Q: A laboratory must prepare 20 litres of a mixture using a 10% solution and a 30% solution. The final mixture must contain 4.8 litres of dissolved substance. Which system of equations correctly models this scenario?

$$\begin{cases} x + y = 20 \\ 0.10x + 0.30y = 4.8 \end{cases}$$

Question 1

A laboratory must prepare 20 litres of a mixture using a 10% solution and a 30% solution. The final mixture must contain 4.8 litres of dissolved substance. Which system of equations correctly models this scenario?

Accepted Answer

$$\begin{cases} x + y = 20 \ 0.10x + 0.30y = 4.8 \end{cases}$$

Question 2

Given the system $$\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$$, what is the geometric interpretation of its solution?

Accepted Answer

The intersection point of two lines in the plane.

Question 3

Which of the following statements about determinants is TRUE for a square matrix $$A$$?

Accepted Answer

If $$\det(A) = 0$$, the system $$Ax = b$$ does not have a unique solution.

Question 4

Consider the matrix $$A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix}. $$What is $$\det(A)$$?

Accepted Answer

$$-3$$

Question 5

Which operation does NOT preserve the solution set of a system of equations?

Accepted Answer

Multiplying an equation by zero.

Question 6

What is the main advantage of using the augmented matrix form when performing elimination?

Accepted Answer

It systematically records equation operations without repeating variable names.

Question 7

Given the upper triangular system $$\begin{bmatrix} 2 & -1 & 1 \ 0 & 3 & -2 \ 0 & 0 & 4 \end{bmatrix} \begin{bmatrix} x_1$ \ x_2$ \ x_3$ \end{bmatrix} = \begin{bmatrix} 5 \ 1 \ 8 \end{bmatrix}$$, what is $$x_3$$?

Accepted Answer

$$2$$

Question 8

Why is partial pivoting used in Gaussian elimination?

Accepted Answer

To avoid dividing by zero or very small numbers, improving numerical stability.

Question 9

Which of the following is a sufficient condition for convergence of the Jacobi and Gauss–Seidel methods?

Accepted Answer

Strict diagonal dominance of the coefficient matrix.

Question 10

Suppose you have a tridiagonal system. Which algorithm is most efficient for solving it?

Accepted Answer

Thomas algorithm

Question 11

If a system is ill-conditioned, what does this imply about its solutions?

Accepted Answer

Small changes in the data can cause large changes in the solution.

Question 12

What does the condition number $$\kappa(A) of a $$matrix measure?

Accepted Answer

The sensitivity of the solution to changes in the data.

Question 13

Which of the following best describes the difference between Jacobi and Gauss–Seidel methods?

Accepted Answer

Jacobi uses only old values for all updates; Gauss–Seidel uses new values as soon as they are available.

Question 14

Given $$A = \begin{bmatrix} 4 & 2 & 0 \ 4 & 4 & 2 \ 2 & 2 & 3 \end{bmatrix}$$, which factorisation method is being used if the diagonal of $$L is $$not all ones?

Accepted Answer

Crout factorisation

Question 15

For the system $$4x_1 - x_2$ = 3$$, $$-x_1$ + 3x_2 = 2$$, what is the Jacobi iteration formula for $$x_1$$?

Accepted Answer

$$x_1$$^{(k+1)} = \frac{3 + $$x_2$$^{(k)}}{4}$$

Question 16

What is the main purpose of calculating the residual $$r = b - Ax$$ after solving a system?

Accepted Answer

To check how well the computed solution satisfies the original equations.

Question 17

Which of the following is TRUE about the spectral radius $$\rho(T) of an $$iteration matrix $$T$$?

Accepted Answer

If $$\rho(T) \u003c 1$$, the iterative method converges for any initial guess.

Question 18

In the Successive Over-Relaxation (SOR) method, what does a relaxation parameter $$\omega \u003e 1$$ represent?

Accepted Answer

Over-relaxation, which may accelerate convergence.

Question 19

Which method is most appropriate for a symmetric positive-definite matrix?

Accepted Answer

Cholesky factorisation

Question 20

Why must every numerical answer be verified by calculating a residual?

Accepted Answer

To ensure the computed solution actually satisfies the original equations within tolerance.

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