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Multiple choice business maths applications of matrices and determinants non-homogeneous linear equations system of simultaneous equations matrices

The system of equation $5x+2y=4$,$7x+3y=5$ are inconsistent.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$5x + 2y - 4 = 0$, $7x + 3y - 5 = 0$
$\dfrac{{{a _1}}}{{{a _2}}} = \dfrac{5}{7}$
$\dfrac{{{b _1}}}{{{b _2}}} = \dfrac{2}{3}$
$\dfrac{{{c _1}}}{{{c _2}}} = \dfrac{{ - 4}}{{ - 5}} = \dfrac{4}{5}$
Since $\dfrac{{{a _1}}}{{{a _2}}} \ne \dfrac{{{b _1}}}{{{b _2}}}$
The system of equations has unique solution
Therefore it is consistent
Thus it is false
Multiple choice business maths applications of matrices and determinants non-homogeneous linear equations system of simultaneous equations matrices

The number of values of $k$ for which the system of equations 
$(k+1)x+8y = 4 $
$kx+(k+3)y = 3k-1$
has infinitely many solutions is

  1. $0$
  2. $1$
  3. $2$
  4. $infinite$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For infinitely many solutions,
$\displaystyle \frac { k+1 }{ k } =\frac { 8 }{ k+3 } =\frac { 4 }{ 3k-1 } $

$\Rightarrow k^{2}-4k+3=0 $ and $24k-8=4k+12$
$\Rightarrow k=3,k=1$ and $k=1$

Hence, $k=1$

Multiple choice business maths applications of matrices and determinants non-homogeneous linear equations system of simultaneous equations matrices

If $f(x),g(x)$ and $h(x)$ are three polynomials of degree $2$ and $\Delta(x) \left| \begin{matrix} f\left( x \right)  \ f'\left( x \right)  \ f"\left( x \right)  \end{matrix}\begin{matrix} g\left( x \right)  \ g'\left( x \right)  \ g"\left( x \right)  \end{matrix}\begin{matrix} h\left( x \right)  \ h'\left( x \right)  \ g"\left( x \right)  \end{matrix} \right|$ then polynomial of degree (whenever defined)

  1. $2$
  2. $3$
  3. $0$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\triangle \left( x \right) =\begin{vmatrix} f\left( x \right)  & g\left( x \right)  & h\left( x \right)  \\ f^{ ' }\left( x \right)  & g^{ ' }\left( x \right)  & h^{ ' }\left( x \right)  \\ f^{ '' }\left( x \right)  & g^{ '' }\left( x \right)  & h^{ '' }\left( x \right)  \end{vmatrix}$
$\Rightarrow f\left( x \right) \left[ g^{ ' }\left( x \right) .h^{ '' }\left( x \right) -h^{ ' }\left( x \right) g^{ '' }\left( x \right)  \right] -g\left( x \right) \left[ f^{ ' }\left( x \right) h^{ '' }\left( x \right) -h^{ ' }\left( x \right) f^{ '' }\left( x \right)  \right] +h\left( x \right) \left[ f^{ ' }\left( x \right) g^{ '' }\left( x \right) -g^{ ' }\left( x \right) f^{ '' }\left( x \right)  \right] $
$\because f\left( x \right) ,g\left( x \right) ,h\left( x \right) $ are polynomial of degree $2$
$\therefore f^{ ' }\left( x \right) ,g^{ ' }\left( x \right) ,h^{ ' }\left( x \right) $ are of degree $1$
$\therefore f^{ '' }\left( x \right) ,g^{ '' }\left( x \right) ,h^{ '' }\left( x \right) $ are of degree $0$
$\therefore \triangle \left( x \right) $ is polynomial of degree $3$
Multiple choice business maths applications of matrices and determinants non-homogeneous linear equations system of simultaneous equations matrices

Solve the equation for $x$.
$\left|\begin{matrix} a^2 & a & 1 \ \sin(n+1)x & \sin{nx} & \sin(n-1)x \ \cos(n+1)x & \cos{nx} & \cos(n-1)x\end{matrix}\right| = 0$. Given that $ a>0$

  1. $x = n\pi$
  2. $x = (n-1)\pi$
  3. $x = (n+1)\pi$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\left|\begin{matrix} a^2 & a & 1 \ \sin(n+1)x & \sin{nx} & \sin(n-1)x \ \cos(n+1)x & \cos{nx} & \cos(n-1)x\end{matrix}\right| = 0$
 
$ \Rightarrow \displaystyle { a }^{ 2 }\left[ \sin { nx } .\cos { \left( n-1 \right) x } -\cos { nx } .\sin { \left( n-1 \right) x }  \right] +a\left[ \sin  (n-1)x.\cos  (n+1)x-\cos  (n-1)x.\sin  (n+1)x \right] \ +1\left[ \sin  (n+1)x.\cos { nx } -\cos  (n+1)x.\sin { nx }  \right] =0$

$ \displaystyle \Rightarrow a^{ 2 }\sin  x-a\sin  2x+\sin  x=0\ \displaystyle \Rightarrow \sin  x\left( { a }^{ 2 }+1-2a\cos { x }  \right) =0\ \displaystyle \Rightarrow \sin  x=0\; { or }\; \cos { x } =\frac { { a }^{ 2 }+1 }{ 2a } \ \displaystyle \Rightarrow \sin  x=0\; { or }\; \cos { x } =1\quad \left( \because { a }^{ 2 }+1\ge 2a,\; a>0\Rightarrow \frac { { a }^{ 2 }+1 }{ 2a } \ge 1 \right) \ \displaystyle \Rightarrow x=n\pi \; or\; x=2n\pi, \; n \in I \ \displaystyle\Rightarrow x=n\pi $

Multiple choice business maths applications of matrices and determinants non-homogeneous linear equations system of simultaneous equations matrices

If the system of linear equations 
$x+ay+z=3$
$x+2y+2z=6$
$x+5y+3z=b$
Has infinitely many solutions, then 

  1. $a=1, b\neq 9$
  2. $a \neq-1, b=9$
  3. $a=-1, b=9$
  4. $a=-1, b \neq 9$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For the system to have infinitely many solutions, the augmented matrix must lead to a row of zeros equal to zero (0 = 0). This occurs when a = -1 and b = 9.