Here’s what I remember - family (possibly orphans) are being tested by a parapsychologist for ESP. One of the kids tries to throw the test, but gives the wrong answer for every card, so the researcher knows she has ESP.
One of the features of the book is some really excellent pencil sketches, including one of a very very close up view of a cat’s eye.
The author put out several novels along these lines - pre-teen protagonists, paranormal themes and illustrated with pencil sketches. This is the only book I remember any plot details from.
I read these 74-76* so they had to have been published before then, and IIRC, the author didn’t characterize the girls as taking on traditional female roles, so probably no earlier than the late 60s.
Final possible clue - I think the author’s last name started with a letter in the middle of the alphabet, possibly M or N
When I was in middle school, and I identified books by which shelf they were on in that library. Heck, I can still mentally locate them in that library. Which doesn’t help anyone unless they also went to Daggett Middle School in Fort Worth in the mid-70s.
I don’t recall the pencil sketches (possibly we read different editions of the same title) but I believe you’re thinking of A Gift of Magic (pub. 1971) by Lois Duncan. Nancy’s brother and sister each have a “gift” (music and dance), and Nancy’s gift is ESP. There’s a scene where investigators test her ESP using playing cards and she purposely reverses her answer for each.
I think I’ve seen the “deliberate failure of ESP test makes tester suspicious” bit a few times; it happened in Robert Silverberg’s “Dying Inside”, for instance. (Not the book the OP is looking for, though.)