A system of linear simultaneous equations is given as $$AX=b$$ where $$$$A = left[ { matrix{ 1 & 0 & 1 & 0 cr 0 & 1 & 0 & 1 cr 1 & 1 & 0 & 0 cr 0 & 0 & 0 & 1 cr } } ight] , , & , ,b = left[ { matrix{ 0 cr 0 cr 0 cr 1 cr } } ight]$$$$ Which of the following statement is true?

$$x$$ has infinitely many values

The system of simultaneous linear equations $$ begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 & 2 x+3 y+z=6 ext { has } end{aligned} $$

unique solution having $$z= frac{1}{2}$$.

unique solution having $$z=2$$.

If $$A X=D$$ represents the system of simultaneous linear equations $$x+y+z=6, 5 x-y+2 z=3$$ and $$2 x+y-z=-5$$, then (Adj $$A$$) $$D=$$

$$ left[ begin{array}{c}8 -16 40 end{array} ight]$$

$$ left[ begin{array}{c}32 64 -160 end{array} ight]$$

$$ left[ begin{array}{c}-16 32 80 end{array} ight]$$

$$ left[ begin{array}{l}12 24 60 end{array} ight]$$

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Simultaneous linear differential equations

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Question 1

A system of linear simultaneous equations is given as [Tex]AX=b[/Tex]
where [Tex][/Tex]A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right][Tex][/Tex]

Which of the following statement is true?

[Tex]x[/Tex] is a null vector
[Tex]x[/Tex] is unique
[Tex]x[/Tex] does not exist
[Tex]x[/Tex] has infinitely many values

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Question 2

The system of simultaneous linear equations

[Tex] \begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 \\ & 2 x+3 y+z=6 \text { has } \end{aligned} [/Tex]

infinitely many solutions.
no solution.
unique solution having [Tex]z=2[/Tex].
unique solution having [Tex]z=\frac{1}{2}[/Tex].

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Question 3

If [Tex]A X=D[/Tex] represents the system of simultaneous linear equations [Tex]x+y+z=6, 5 x-y+2 z=3[/Tex] and [Tex]2 x+y-z=-5[/Tex], then (Adj [Tex]A[/Tex]) [Tex]D=[/Tex]

[Tex]\left[\begin{array}{c}8 \\ -16 \\ 40\end{array}\right][/Tex]
[Tex]\left[\begin{array}{c}32 \\ 64 \\ -160\end{array}\right][/Tex]
[Tex]\left[\begin{array}{c}-16 \\ 32 \\ 80\end{array}\right][/Tex]
[Tex]\left[\begin{array}{l}12 \\ 24 \\ 60\end{array}\right][/Tex]

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Question 4

Simultaneous linear differential equations are linear only when:

Coefficients are constants
Coefficients are functions of t
x and y appear with power 1 (not multiplied together)
Both A and C

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Question 5

Which of the following methods is commonly used to solve simultaneous linear differential equations?

Separation of variables
Laplace transform
Elimination or matrix (eigenvalue) method
Integrating factor

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Question 6

Simultaneous linear differential equations involve:

One dependent and one independent variable
Two or more dependent variables with one independent variable
Two or more independent variables
None of these

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Mathematics

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