CAT 2022 Slot 2 · Quant · Question 15

Let c be the correct answers, w the wrong ones and u the unattempted.

Every question falls into one of those, so c+w+u=75.

The scoring gives 3c-w+u=97.

Subtracting the first equation from the second removes u entirely: 2c-2w=22.

So c-w=11, that is w=c-11.

The attempted questions number c+w=2c-11.

So the unattempted number u=75-\left( 2c-11 \right)=86-2c.

The condition is that unattempted exceed attempted: 86-2c>2c-11.

Rearranging, 97>4c, so c<24.25.

Since c is a whole number, the largest value is c=24.

Checking at c=24: w=13, u=86-48=38, and the attempted total is 37, so 38>37 holds.

The score is 72-13+38=97, exactly as required.

Shortcut. Subtracting the count equation from the score equation is what makes this quick — the unattempted term has coefficient 1 in both, so it cancels and leaves the clean relation c-w=11. Everything else follows in one variable.