Advertisements
Advertisements
Question
A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission
| Month | Sale of units | Total commission drawn (in Rs) |
||
| A | B | C | ||
| Jan | 90 | 100 | 20 | 800 |
| Feb | 130 | 50 | 40 | 900 |
| March | 60 | 100 | 30 | 850 |
Find out the rates of commission on items A, B and C by using determinant method.
Advertisements
Solution
Let x, y and z be the rates of commission on items A, B and C respectively. Based on the given data, we get
\[90x + 100y + 20z = 800\]
\[130x + 50y + 40z = 900\]
\[60x + 100y + 30z = 850\]
Dividing all the equations by 10 on both sides, we get
\[9x + 10y + 2z = 80\]
\[13x + 5y + 4z = 90\]
\[6x + 10y + 3z = 85\]
\[D = \begin{vmatrix}9 & 10 & 2 \\ 13 & 5 & 4 \\ 6 & 10 & 3\end{vmatrix} \left[\text{ Expressing the equation as a determinant }\right]\]
\[ = 9(15 - 40) - 10(39 - 24) + 2(130 - 30)\]
\[ = 9( - 25) - 10(15) + 2(100)\]
\[ = - 175\]
\[ D_1 = \begin{vmatrix}80 & 10 & 2 \\ 90 & 5 & 4 \\ 85 & 10 & 3\end{vmatrix}\]
\[ = 80(15 - 40) - 10(270 - 340) + 2(900 - 425)\]
\[ = 80( - 25) - 10( - 70) + 2(475)\]
\[ = - 350\]
\[ D_2 = \begin{vmatrix}9 & 80 & 2 \\ 13 & 90 & 4 \\ 6 & 85 & 3\end{vmatrix}\]
\[ = 9(270 - 340) - 80(39 - 24) + 2(1105 - 540)\]
\[ = 9( - 70) - 80(15) + 2(565)\]
\[ = - 700\]
\[ D_3 = \begin{vmatrix}9 & 10 & 80 \\ 13 & 5 & 90 \\ 6 & 10 & 85\end{vmatrix}\]
\[ = 9(425 - 900) - 10(1105 - 540) + 80(130 - 30)\]
\[ = 9( - 475) - 10(565) + 80(100)\]
\[ = - 1925\]
Thus,
\[x = \frac{D_1}{D} = \frac{- 350}{- 175} = 2\]
\[y = \frac{D_2}{D} = \frac{- 700}{- 175} = 4\]
\[z = \frac{D_3}{D} = \frac{- 1925}{- 175} = 11\]
Therefore, the rates of commission on items A, B and C are 2, 4 and 11, respectively.
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
\[∆ = \begin{vmatrix}\cos \alpha \cos \beta & \cos \alpha \sin \beta & - \sin \alpha \\ - \sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
Without expanding, prove that
\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]
If a, b, c are real numbers such that
\[\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0\] , then show that either
\[a + b + c = 0 \text{ or, } a = b = c\]
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Using determinants show that the following points are collinear:
(2, 3), (−1, −2) and (5, 8)
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
2x − y = 17
3x + 5y = 6
9x + 5y = 10
3y − 2x = 8
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
If \[A = \begin{bmatrix}0 & i \\ i & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\] , find the value of |A| + |B|.
Write the value of
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
If \[\begin{vmatrix}2x & x + 3 \\ 2\left( x + 1 \right) & x + 1\end{vmatrix} = \begin{vmatrix}1 & 5 \\ 3 & 3\end{vmatrix}\], then write the value of x.
If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.
Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant
The other factor in the value of the determinant is
If \[x, y \in \mathbb{R}\], then the determinant
Solve the following system of equations by matrix method:
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0
The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals ______.
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
