Consider the system of equations ax + by = 0; cx + dy = 0, where `a, b, c, d in {0,1}`)STATEMENT-1: The probability that the system of equations has a

Question

Consider the system of equations ax + by = 0; cx + dy = 0, where a,b,c,d∈{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 of equations has a solution is 1 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. Select the correct answer from above options

Answer 1

Correct Answer - B 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≠0 As a,b,c,d ∈{0,1) . So, each of ,a,b,c and d can assume tow values. Therefoe, there are 24 sets aof values of a,b,c and d. Clearly ,ad -bc ∈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 24 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.