Ansar Beit al-Maqdis, which has pledged allegiance to the extremist Islamic State group, claimed responsibility for both attacks.
The group has carried out a slew of attacks in Sinai since the army's ouster of Islamist president Mohamed Morsi in July 2013.
A roadside bomb attack on an army vehicle killed six soldiers and wounded two near the North Sinai town of Sheikh Zuweid, a jihadist stronghold.
"An armoured personnel carrier for the army of the apostates was destroyed... Killing and wounding all aboard," Ansar Beit al-Madqis said on a Twitter account attributed to it.
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"Their vehicle was targeted by an explosive device placed by the extremist terrorist elements which left an officer, a sergeant, and four soldiers martyred, and two other soldiers wounded," it said on its Facebook page.
Hours later a suicide car bomb targeting a police station in North Sinai's provincial capital of El-Arish killed five policemen and a civilian, the interior ministry said in a statement.
Medics said the attack left another 44 people wounded, including at least 15 policemen.
The interior ministry said security forces fired at the vehicle before it was able to ram the police station.
A police officer said the truck was loaded with explosives but covered with straw, and blew up close to the police station.
"A den of the apostate police was targeted in El-Arish by a car bomb driven by a martyr," Ansar Beit al-Maqdis said as it claimed the second attack.
Ansar Beit al-Maqdis changed its name last year to the Sinai Province after pledging allegiance to IS.
It has claimed several sophisticated attacks in Sinai and the Nile valley and now wants to establish a province of the self-declared IS "caliphate".