Consider a system of linear equations 3x+y−z=0, x−py4+z=2 and 2x−y+2z=q where p,q∈I and p,q∈[1,10], then which of the following statements is/are correct ?

3 x + y − z = 0 x − p y 4 + z = 0 2 x − y + 2 z = q E q n . ( 2 ) × ( 2 ) − e q n . ( 3 ) ⇒ ( 1 − p 2 ) y = 4 − q For unique solution, p ≠ 2 , q ∈ N ⇒ Number of ordered pairs ( p , q ) i n [ 1 , 10 ] are 90 For unique solution, p = 2 a n d q = 4 ⇒ exactly one ordered pair For no solution, p = 2 a n d q ≠ 4 ⇒ Number of ordered pairs ( p , q ) i n [ 1 , 10 ] a r e 9

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Consider a system of linear equations $3 x + y - z = 0, x - \frac{p y}{4} + z = 2 a n d 2 x - y + 2 z = q$ where $p, q \in I a n d p, q \in [1, 10],$ then which of the following statements is/are correct ?

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Number of ordered pairs (p, q) for which system of equation has no solution is 9.

Number of ordered pairs (p, q) for which system of equation has infinite solutions is one (1) pair.

Number of ordered pair (p, q) for which system of equation has unique solution is 90.

Number of ordered pairs (p, q) for which system of equation has at least one solution is 91.

answer is A, B, C, D.

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Detailed Solution

$3 x + y - z = 0$
$x - \frac{p y}{4} + z = 0$
$\begin{array}{l} 2 x - y + 2 z = q \\ E q n . (2) \times (2) - e q n . (3) \\ \Rightarrow (1 - \frac{p}{2}) y = 4 - q \end{array}$
For unique solution, $p \neq 2, q \in N \Rightarrow$ Number of ordered pairs $(p, q) i n [1, 10]$ are 90
For unique solution, $p = 2 a n d q = 4 \Rightarrow$ exactly one ordered pair
For no solution, $p = 2 a n d q \neq 4 \Rightarrow$ Number of ordered pairs $(p, q) i n [1, 10] a r e 9$

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