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Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to THREE significant figures. 1. (a) Let p and q be given propositions. (i) Copy and complete the table below to show the truth tables of p → q and ~ p ∨ q. equivalent, giving reasons for your answer. [2 marks] (iii) Use the algebra of propositions to show that p ∧ ሺ ሺሺ ሺp → qሻ ሻሻ ሻ = p ∧ q. [3 marks] (b) The binary operation * is defined on the set of real numbers, ℝ, as follows: ݔ ∗ ݕ = ݔ + ݕ −1 for all x, y in ℝ. (i) is closed in ℝ, [3 marks] (ii) is commutative in ℝ, [2 marks] (iii) is associative in ℝ. [4 marks] stating the value of θ for which g (θ ) is a minimum. [4 marks] (c) Let A = (2, 0, 0), B = (0, 0, 2) and C = (0, 2, 0). (ii) Show that the vector r = i + j + k is perpendicular to the plane through A, B, and C. (iii) Hence, find the Cartesian equation of the plane through A, B and C. [3 marks] meets the curve at P(3, 27). b) the coordinates of Q. [1 mark ] a) the area of the shaded region, [5 marks] (b) (iii) ሺݔ ∗ ݕሻ ∗ ݖ = ሺݔ + ݕ −1ሻ ∗ ݖ for x, y, ݖ ∈ ℝ ⇒ ሺݔ ∗ ݕሻ ∗ ݖ = ݔ ∗ ሺݕ ∗ ݖሻ for all ݔ, ݕ, ݖ ∈ℝ = ሺܑ + ܒ +ܓሻ. ሺ2ܒ −2ܓሻ = 0 +2 −2 = 0 (1 mark) = ሺܑ + ܒ + ܓሻ. ሺ2ܑ − 2ܓሻ = 2 +0 − 2 = 0 (1 mark) So n is perpendicular to the plane through A, B and C. represent the plane through A, B and C.
Pages: 689
Publisher: Butterworth-Heinemann; 5 edition (September 11, 2000)
ISBN: 0750650494
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We may therefore write sin d = sin {(2 71 + 1) tt - ^}, and sin d = sin {2 nw + d}. We observe that when 6 has a negative sign, the multiple of TT is odd, and when 6 has a positive sign, the multiple of IT is evvn. Moreover, ( - 1)** is - or + according as 71 is odd or even Applied Finite Mathematics download epub
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