Advertisements
Advertisements
Question
A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.
Advertisements
Solution
Let the award money given for Honesty, Regularity and Hard work be x, y and z respectively.
Since total cash award is Rs 6,000.
∴ x + y + z = Rs 6,000 ...(1)
Three times the award money for Hard work and Honesty is Rs 11,000.
∴ x + 3 z = Rs 11,000
⇒ x + 0.y + 3 z = Rs 11,000 ...(2)
Award money for Honesty and Hard work is double the one given for regularity.
∴ x + z = 2y
⇒ x − 2y + z = 0 ...(3)
The above system can be written in matrix form as,
`[(1,1,1),(1,0,3),(1,-2,1)][(x),(y),(z)]=[(6000),(11000),(0)]`
Or AX = B, where
`A=[(1,1,1),(1,0,3),(1,-2,1)], X=[(x),(y),(z)] and B =[(6000),(11000),(0)]`
`|A|=6!=0`
Thus, A is non-singular. Hence, it is invertible.
Adj A = `[(6,-3,3),(2,0,-2),(-2,3,-1)]`
`thereforeA^-1=1/|A|(adjA)=1/6[(6,-3,3),(2,0,-2),(-2,3,-1)]`
`X = A^-1B=1/6[(6,-3,3),(2,0,-2),(-2,3,-1)][(6000),(11000),(0)]`
`=>[(x),(y),(z)]=[(500),(2000),(3500)]`
`Hence, x= 500, y=2000, andz=3500`
Thus, award money given for Honesty, Regularity and Hard work are Rs 500, Rs 2000 and Rs 3500 respectively.
School can include sincerity for awards.
APPEARS IN
RELATED QUESTIONS
If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of x.
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Solve the system of linear equations using the matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Evaluate
\[\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\] by two methods.
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}a + b & 2a + b & 3a + b \\ 2a + b & 3a + b & 4a + b \\ 4a + b & 5a + b & 6a + b\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}\]
Prove the following identities:
\[\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Using determinants prove that the points (a, b), (a', b') and (a − a', b − b') are collinear if ab' = a'b.
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Prove that :
Prove that :
2x − y = 17
3x + 5y = 6
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
x + 2y = 5
3x + 6y = 15
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0
Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = −1
Solve the following system of equations by matrix method:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0
Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13
Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
If \[A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\] ,find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7.
The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______
System of equations x + y = 2, 2x + 2y = 3 has ______
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 has non-trivial solution, then the value of k is ____________.
Let `θ∈(0, π/2)`. If the system of linear equations,
(1 + cos2θ)x + sin2θy + 4sin3θz = 0
cos2θx + (1 + sin2θ)y + 4sin3θz = 0
cos2θx + sin2θy + (1 + 4sin3θ)z = 0
has a non-trivial solution, then the value of θ is
______.
The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ2 – |λ|z) = 16 has no solution, is ______.
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.
