Hi everyone, so my dear mum (76) was told yesterday that she has cancer

Its all happened relatively quickly and I can't quite remember why it all started. She lives with me, my husband and 2 daughters (12 and 6).

She had a stomach X-ray and was informed there's nothing sinister but something was enlarged (she can't remember what), she had a blood test which showed she was anaemic (prescribed iron tablets), GP rang and asked questions about if she'd had a cough, coughed any blood etc. She replied no (I've not heard any cough tbh) but decided to send her for a chest x ray. He rang her a few hours later to say they'd found a 3cm lump in her lung and was being referred for a CT of thorax, abdomen and pelvis. That when we got the results yesterday (week after CT). The doctor said it's a 3.5cm x2.5 cm lump in her right lung and by looking at the CT he couldn't see it had spread but she's booked for a PET scan next week. Then the following week the MDT will meet and discuss the results and will hopefully be able to book a biopsy. My question is, even though they can't see it's spread on the CT what's the likely hood that the PET scan will pick something up. Has anyone had any experience where CT said no to spread but PET scan showed it had spread. 
My mum is taking it in her stride, and I keep asking if she's being honest with her symptoms and she said yes, no cough, no blood from chest, no blood in stools. 
thanks for reading.