Tare, who replaced an injured Sachin Tendulkar, paid back for the faith showed in him by the team management, with a 37-ball-59 that was studded with eight boundaries and a six.
The Rajasthan bowlers put up an impressive performance especially in the death overs as MI managed only 34 runs in the final five overs.
While James Faulkner (2/32) and Shane Watson (2/30) were the most successful bowlers, Kevon Cooper (1/27) and 42-year-old leggie Praveen Tambe (1/38) also did well in patches.
The new-look Mumbai opening pair of Tare and Maxwell, put on 25 runs out of which 23 were scored by the Australian batsman with the help of three sixes in 17 balls.
Maxwell, who smacked compatriot Faulkner for his first six over long on and Ankit Chavan for two in three balls, was beaten squarely by Waston who trapped him leg before with his second ball.
After the first six overs of power play MI were 49 for 1.
The wicketkeeper-batsman completed his half century off only 30 balls after hitting Faulkner for successive fours in the lanky left-armer's second spell and the first 10 overs yielded 81 runs.
Karthik departed in the 12th over soon after MI reached 100, caught in the deep off leg spinner Pravin Tambe who bowled quite well to restrict the run-rate.
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