Advanced Level Mathematics Revision: Pure Mathematics & Combinatorics

Mastering structural algebraic equations, series expansions, and vector geometry is essential for succeeding in Advanced Level Pure Mathematics. Test your problem-solving skills with these detailed structured problems from image_35487e.jpg.

1. Polynomials & Rational Inequalities

Question 1

• (i) The polynomial f(x), where f(x) = 2x^3 + ax^2 + bx – 3, is exactly divisible by (x – 1) and has a remainder of 9 when divided by (x + 2). Find the values of the constants a and b. Hence solve the equation f(x) = 0. (7 marks)

• (ii) Find the set of real values of x for which:

  2 / (x – 2) < 1 / (x + 1)

  (4 marks)

2. Complex Numbers & Quadratic Equations

Question 2

• (i) If the roots of the quadratic equation ax^2 + 2bx + c = 0 are alpha and alpha + 4, prove that 4a^2 + ca – b^2 = 0. Hence show also that 8a = -c +/- square_root(c^2 + 16b^2). (6 marks)

• (ii) Given that beta = -1 + 2i, express beta^2 and beta^3 in the form a + b*i, where a and b belong to the set of real numbers. Hence show that beta is a root of the equation z^3 + 7z^2 + 15z + 25 = 0 and state a second complex root of this equation. (6 marks)

3. Binomial Expansions & Relation Theory

Question 3

• (i) The first three terms in the expansion of (1 + b*x)^n in ascending powers of x are:

  1 – (3/5)*x – (27/100)*x^2

  Find the values of the constants b and n. (5 marks)

• (ii) Consider the relation R on a non-empty set A. State the conditions that would make R a partial order on A. Given that A = {1, 2, 3, 4, 6, 12} and that R is the relation defined by:

  a R b <=> a divides b

  show that R defines a partial order on A. (6 marks)

4. Vector Geometry & Combinatorics

Question 4

The vector equations of two lines L1 and L2 are:

• L1: r = (j – k) + lambda*(i + 2j + k)

• L2: r = (i + 7j – 4k) + mu*(i + 3k), where lambda and mu are scalar variables.

Tasks:

• (a) Find the values of lambda and mu for which L1 and L2 intersect, stating the position vector of the point of intersection. (4 marks)

• (b) Find the Cartesian equation of the plane containing L1 and L2. (4 marks)

Question 5

A bus driver has 8 passengers to carry but he only has room for 4. In how many ways can he choose the 4 passengers, if two of them are sisters who must not be separated? (3 marks)

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