CAT 2023 Slot 3 · Quant · Question 22

The function is only ever described through the two combinations 3x+2y and 2x-5y, so name them.

Let a=3x+2y and b=2x-5y, so the given statement is f\left( a,b \right)=19x.

The task is to express 19x in terms of a and b alone.

Look for constants \alpha and \beta with \alpha a+\beta b=19x, that is \alpha \left( 3x+2y \right)+\beta \left( 2x-5y \right)=19x.

Matching the y terms, 2\alpha -5\beta =0, so \alpha =\frac{5\beta}{2}.

Matching the x terms, 3\alpha +2\beta =19.

Substituting, \frac{15\beta}{2}+2\beta =\frac{19\beta}{2}=19, so \beta =2 and \alpha =5.

Therefore f\left( a,b \right)=5a+2b, which is the function in plain form.

Now evaluate at the required point: f\left( x,2x \right)=5x+2\left( 2x \right)=9x.

Setting 9x=27 gives x=3.

Shortcut. The function must be linear, so write f\left( a,b \right)=\alpha a+\beta b and match coefficients — the y terms must cancel, which fixes the ratio \alpha :\beta =5:2 at once. Then f\left( x,2x \right)=5+4=9 times x, and the rest is one division.