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Practice Questions End of Term 3
Q1.
The quadratic equation x^2 − 2 x + 3 = 0 has roots α and β. Without solving the equation, (a) (i) write down the value of ( α + β ) and the value of αβ (ii) show that α^2 + β^2 = − 2 (iii) find the value of α^3 + β^3 (5) (b) (i) show that α^4 + β^4 = ( α^2 + β^2 )^2 − 2( αβ )^2 (ii) find a quadratic equation which has roots ( α^3 − β ) and ( β^3 − α ) giving your answer in the form px^2 + qx + r = 0 where p , q and r are integers. (6)
Use algebra to find the set of values of x for which (7)
z = 3 + 2i, w = 1 − i Find in the form a + b i, where a and b are real constants, (a) zw (2) (b) , showing clearly how you obtained your answer. (3) Given that | z + k | = √53, where k is a real constant (c) find the possible values of k. (4)
f( z ) = z^4 + 4 z^3 + 6 z^2 + 4 z + a where a is a real constant. Given that 1 + 2i is a complex root of the equation f( z ) = 0 (a) write down another complex root of this equation. (1) (b) (i) Hence, find the other roots of the equation f( z ) = 0 (ii) State the value of a. (7)
(a) Describe fully the single geometrical transformation represented by the matrix A. (3) (b) Hence find the smallest positive integer value of n for which A n^ = I where I is the 2 × 2 identity matrix. (1) The transformation represented by the matrix A followed by the transformation represented by the matrix B is equivalent to the transformation represented by the matrix C. Given that C = , (c) find the matrix B. (4)
(a) Find M –^1 in terms of k. (5) Hence, given that k = 0 (b) find the matrix N such that (4)
Given that (a) find the matrix AB , (2) (b) find the exact value of k for which det( AB ) = 0 (2)
(a) Show that the equation f( x ) = 0 has a root α in the interval [1.6, 1.7] (2) (b) Taking 1.6 as a first approximation to α apply the Newton-Raphson process once to f( x ) to find a second approximation to α. Give your answer to 3 decimal places. (5)
