Question

Number of triplets of a, b and c for which the system of equations by - ax = b - 2a and (c + 1)x + cy + a = 10 + 3b has infinitely many solutions and x = 1, y = 3 is one of the solution, is:

• A. Exactly one
• B. Exactly two
• C. Exactly three
• D. Infinitely many

Answer 1

The correct option is B Exactly two

ax - by = 2a - b
(c + 1)x + cy = 10 + 3b - a
Both the equations will satisfy (1, 3)
a - 3b = 2a - b
a = - 2b
c + 1 + 3c = 10 + 3b - a
4c = 9 + 5b
The given equations have infinite solutions so
$\left|\begin{array}{cc}a& -b\\ c+1& c\end{array}\right|=0\Rightarrow ac+bc+b=0\Rightarrow bc=b\Rightarrow b=0,c=1(a,b,c)\equiv (0,0,\frac{9}{4})or(2,-1,1)$
Exactly two values of (a, b, c)