company's property, e.g., Udaipur, Jhansi.
(iv) Introduction of 'Subsidiary Alliance' : According to the terms of subsidiary alliance, Indian rulers were not allowed to have their independent armed forces. They were to be protected by the company. Also, they had to pay for the 'subsidiary forces' that the company was supposed to maintain for the purpose of their protection. If the Indian rulers failed to make the payment, then a part of their territory was taken away as penalty. 

Beginning of trade in Bengal by East India Company
(i) The first English factory was set up on the banks of the river Hugli in 1651. This was the base, from which the Company traders, known at that time as "factors", operated.
(ii) The factory had a warehouse where goods for export were stored, and it had offices where company officials used to sit. As trade expanded, the company persuaded merchants and traders to come and settle near the factory.
(iii) By 1696, it began building a fort around the settlement. Demands put up by East India Company:
(i)Two years later, EIC bribed Mughal officials to give the Company, zamindari rights over three villages.
(ii) One of these was Kalikata, which later grew into the city of Calcutta or Kolkata (as it is known today).
(iii) It also persuaded the Mughal emperor Aurangzeb to issue a farman, granting the Company the right to trade duty free.

The English developed a strong army due to the following reasons :
(i)The army was needed to establish control upon the vast empire.
(ii) It was also required to suppress internal revolts against the British.
(iii) The army consisted of Indian soldiers who were known as sepoys.
(iv) The army was well disciplined, well trained and regularly paid.

This toy tiger was taken by the British when Tipu Sultan died defending he capital Seringapatam in 1799. The toy was then kept in the Victoria and Albert Museum in London.

Tipu Sultan, the ruler of Mysore, possessed the mechanical toy in the above picture. 

Governor -General Lord Hastings (1813-1828) introduced a new policy of paramountcy' in India. It meant that the power of the Company was paramount or supreme, i.e. the power of the Company was greater than that of Indian states. In order to protect its business interest, the Company could annex or threaten to annex any Indian kingdom.
This policy of the British was followed by the later rulers. But this policy of the British did not go unchallenged. For example, when the British tried to capture the small state of Kitoor, Rani Channamma revolted against this annexation. As a result, she was arrested and later died in the prison in 1829. This resistance movement did not stop. It was carried on by Rayanna, a poor chowkidar of Sangoli in Kitoor.
He got support from the common people and destroyed many British camps and records.In 1830, he was caught and hanged by the British

In the above map, the areas which are under the British rule are-some parts of East UP, Bihar, Bengal, Coastal Andhra Pradesh, Madras city, some parts of Tamil Nadu and Kerala.