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After Australia made a competitive 326 in their first innings, spearheads Mitchell Starc and Josh Hazlewood started spectacularly with the ball by dismissing India openers Murali Vijay and Lokesh Rahul inside the opening six overs.
But Kohli rescued India through half-century partnerships with the in-form Cheteshwar Pujara and Ajinkya Rahane.
India reached stumps on a more comfortable 172 for three with Kohli 82 not out and his deputy Rahane unbeaten on 51 to leave the match evenly poised.
Australia bowled accurately but the pitch, marked with widening cracks, settled down late in the day.
Australia batsman Usman Khawaja believed there was enough spice in the wicket for the bowlers to exploit in the crucial third morning session.
“It had (some) patches today, which were still doing a bit,” Khawaja told reporters. “I think we bowled well but we probably could have been more disciplined. Hopefully tomorrow we can be just a little bit better.”
Beyond Kohli and Rahane, who have shared an unbeaten 90-run partnership, India’s batting order contains inexperienced pair Hanuma Vihari and Rishabh Pant, and a fragile lower-order.
“I think one or two wickets in the first session tomorrow will be pretty big for us,” Khawaja said. “It isn’t an easy wicket to bat on. We have to start well and then we can have a chance at that tail.”
Khawaja praised Kohli's determined 181-ball knock, but reckoned Australia would have opportunities to knock over the India talisman.
“He was on early today and we gave him a couple of full balls and he hit us,” Khawaja said. “He got away but he will have to start again tomorrow and we will have another chance.”
India lead the four-match series after a nail-biting 31-run victory in Adelaide.