Consider the system of linear equation ax+by=0, cx+dy=0, a, b, c, d `in` {0, 1}
Statement-1 : The probability that the system of equations has a unique solution is `3/8`.
Statement-2 : The probability that the system has a solution is 1.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

B

Explanation

The given system of equation s is a homogenous system of equations which is always consistent. So, the propability that the system has a solution is 1.
The given system of equations will have a unieque solution iff
` |{:(a,b),(c,d):}| = ad - bc ne 0`
As a,b,c,d ` in { 0,1) ` . So, each of ,a,b,c and d can assume tow values. Therefoe, there are ` 2^(4)` sets aof values of a,b,c and d.
Clearly ,ad -bc ` in 0 ` iff ad=1 and bc=0 or ad =0 and bc =1
Now, ad=1 and bc=0 iff ( a =1, d=1 and b=1 ,c =0)
or ( a=1 , d =1 and b=0,c =1 ) or a =1 ,d =1 and b=0, c=0)
so, there are three sets of values of a,b,c,d satisfyfing ad =1 and bc=0
Similarly, there sets of vallues of a,b,c,d satisfying bc=1 and ad=0
Thus out of ` 2^(4)` sets of values of a,b,c and d. therefore, six sets for which the given system has a unique solution
Probability that the system has a unique soltution = `6/16 = 3/8`
Hence, statement -1 is true. But , statement -2 is not a correct explanation for statement -1.