<p> Further Calculus, Vectors, Revision of Units 1 & 2</p><p>1. Differentiate, expressing your answers with positive indices:</p><p>4 1 (a) (3x  2) (b) (x 2  4) 2 (c) 5  4x</p><p>1  4 2 (d) 2 3 (e) (f) (x  3x) x  2 x 2  7</p><p>2. Differentiate, expressing your answers with positive indices: (a) 2sin 3x (b) cos3 x (c) 5sin2 x 1 1 (d) cos (3  2x) (e) (f) sin x cos x 5 3. A curve has equation y  , where x   1 . 4x 1 4 Find the equation of the tangent to this curve at the point where x = 1.</p><p>4. A curve has equation y = 2sin ½  , 0    .</p><p>2 (a) Find the exact value of y when  = 3  . dy (b) Find and evaluate it for  = 2  . d 3</p><p>2 (c) Hence show that the equation of the tangent to the curve at the point where  = 3  2 can be written in the form 2y    (2 3  3  ). 5. Find: (a)  (x 1) 4 dx (b)  (2x 1)5 dx (c)  (3  2x)3 dx (d)  (x 1) 4 dx 1  3 (e)  (3  x) 2 dx (f)  (x  4) 2 dx (g)  (6x  5) 2 dx (h)  x  3dx</p><p>6. Evaluate: 1 1 4 2 dx (2x 1)3dx (1 x)3dx 4  x dx (a) (b) (c) (d)  1 2  0  1  0 (x  2)  1   4  5        7. a =  2  , b =   3 and c =  2 . (a) Find the components of the vector a  2b + 3c.        2  1   0  (b) Calculate the lengths of a + b and a – b . 8. Find the coordinates of the points dividing the line joining A(0 , 1 , 5) and B(0 , 4 , 5) internally and externally in the ratio 3 : 2. 9. P(0 , 2 , 1) and Q(0 , 1 , 2). Calculate the size of POQ where O is the origin.</p><p> 2   k      10. a =  1 , b =  1  . Find k if a is perpendicular to b.      3 1</p><p>11. In this right angled triangle a = 2 and b = 4</p><p>(a) Find the exact value of c . b c (b) Show that b.(a + c) = 16  3 a</p><p> y 12. The diagram shows the graph of a quadratic function f(x) and a linear function g(x). f(x) g(x) Find, in terms of x: O 1 3  4 x (a) the equation of the parabola f(x). (b) the equation of the line g(x). 4  (c) the values of x for which f(x)  g(x).</p><p>13. When an oil tanker is travelling at a speed of x km/h it uses up fuel at a rate of (1 + 00000625x3) tonnes per hour.</p><p>(a) Prove that the total amount of fuel used on a 16000 km voyage at a speed of x km/h is 16000 given by the function F(x) =  x 2 . x</p><p>The captain of the tanker wants to minimise the amount of fuel used in this voyage.</p><p>(b) Find the optimum speed at which the tanker should travel to keep the fuel consumption at a minimum. (c) Find the actual amount of fuel used for the voyage if the captain manages to maintain the optimum speed throughout the voyage.</p><p>14. Write 2cos2 x – 4cos x + 5 in the form a(cos x + b)2 + c</p>