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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.
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the system of equations ax + by = 0, cx + dy = 0, where a, b,...
Favourable number fo caes is 6 (either ad = 1, bc =0 or ad = 0, bc =1). The probability that system of equation has unique solution is 6/16 = 3/8. Since x = 0 satisfies both the equations, the system has at least one solution.
Most Upvoted Answer
Consider the system of equations ax + by = 0, cx + dy = 0, where a, b,...
Given:
System of equations: ax + by = 0, cx + dy = 0
a, b, c, d ∈ {0, 1}
To find:
The probability that the system of equations has a unique solution.
Solution:
Let's analyze each statement separately.
Statement-1: The probability that the system of equations has a unique solution is 3/8.
To find the probability of a unique solution, we need to analyze the different cases that can occur.
Case 1: a = 0, b = 0, c = 0, d = 0
In this case, both equations become 0x + 0y = 0, which is always true. There are infinitely many solutions in this case.
Case 2: a = 1, b = 1, c = 1, d = 1
In this case, both equations become 1x + 1y = 0 and 1x + 1y = 0, which are equivalent equations. There are infinitely many solutions in this case.
Case 3: a = 0, b = 0, c = 1, d = 1
In this case, the first equation becomes 0x + 0y = 0, which is always true. The second equation becomes 1x + 1y = 0, which has a unique solution (x = 0, y = 0).
Case 4: a = 1, b = 0, c = 0, d = 1
In this case, the first equation becomes 1x + 0y = 0, which has a unique solution (x = 0, y = 0). The second equation becomes 0x + 1y = 0, which also has a unique solution (x = 0, y = 0).
Case 5: All other possible combinations of a, b, c, d
In these cases, either one of the equations becomes 0x + 0y = 0 or both equations become equivalent equations (1x + 1y = 0). In both cases, there are infinitely many solutions.
Out of the 16 possible combinations of a, b, c, d, only 2 cases have a unique solution (Case 3 and Case 4). Therefore, the probability of a unique solution is 2/16 = 1/8.
Hence, Statement-1 is false.
Statement-2: The probability that the system of equations has a solution is 1.
The system of equations will always have a solution because even in the worst-case scenario (Case 1 where a = 0, b = 0, c = 0, d = 0), the equations are always satisfied. Therefore, Statement-2 is true.
Conclusion:
Statement-1 is false, and Statement-2 is true. However, Statement-2 is not a correct explanation for Statement-1.
Therefore, the correct answer is option 'B'.
System of equations: ax + by = 0, cx + dy = 0
a, b, c, d ∈ {0, 1}
To find:
The probability that the system of equations has a unique solution.
Solution:
Let's analyze each statement separately.
Statement-1: The probability that the system of equations has a unique solution is 3/8.
To find the probability of a unique solution, we need to analyze the different cases that can occur.
Case 1: a = 0, b = 0, c = 0, d = 0
In this case, both equations become 0x + 0y = 0, which is always true. There are infinitely many solutions in this case.
Case 2: a = 1, b = 1, c = 1, d = 1
In this case, both equations become 1x + 1y = 0 and 1x + 1y = 0, which are equivalent equations. There are infinitely many solutions in this case.
Case 3: a = 0, b = 0, c = 1, d = 1
In this case, the first equation becomes 0x + 0y = 0, which is always true. The second equation becomes 1x + 1y = 0, which has a unique solution (x = 0, y = 0).
Case 4: a = 1, b = 0, c = 0, d = 1
In this case, the first equation becomes 1x + 0y = 0, which has a unique solution (x = 0, y = 0). The second equation becomes 0x + 1y = 0, which also has a unique solution (x = 0, y = 0).
Case 5: All other possible combinations of a, b, c, d
In these cases, either one of the equations becomes 0x + 0y = 0 or both equations become equivalent equations (1x + 1y = 0). In both cases, there are infinitely many solutions.
Out of the 16 possible combinations of a, b, c, d, only 2 cases have a unique solution (Case 3 and Case 4). Therefore, the probability of a unique solution is 2/16 = 1/8.
Hence, Statement-1 is false.
Statement-2: The probability that the system of equations has a solution is 1.
The system of equations will always have a solution because even in the worst-case scenario (Case 1 where a = 0, b = 0, c = 0, d = 0), the equations are always satisfied. Therefore, Statement-2 is true.
Conclusion:
Statement-1 is false, and Statement-2 is true. However, Statement-2 is not a correct explanation for Statement-1.
Therefore, the correct answer is option 'B'.
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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.Correct answer is option 'B'. Can you explain this answer?
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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.Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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.Correct answer is option 'B'. Can you explain this answer?.
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