Transcription of her NSW birth certificate ex FindMyPast:

First name(s) Mary
Last name Cartwright
Birth year 1814
Registration year 1814
Denomination Church of England
Registration district Windsor, St Matthew's
State New South Wales
Country Australia
Father's first name(s) Robert
Mother's first name(s) Mary
Volume reference V18143405 1A
Record set New South Wales Births

The actual certificate gives the full birth and baptism dates.
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There is a marriage on record for a Mary Cartwright in 1837 in Sydney to a George Langhorne.  And there is a death for a Mary Langhorne with a father named Robert does add wek confirmation that this marriage was of this Mary.  Here the two transcriptions:

First name(s) Mary
Last name Cartwright
Denomination Church of England
Marriage year 1837
Registration year 1837
Registration district Sydney, St James'
State New South Wales
Country Australia
Spouse's first name(s) George
Spouse's last name Langhorne
Volume reference V18371499 21
Record set New South Wales Marriages 1788-1945

And the death in 1861:

First name(s) Mary
Last name Langhorne
Death year 1858
Registration year 1858
Registration district St George
State New South Wales
Country Australia
Father's first name(s) Robert
Mother's first name(s) -
Registration number 2287
Category Birth, Marriage, Death & Parish Records

The Letters of Administration given to George Langhorne on 6 Sep 1861 confirm it was the same Mary that married in 1837, but do not give any more information on her parentage.
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