The number of distinct real values of ' K ' such that the system of equations x + 2 y + z = 1 , x + 3 y + 4 z = K , x + 5 y + 10 z = K 2 has infinitely many solutions is:

Question

The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is:

Smart Achievers · Accepted Answer

(3)

$D =$ $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ $= 0$

$D_{1} =$ $| \begin{matrix} 1 & 2 & 1 \\ K & 3 & 4 \\ K^{2} & 5 & 10 \end{matrix} |$ $= 5 (K^{2} - 3 K + 2) = 5 (K - 1) (K - 2)$

$D_{2} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & K & 4 \\ 1 & K^{2} & 10 \end{matrix} |$ $= - 3 (K^{2} - 3 K + 2) = - 3 (K - 2) (K - 1)$

$D_{3} =$ $| \begin{matrix} 1 & 2 & 1 \\ 1 & 3 & K \\ 1 & 5 & K^{2} \end{matrix} |$ $= K^{2} - 3 K + 2 = (K - 2) (K - 1)$

Solution

Q.
The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is:

1 0

2 4

3 2

4 3

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Solution

Q. The number of distinct real values of 'K' such that the system of equations x+2y+z=1, x+3y+4z=K, x+5y+10z=K2 has infinitely many solutions is:

1 0

2 4

3 2

4 3

Ans. (3) D=|1211341510|=0 D1=|121K34K2510|=5(K2-3K+2)=5(K-1)(K-2) D2=|1111K41K210|=-3(K2-3K+2)=-3(K-2)(K-1) D3=|12113K15K2|=K2-3K+2=(K-2)(K-1)

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Q.
The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is: