Advanced Level Mathematics Practice: Calculus, Vectors & Number Theory

Mastering structural mathematical proofs requires solid algebraic manipulation and a strong grasp of core calculus techniques. Test your problem-solving skills with these selected questions from image_2320eb.jpg.

1. Vector Geometry & Linear Algebra

Question 1:

The vectors a = i – 4j + 2k, b = pi – 3j + qk and c = pi + j – qk, where p and q belong to the set of real numbers, are such that vector a is perpendicular to both vector b and vector c.

• (i) Find the values of the scalars p and q. (5 marks)

• (ii) With these values of p and q, find the Cartesian equation of the plane containing the position vectors a, b and c. (4 marks)

2. Differential Equations

Question 2:

Using the substitution y = x*v, where v is a function of x, show that the differential equation:

x * (dy/dx) = x^2 – xy + y – 2y^2

can be transformed into the differential equation:

(dv/dx) = 1 – v – 2*v^2 (3 marks)

Hence find the solution in the form x = f(v) for which y = 1 when x = 0. (4 marks)

3. Number Theory & Group Theory

Question 3:

• (a) Use the Chinese Remainder Theorem, or otherwise, to solve the system of congruences below from image_2320eb.jpg:

  • x = 5 mod 7

  • x = 3 mod 11

  • x = 8 mod 15 (5 marks)

• (b) Show that C, the set of all non-zero complex numbers under multiplication, is a group. (5 marks)

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