Linear Algebra

Q5. By completing the square, sketch the following quadratic curve:

Q7. The graphs of three quadratic functions f, g and h are shown.

Q8. (1) Construct a quadratic equation with roots 1 and -4. (ii) Construct a quadratic equation with roots x = 5 and x = write your answer in the form of ax² + bx+c = 0, where a, b, c € Z.

Q9. (1) Solve the equation x² 18x + 72 = 0 - (ii) Hence, find the solutions of the equation (t² +t)² − 18(t² + t) + 72 = 0

Q10. The diagram shows a graph of the function f(x) = 5x³ - 24x² + bx + 4. The graph of the function crosses the x-axis at the points a, 1 and 4. Determine the values of a and b. [It is not sufficient to guess a value for a (by eye) - you must show your workings].

3. Quadratic Functions - Part 2 (S7) A. Use algebra to find the zeros of the following functions. (write as points in fractional form, no decimals). 1) x² - 2x - 48 2) 3x²+2x+1 B. If there is a quadratic function that has roots at x = 2 and x = 7, write a possible equation of this function? If there can be more than one possible equation, provide another equation for this function. If there cannot be more than one possible equation, explain why not.

1. Let T: R¹ R³ be the linear transformation given by the formula T (a) What is the standard matrix of T? (b) Find all vectors in R4 such that T(z) =

9. (a) A cuboid has a total surface area of 150 cm² and is such that its base is a square of side x cm. Let the height of the cuboid be h cm. Express h in terms of z. Express the volume, V cm³, of the cuboid in terms of x. Hence, determine, as x varies, its maximum volume and show that this volume is a maximum. (b) The volume of a solid right circular cylinder of radius r cm is 432 cm³. Let S cm² be the total surface area of the cylinder. Express S in terms of r. Calculate the value of r for which S has a stationary value. Determine whether this value of r makes the surface area a maximum or a minimum, and find the corresponding value of the height of the cylinder.

Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v, w. (a) Give a basis for the nullspace and a basis for the column space. (b) Find a particular solution to Ax=v+w. Also, find all solutions to Ax=v+w.

Problem 2. Give an example to show that the eigenvalues can be changed when a multiple of one row is subtracted from another. Why is a zero eigenvalue not changed by the steps of elimination?