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?

$$x + y = 20 \\ 0.10x + 0.30y = 4.8$$

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

$$x + y = 20 \ 0.10x + 0.30y = 4.8$$

Question 2

Given the system $$x + y = 5$$ and $$2x - y = 1$$, what is the geometric interpretation of the solution?

Accepted Answer

The intersection point of two lines

Question 3

Which of the following statements about determinants is TRUE?

Accepted Answer

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

Question 4

What is the determinant of the matrix $$A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix}$$?

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

A system is said to be ill-conditioned if:

Accepted Answer

A small change in the data causes a large change in the solution.

Question 7

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

Accepted Answer

Strict diagonal dominance

Question 8

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 the value of $$x_3$$?

Accepted Answer

$$2$$

Question 9

What is the main advantage of the Thomas algorithm for tridiagonal systems?

Accepted Answer

It requires $$O(n)$$ operations and exploits the tridiagonal structure.

Question 10

In the context of iterative methods, what does the spectral radius $$\rho(T) of $$the iteration matrix $$T$$ indicate?

Accepted Answer

The method converges for every initial guess if $$\rho(T) \u003c 1.$$

Question 11

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

Accepted Answer

Cholesky factorisation

Question 12

What is the effect of over-relaxation ($$\omega \u003e 1) in $$the SOR method?

Accepted Answer

It moves the update beyond the ordinary Gauss-Seidel candidate.

Question 13

Which of the following best describes the PLU factorisation?

Accepted Answer

$$PA = LU$$, where $$P is a $$permutation matrix, $$L is $$lower triangular, and $$U is $$upper triangular.

Question 14

If a computed solution $$\hat{x} to$$ $$Ax = b$$ has a small residual $$r = b - A\hat{x}$$ but the system is ill-conditioned, what can you conclude?

Accepted Answer

The solution $$\hat{x}$$ may still have a large error.

Question 15

Which of the following is NOT a reason to prefer an iterative method over a direct method?

Accepted Answer

The system is small and dense.

Question 16

What is the main purpose of partial pivoting in Gaussian elimination?

Accepted Answer

To avoid division by zero or small pivots and improve numerical stability.

Question 17

Given $$A = \begin{bmatrix} 4 & 2 & 0 \ 4 & 4 & 2 \ 2 & 2 & 3 \end{bmatrix}$$, which method would be most efficient for solving $$Ax = b if$$ $$A is $$symmetric positive-definite?

Accepted Answer

Cholesky factorisation

Question 18

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

Accepted Answer

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

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