Matrices & Determinants
System of Linear Equations
Grade Class 12
Question:
If system of equation a<sub>1</sub>x + b<sub>1</sub>y = c<sub>1</sub> & a<sub>2</sub>x + b<sub>2</sub>y = c<sub>2</sub> (where a<sub>1</sub>, b<sub>1</sub>, c<sub>1</sub>, a<sub>2</sub>, b<sub>2</sub>, c<sub>2</sub> ≠ 0) has infinite solutions, then-<br>(A) a<sub>1</sub>/a<sub>2</sub> = b<sub>1</sub>/b<sub>2</sub> = c<sub>1</sub>/c<sub>2</sub><br>(B) (a<sub>1</sub>+a<sub>2</sub>)/(a<sub>1</sub>-a<sub>2</sub>) = (b<sub>1</sub>+b<sub>2</sub>)/(b<sub>1</sub>-b2) = (c<sub>1</sub>+c<sub>2</sub>)/(c<sub>1</sub>-c<sub>2</sub>)<br>(C) the quadratic equations a<sub>1</sub>x<sup>2</sup> + b<sub>1</sub>x + c<sub>1</sub> = 0 & a<sub>2</sub>x<sup>2</sup> + b<sub>2</sub>x + c<sub>2</sub> = 0 have no common root<br>(D) system of equation a<sub>1</sub><sup>2</sup> a<sub>2</sub>x + b<sub>1</sub><sup>2</sup> b<sub>2</sub>y = c<sub>1</sub><sup>2</sup> c<sub>2</sub> & a<sub>1</sub> a<sub>2</sub><sup>2</sup>x + b<sub>1</sub> b<sub>2</sub><sup>2</sup>y = c<sub>1</sub> c<sub>2</sub><sup>2</sup> will also have infinite number of solutions
(A) a<sub>1</sub>/a<sub>2</sub> = b<sub>1</sub>/b<sub>2</sub> = c<sub>1</sub>/c<sub>2</sub>
(B) (a<sub>1</sub>+a<sub>2</sub>)/(a<sub>1</sub>-a<sub>2</sub>) = (b<sub>1</sub>+b<sub>2</sub>)/(b<sub>1</sub>-b<sub>2</sub>) = (c<sub>1</sub>+c<sub>2</sub>)/(c<sub>1</sub>-c<sub>2</sub>)
(C) the quadratic equations a<sub>1</sub>x<sup>2</sup> + b<sub>1</sub>x + c<sub>1</sub> = 0 & a<sub>2</sub>x<sup>2</sup> + b<sub>2</sub>x + c<sub>2</sub> = 0 have no common root
(D) system of equation a<sub>1</sub><sup>2</sup> a<sub>2</sub>x + b<sub>1</sub><sup>2</sup> b<sub>2</sub>y = c<sub>1</sub><sup>2</sup> c<sub>2</sub> & a<sub>1</sub> a<sub>2</sub><sup>2</sup>x + b<sub>1</sub> b<sub>2</sub><sup>2</sup>y = c<sub>1</sub> c<sub>2</sub><sup>2</sup> will also have infinite number of solutions
Step-by-Step Solution
Key Concept: For a system of two linear equations in two variables to have infinite solutions, the ratios of the coefficients must be equal (a1/a2 = b1/b2 = c1/c2 = k). This implies the equations are proportional. Option (C) is the correct statement because if the equations are proportional, they represent the same line, and thus any root of one is a root of the other, meaning they have common roots, not no common roots.
For infinite solutions, a<sub>1</sub>/a<sub>2</sub> = b<sub>1</sub>/b<sub>2</sub> = c<sub>1</sub>/c<sub>2</sub> = k. This means a<sub>1</sub> = ka<sub>2</sub>, b<sub>1</sub> = kb<sub>2</sub>, c<sub>1</sub> = kc<sub>2</sub>. Substituting these into the quadratic equations: a<sub>1</sub>x<sup>2</sup> + b<sub>1</sub>x + c<sub>1</sub> = k(a<sub>2</sub>x<sup>2</sup> + b<sub>2</sub>x + c<sub>2</sub>) = 0. Since k ≠ 0, the roots are identical, so they have common roots. Thus, (C) is the correct statement as it is false.
Correct Answer: C