Tag: system of simultaneous equations

Questions Related to system of simultaneous equations

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

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

  1. $0$

  2. $1$

  3. $2$

  4. $infinite$

Show answer
Correct Option: B
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, $k=1$

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)

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

  1. $2$

  2. $3$

  3. $0$

  4. $1$

Show answer
Correct Option: C
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 $3$

The system of linear equations$X-Y+Z=1$$X+Y-Z=3$$X-4Y+4Z=\alpha $ has:

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

  1. A unique solution when $\alpha =2$

  2. A unique solution when $\alpha \neq 2$

  3. An infinite number of solutions, when $\alpha =2$

  4. An infinite number of solution, when $\alpha =-2$

Show answer
Correct Option: A

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$

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

  1. $x = n\pi$

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

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

  4. None of these

Show answer
Correct Option: A
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 $

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

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

  1. $a=1, b\neq 9$

  2. $a \neq-1, b=9$

  3. $a=-1, b=9$

  4. $a=-1, b \neq 9$

Show answer
Correct Option: A

If $A,B,C$ are the angles of a triangle, the system of equations, $(\sin A)x+y+z=\cos Ax+(\sin B)y+z=\cos B$
$x+y+(\sin C)z=1-\cos C$ has 

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

  1. No solutions

  2. Unique solution

  3. Infinitely many solutions

  4. Finitely many solutions

Show answer
Correct Option: A

The number of solutions of the equation $3x+3y-z=5,\ x+y+z=3,\ 2x+2y-z=3$

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

  1. $1$

  2. $0$

  3. $infinite$

  4. $Two$

Show answer
Correct Option: B

If the system of equation $x-ky-z=0,kx-y-z=0,x+y-z$ has a non-zero solution, the possible values of $k$ are

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

  1. $-1,2$

  2. $-1,1$

  3. $1,2$

  4. $0,1$

Show answer
Correct Option: A

The straight lines
$\left.\begin{matrix}
2kx-2y+3=0\
x+ky+2=0\
2x+k=0
\end{matrix}\right}k\in R$  pass through the same point for

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

  1. no real value of $k$

  2. exactly one real value of $k$

  3. three real values of $k$

  4. all real values of $k$

Show answer
Correct Option: B
Explanation:

Given equation of lines passes through same point i.e. lines are concurrent
$kx-2y+3=0\ 
x+ky+2=0\ 
2x+k=0$
$\Rightarrow \left| \begin{matrix} 2k & -2 & 3 \ 1 & k & 2 \ 2 & 0 & k \end{matrix} \right| =0$
$\Rightarrow k^{3}-2k-4=0$
$\Rightarrow (k-2)(k^2+2k+2)=0$
The discriminant of the quadratic expression is negative.
Hence there is only one real value of $k$

The system of equations

$\displaystyle 
\begin{matrix}kx +y+z=1&  & \
 x+ky+z=k&  & \
 x+y+kz=k^{2}&  &
\end{matrix}$
have no solution,if k equals ?

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

  1. 0

  2. 1

  3. -1

  4. -2

Show answer
Correct Option: D
Explanation:

$\Delta =\begin{vmatrix} k & 1 & 1 \ 1 & k & 1 \ 1 & 1 & k \end{vmatrix}$

$\Rightarrow \Delta =(k-1)^{2}(k+2)$

Taking $\Delta=0$
$\Rightarrow (k-1)^{2}(k+2)=0$
$\Rightarrow k=1, k=-2$
At these values of k, system can have either no solution or infinitely many solution.

For k=1, equations takes the form $x+y+z=1$
Hence, infinitely many solution for k=1.

For k=-2,
$D _{1}=\begin{vmatrix} 1 & 1 & 1 \ -2 & -2 & 1 \ 4 & 1 & -2 \end{vmatrix}$
$D _{1}=9 \ne 0$

So, $D=0$, at least one $D _{1}\ne 0$
Hence, system has no solution for k=-2