Answer the following questions correctly, Submit your work at 07.00 a.m. on 25th December, 2010.
CAMBRIDE INTERNATIONAL EXAMINATION 2011 - REVISION EXERCISES 2
MATERIAL : TRIGONOMETRY + DIFFERENTIATION and ITS APPLICATIONS
1. By factorising 6 sin2x – 7 sin x + 2, show that the equation 6 sin^2(x) – 7 sin(x) + 2 = 0 is satisfied when sin x = ½ and sin x = 0.67.
Hence find all solutions in the range 00 ≤ x ≤ 3600.
2. Prove that : sin2 2p – sin2 p = sin p sin 3p. Hence or otherwise find all the values of p, such that sin^2(2p) – sin(2p) = 0, for which 0 ≤ p ≤ 360.
3. Find the greatest and the smallest values of the expression :
1/(5 sinθ+12 cosθ + 20)
4. Find the 1st and 2nd derivatives of the following function :
F(x) = 2 cos ((x-2)/(x+1))
5. A curve has the equation : y = x^3 – 3x^2 + 12 x, hence by completing the square or otherwise, prove that the gradient of the curve is never less than 9.
6. Find the equation of the tangent at the point (4,2) to the curve with equation y = √x .
7.For the following function, find f'(x), and any intervals in which f(x) is increasing :
x^3 – 3x^2 + 3x – 1
8. An open cylindrical wastepaper bin, radius p cm and capacity V cm3, is to have a surface area of 5000 cm2.
(a)Show that V= 0.5p(5000- π p^2)
(b)Calculate the maximum possible capacity of the bin.