PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

3x + y = 5
− 6x − 2y = 9

3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1

3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.

x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1

x + 2y = 5
3x + 6y = 15

x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0

2x + y − 2z = 4
x − 2y + z = − 2
5x − 5y + z = − 2

x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission 

Find out the rates of commission on items A, B and C by using determinant method.

An automobile company uses three types of steel S₁, S₂ and S₃ for producing three types of cars C₁, C₂and C₃. Steel requirements (in tons) for each type of cars are given below : 

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0

Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0

Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0

Find the real values of λ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions
\[2 \lambda x - 2y + 3z = 0\] 
\[ x + \lambda y + 2z = 0\] 
\[ 2x + \lambda z = 0\]

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 prove that ab + bc + ca = abc.

For what value of x, the following matrix is singular?

Write the value of the determinant 
\[\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .\]

State whether the matrix 
\[\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}\] is singular or non-singular.

Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]