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Exercises for Question Type 4: For given parameters in the linear equations, find the value of the parameter that will result in no solution, one solution or infinite number of solutions - IB | RevisionDojo

Question 1

Classify the solutions in terms of $a$ and $b$ for the system

\begin{cases} a x + y = 2 \\ 2x + a y = b \end{cases}

[6]

Question 2

Determine all $a$ for which the system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} a x + y + z = 1 \\ x + a y + z = 1 \\ x + y + a z = 1 \end{cases}

[9]

Question 3

Find all $a$ and $b$ for which the $3 \times 3$ system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} x + y + z = 1 \\ ax + y + z = 1 \\ x + ay + z = b \end{cases}

[7]

Question 4

Find all values of the parameters $a$ and $k$ for which the system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} a x + 2y = 3 \\ 2x + 4y = k \end{cases}

[6]

Question 5

Determine for which parameters the system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} x + y + z = 1 \\ 2x + a y + z = 2 \\ 3x + 2y + b z = c \end{cases}

[6]

Question 6

Find all $a$ and $k$ yielding no solution, one solution, or infinitely many solutions:

\begin{cases} (2-a)x + y = 3 \\ 4x + (2-a)y = k \end{cases}

[7]

Question 7

Find all $k$ and $b$ so that the system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} (1-k)x + y = 2 \\ 2x + (2-2k)y = b \end{cases}

[7]

Question 8

For which values of $a$ and $b$ does the system have no solution, exactly one solution, or infinitely many solutions?

\begin{cases} (a-1)x + y = 2 \\ 2x + (b-2)y = 3 \end{cases}

[6]

Question 9

Classify the solutions in terms of $a$ and $b$ for the system

\begin{cases} (a+1)x + 2y = 5 \\ 2(a+1)x + (4+b)y = 10 \end{cases}

[5]

Question 10

Determine all $a$ and $b$ for which the system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} (2a)x + 3y = 1 \\ (4a)x + by = 2 \end{cases}

[7]

Question 11

Classify all $(a,b)$ for which the following system has no solution, exactly one solution, or infinitely many solutions:

\begin{cases} (3a)x + 6y = 9 \\ 6x + (12 + b)y = 18 \end{cases}

Classify the solutions.

[6]

Question Type 4: For given parameters in the linear equations, find the value of the parameter that will result in no solution, one solution or infinite number of solutions Exercises