Jadhav guided Rest of India to 264 from a precarious 102 for five at the Chinnawsamy Stadium. The other handy contributions came from Naman Ojha (34), Jayant Yadav (26), Paras Dogra (26) and tail-ender Varun Aaron (25).
The effort was enough for Rest of India to take a 20-run lead over Karnataka, who had succumbed to Aaron's lethal bowling to be bowled out for 244 on day one.
While Rest of India lower-order did a decent repair job, the top-order could not do much against the pace trio of R Vinay Kumar (3/76), Abhimanyu Mithun (3/46) and Sreenath Aravind (1/40).
Unmukt Chand lasted for a ball yesterday evening while his opening partner Jiwanjot Singh (17) was dismissed early morning today. He tickled a regulation leg-side ball straight into the hands of wicketkeeper Naman Ojha.
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The following period of play saw Ojha and Dogra defending against Karnataka pacemen, who maintained a nagging off-stump line.
Ojha departed soon after and with the subsequent fall of skipper Manoj Tiwary (14), Rest of India were in a spot of bother at 102 for five.
The 80-run sixth-wicket stand between Jadhav and Yadav resurrected the innings and took the team closer to Karnataka's total before Mithun struck twice in the 63rd over, sending Rishi Dhawan and Jadhav back in the hut.
Aaron, who took a six wicket haul yesterday, then eked out a 20-run lead for his team with a 52-ball stay.