The Australian finished 0.606 seconds quicker than Lewis Hamilton, who earlier set the pace in first practice.
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The world champion said the shape of the race will be decided by qualifying on Saturday.
After two controversial race starts in a row – where the Russian would somehow collide three times with Sebastian Vettel – Kvyat would be swapped out for Dutch sensation Max Verstappen.
Red Bull struggled for consistency last season, and came close to cutting ties with engine supplier Renault, but the car’s pace is clearly improving.
“It’s not so easy to find that out of the vehicle, and I’m looking forward to using it in Canada”.
“That was a massively impressive session for Daniel”, said Red Bull team chief Christian Horner.
Red Bull’s Daniel Ricciardo is confident he can challenge Mercedes for victory in Sunday’s Monaco Grand Prix.
Ricciardo, who finished the session 0.606 seconds ahead of Hamilton, said winning was “definitely my aim”.
Verstappen was fifth and fourth in the respective sessions.
Ferrari driver Sebastian Vettel was third quickest.419 behind Hamilton, who has gone eight races without a win.
“I brought them the first podium of the year”.
The cover clipped Jenson Button’s McLaren, damaging the front wing and halting the session.
Track workers were then deployed to Sainte Devote to weld down the offending cover. You’ll see us within tenths of a second.
Meanwhile, Williams watched in frustration as veteran driver Felipe Massa lost control turning up the hill at Sainte Devote and slammed into the barrier, damaging the left side of his auto.
Mexican Esteban Gutierrez stopped his Haas before Portier with electrical problems, and Renault’s British rookie Jolyon Palmer crashed at Tabac.
Dane Kevin Magnussen followed suit at the final corner when he went into the barriers at the exit of Rascasse just before Vettel “kissed” the wall at Ste Devote again.
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There was no immediate response from the FIA, whose report into the accident said that the Frenchman had not slowed sufficiently under warning flags before crashing.
