Download BP15 Trig, Polynomials, Quad, Diff, RR, Int

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) = 35 , (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 = 725 , (0  x  90)
2.
Show that the equation x3 – 6x2 + 2 = 0 has a root which lies between x = 05 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 123 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.