JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 5)
Consider the system of equations $$ax+by=0; cx+dy=0,$$
where $$a,b,c,d$$ $$ \in \left\{ {0,1} \right\}$$
STATEMENT - 1 : The probability that the system of equations has a unique solution is $${3 \over 8}.$$ and
STATEMENT - 2 : The probability that the system of equations has a solution is $$1.$$
where $$a,b,c,d$$ $$ \in \left\{ {0,1} \right\}$$
STATEMENT - 1 : The probability that the system of equations has a unique solution is $${3 \over 8}.$$ and
STATEMENT - 2 : The probability that the system of equations has a solution is $$1.$$
STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is False.
STATEMENT - 1 is False, STATEMENT - 2 is True.
Explanation
We have,
$$ax + by = 0$$
$$cx + dy = 0$$
since, the system of homogeneous equation is always consistent and has a solution.
Therefore, statement 2 is true.
Now, $$\Delta = \left| {\matrix{ a & b \cr c & d \cr } } \right|$$ and $$a,b,c,d \in \{ 0,1\} $$
$$ = ad - bc$$
No. of ways of selecting $$a,b,c,d$$ from the set {0, 1} is 2 $$\times$$ 2 $$\times$$ 2 $$\times$$ 2 = 16
If the system has unique solution,
Then $$\Delta\ne0$$
$$\Rightarrow$$ either $$ad = 1,bc = 0$$ or $$ad = 0,bc = 1$$ favourable case = 6
Therefore, probability that system of equation has unique solution is $${6 \over {16}} = {3 \over 8}$$. Statement 1 is True.
Hence, the Statement 2 is True but is not a correct explanation of statement 1.
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