Question 3 10 marks You should be able to answer this question after studying Unit 9....
Question 4 10 marks You should be able to answer this question after studying Unit 9. (a) Use proof by contraposition to prove that the following statement is true for all integers n: If n2 + 2n + 7 is odd, then n is even. [6] (b) Hence prove that the following statement is true for all integers n: na + 2n + 7 is odd if and only if n is even. [4]
Question 3 You should be able to answer this question after studying Unit 9. (a) Use mathematical induction to prove that the following statement is 10 marks true: [9 n - n is a multiple of 3 for all n 2 1. (b) Show that the following statement is false: n-n is a multiple of 4 for all n 2 1. Question 4-10 ma You should be able to answer this question after studying Unit 9. (a) Use proof by...
Question 2 10 marks You should be able to answer this question after studying Unit 9. (a) State whether the following statement is true or false. If it is true, give a proof; if it is false, show that it is false. х The real function f(x) = is an even function. X + 5 [3] (Recall from Unit 7 that a function f is even if f(-x) = f(x) for all x in the domain of f.) (b) Prove...
uestion 6 20 marks You should be able to answer this question after studying Unit 5 A mountain rescue team is preparing to move a stretcher (with an injured climber) down a flat slope. The combined mass of the climber and stretcher is 110 kg, and the slope makes an angle of 15° with the horizontal. The stretcher is held at rest by a rope that makes an angle of 20° with the upward slope. The stretcher is on the...
Question 3 - 15 mai You should be able to answer this question after studying Unit 7. (a) Find the partial fraction expansion of the rational expression 2a3- 3a2 - 18x 17 a2 -3r- 4 10 (b) Use the partfrac command in Maxima to verify your answer to part (a). Include a screenshot or printout of your Maxima worksheet in your answer (c) Hence (without using Maxima) find the integral 2r332 -18x +17 [4
Question 3 - 15 mai You...
Question 5 15 marks You should be able to answer this question after studying Unit 10 A crate, initially at rest, is pushed along a straight line up a rough ramp that makes an angle of 18° with the horizontal. The magnitude of the pushing force is 33 N, in a direction parallel to and up the ramp, and the acceleration of the crate is 0.16 ms-2. The coefficient of sliding friction between the crate and the ramp is 0.35....
Question 3 Use mathematical induction to prove 3 + 7 + 11 +. + (4n - 1) = n (2n + 1). Show P1 is true. Assume Pk is true. Show Pk11 is true.
Question 615 marks You should be able to answer this question after studying Unit 10 At a battle re-enactment, a catapult launches a missile with a speed of 38 ms-, at an angle of 40° to the horizontal. The height from which the projectile is launched is 10m. Set up coordinate axes at the point where the missile leaves the catapult, as shown below 40° 10 m Let i and j be the Cartesian unit vectors in the positive directions...
Question 9 You should be able to answer this question after studying Unit 8. An object moves along a straight line, and its speed v (in metres per second) when at a position r (in metres) from its starting point can be modelled by the differential equation 10 marks 0 dr0, 20), where k is a positive constant (a) Find the general solution of this differential equation in explicit form (b) The speed of the object at its starting point...
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Question 3 Use mathematical induction to prove 3+7+11+ ... +(4n – 1) = n (2n + 1). • Show P1 is true. • Assume Pk is true. • Show Pk+1 is true.