Australian citizen can be trustee in india and have authority sign cheque 

Yes, no such discrimination or bar to hold trust office.

A foreign national or non resident cannot be trustee in Indian trust.

1. if the nature of the trust is such that the trustee is not required to remain personally present to take care and manage the subject matter of trust, then i do not see any reason why such a person cannot be a trustee in either country

Every person capable of holding property and competent to contract can be a trustee.

Regards 

G.Rajaganapathy Lawyer 

High Court of Madras 

Yes he can become trustee in India and sign cheques

1.  Yes.  It is possible, PROVIDED proper permission is taken from RBI, by following due procedure of law.

2.  There would be other compliance's under other Indian laws, like FERA etc....

A trustee has certain powers conferred on him or her by the Trustees Act (Cap 337), ... Trustees will usually be appointed by the instrument that creates the trust.

A “trust” is an obligation annexed to the ownership of property, and arising out of a confidence reposed in and accepted by the owner, or declared and accepted by him, for the benefit of another, or of another and the owner: “author of the trust”; “trustee”; “beneficiary”; “trust property”; “beneficial interest”; “instrument of trust”.—The person who reposes or declares the confidence is called the “author of the trust”; The person who accepts the confidence is called the “trustee”

Every person capable of holding property may be a trustee; but, where the trust involves the exercise of discretion, he cannot execute it unless he is competent to contract

The trustee is bound to fulfil the purpose of the trust, and to obey the directions of the author of the trust given at the time of its creation, except as modified by the consent of all the beneficiaries being competent to contract