YOKOHAMA, Japan — Within the high stone walls of St. Xavier’s College in Mumbai, Ashreen Irani, a 19-year-old management major, asked President Barack Obama a question that left her fellow students murmuring nervously over its bluntness.

“Why is Pakistan such an important ally of the United States?” Irani said in the city where, two years ago, militants trained in Pakistan came ashore and killed more than 160 people in a days-long siege. “Why hasn’t America called it a terrorist state?”

Obama, his shirt collar open to the morning heat, told the student audience that he expected the question. Pakistan’s stability matters to the United States, Obama explained, and it should matter to India’s young generation also.

The moment, which came early in Obama’s trip, announced a shift in how the world views the first African-American U.S. president two years into his term.

A leader whose biography inspired millions outside the country when he took office, Obama is judged today by foreign leaders and audiences less on the power of his personal story than his policies, a more challenging sell in many parts of the world.

In seeking to repair the U.S. image abroad during his first year in office, Obama often used himself as a parable of America’s ability to learn from its mistakes, telling audiences from Turkey to Trinidad that he was a different mix of national advocate and citizen of the world than his predecessor.

But Obama is now pushing policies he argues would strengthen American security and accelerate the U.S. economic recovery, unfolding more slowly than in many other nations, that have met resistance overseas.

He encountered criticism in India over U.S. aid to Pakistan, resistance to his trade ambitions in South Korea and frustration from leaders of the Group of 20 nations with U.S. monetary policy. Those challenges made for some unscripted encounters with the public and some awkward ones with some heads of state.

But there were also flashes of the first-year magic — from his appearance inside India’s parliament, where he spoke beneath a portrait of Mohandas Gandhi, to his ecstatic reception at the University of Indonesia in Jakarta, a childhood home. National security adviser Thomas Donilon called the visits “seminal events” in the U.S. relationship with those countries.

Successes and setbacks

At the end of his 10-day trip, which he concluded today with a visit to the Great Buddha statue outside this city, Obama returns home with a mix of successes in raising America’s profile in Asia and setbacks in transforming the growing markets of the region into ones more amendable to U.S. exports.

“It’s not just a function of personal charm,” Obama said after the Group of 20 meeting in Seoul, referring to the economic issues that remain largely unresolved. “It’s a function of countries’ interests and seeing if we can work through to align them.”

Obama left Washington weakened after midterm voters punished his party for the anemic U.S. economy. His mission in Asia, home to some of the world’s fastest-growing economies, was primarily an economic one.

Over three days in India, his longest stay in any country as president, Obama supported India’s bid for a permanent seat on an expanded U.N. Security Council and left with the message that India generates U.S. jobs, not just takes them.

Ben Rhodes, the deputy national security adviser for strategic communication, said few foreign trips have relied on Obama’s public appearances as much as the one to India. “Part of what captivated them was his biography,” Rhodes said. “The connection that, frankly, they just draw from Gandhi to the (American) civil rights movement to the president is something that had huge resonance in India.”

But in India’s querulous democracy, Obama was also the target of tough commentary over his policy in Pakistan, against which India has fought three major wars.

Relationship with Russia: still uneasy

There is an uneasiness within Russia on the START issue, and Russian President Dmitry Medvedev is being well briefed about the dynamic of the U.S. Senate, said an official who spoke on condition of anonymity to discuss details of the closed-door meeting.The START treaty, which has been pending in the Senate for months, would reduce the limit on strategic warheads to 1,550 for each country from the current ceiling of 2,200. The treaty has drawn resistance, principally from Republicans.

On other issues:

• Obama praised Medvedev for strongly condemning the beatings of Russian journalists.

• The two leaders discussed containment of Iran in its purported pursuit of nuclear weapons, and Obama and Medvedev have no disagreement about how to proceed, according to the Obama official’s account.

• Obama said he believes Medvedev is bringing about reforms in the former Soviet Union and is moving the country forward. He said he supports Medvedev’s pursuit of membership in the World Trade Organization — a point the Russian leader reinforced as he and Obama appeared briefly before reporters and camera crews. He said he is “working closely” to achieve that end.