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Trigonometry, Polynomials, Quadratic Theory, Differentiation, Recurrence Relations & Integration 1. Solve (a) 3 tan x 1 , (0 x 360) (c) 4 = 5 – tan 2x , (0 x 180) (e) 2 cos2 x = 1 , (0 x 360) (g) 2 cos (x + 60) = 3 , (0 x 360) (i) 8 tan (x + 30) = 35 , (0 x 360) (b) 2 cos x 1 0 , (0 x 180) (d) 4 tan 3x + 4 = 0 , (0 x 90) (f) 4 sin2 x – 1 = 0 , (0 x 360) (h) 5 sin (2x – 20) = 3 , (0 x 360) (j) 5 cos (6x – 20) + 3 = 725 , (0 x 90) 2. Show that the equation x3 – 6x2 + 2 = 0 has a root which lies between x = 05 and x = 1 and find the value of this root correct to two decimal places. 3. A function is defined by g(x) = x3 + a , where a is a constant. When g(x) is divided by x – 3 , the remainder is 36. Find a , and hence solve g(2x) = 18 4. Show that 2 is a root of x3 – 9x2 + 20x – 12 = 0 and find the other roots. 5. Find m, given that x2 + (mx – 5)2 = 9 has equal roots. 6. The point (2 , 1) lies on the graph of a function whose derivative f ( x) 3x 2 12 . Find f(x). 7. Find the point on the curve y 8. 1 2 8 x at which the tangent is parallel to the x-axis. 2 x An unstable atomic particle decays in mass by half every two hours. At the end of each two hour period it is bombarded by atomic matter which allows it to recover a mass of 6 a.m.u. (a) By considering a suitable recurrence relationship and taking the original mass of the particle to be 40 a.m.u., calculate the number of hours it will take to decay to a value which is below 32% of its original mass. (b) If the mass of the particle falls below 123 a.m.u. it becomes highly unstable resulting in an explosion. Should the scientists allow this experiment to continue over a long period of time? Explain your answer. 9. a 1 1 p 2 dp 42 , find a.