Colegio del mundo IB. Programa Diploma REPASO The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

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1 REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f() = 6 +, for. Write f() in the form ( a) + b. Find the inverse function f. (c) State the domain of f.. The equation k + + = 0 has eactl one solution. Find the value of k.

2 REPASO 4. q The diagram shows part of the graph of the function f() =. p through the point A (, 0). The line (CD) is an asmptote. C 5 The curve passes 0 A D Find the value of (i) p; (ii) q.

3 REPASO The graph of f() is transformed as shown in the following diagram. The point A is transformed to A (, 0). 5 C A 5 D Give a full geometric description of the transformation.

4 4 REPASO 5. The diagram shows part of the graph of the curve = a ( h) + k, where a, h, k The verte is at the point (, ). Write down the value of h and of k. The point P(5, 9) is on the graph. Show that a =. () () (c) Hence show that the equation of the curve can be written as = + 9. () (d) (i) Find d. d A tangent is drawn to the curve at P (5, 9). (ii) (iii) Calculate the gradient of this tangent, Find the equation of this tangent. (4) (Total 0 marks) 4

5 5 REPASO 6. A famil of functions is given b f() = + + k, where k {,,, 4, 5, 6, 7}. One of these functions is chosen at random. Calculate the probabilit that the curve of this function crosses the -ais. 7. Let f() = e, and g() = +,. Find f () (g f)(). 8. The sketch shows part of the graph of = f() which passes through the points A(, ), B(0, ), C(l, 0), D(, ) and E(, 5) E 4 A B D C

6 6 REPASO A second function is defined b g() = f( ). Calculate g(0), g(), g() and g(). On the same aes, sketch the graph of the function g(). 9. Let g() = 4 +. Solve g() = 0. () Let f() = +. A part of the graph of f() is shown below. g( ) C A 0 B The graph has vertical asmptotes with equations = a and = b where a < b. Write down the values of (i) a; (c) (ii) b. The graph has a horizontal asmptote with equation = l. Eplain wh the value of f() approaches as becomes ver large. () () (d) The graph intersects the -ais at the points A and B. Write down the eact value of the - coordinate at (i) A; (ii) B. () 6

7 7 REPASO (e) The curve intersects the -ais at C. Use the graph to eplain wh the values of f () and f () are zero at C. () (Total 0 marks) 0. Given that f() = e, find the inverse function f ().. Consider the functions f : a 4 ( ) and g : a 6. Find g. Solve the equation (f g ) () = 4.. Solve the equation e = 5, giving our answer correct to four significant figures. á. Consider the function f() = Epress f() in the form a ( p) + q, where a, p, q. Find the minimum value of f(). 4. Consider functions of the form = e k Show that e d = Let k = k ( e k ). k () (i) Sketch the graph of = e 0.5, for, indicating the coordinates of the -intercept. (ii) Shade the region enclosed b this graph, the -ais, -ais and the line =. 7

8 8 REPASO (iii) Find the area of this region. (5) (c) (i) Find d in terms of k, where = e k. d The point P(, 0.8) lies on the graph of the function = e k. (ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P. (5) (Total marks) 5. The diagram shows part of the graph of = a ( h) + k. The graph has its verte at P, and passes through the point A with coordinates (, 0). P A 0 Write down the value of (i) h; (ii) k. Calculate the value of a. 6. Let f() =, and g() =, ( ). 8

9 9 REPASO Find (g f) (); g (5). 7. The diagram below shows part of the graph of the function f : a A P 5 0 Q B The graph intercepts the -ais at A(,0), B(5,0) and the origin, O. There is a minimum point at P and a maimum point at Q. The function ma also be written in the form where a < b. Write down the value of (i) a; f : a ( a) ( b), (ii) b. () Find (i) f (); 9

10 0 REPASO (ii) the eact values of at which f '() = 0; (iii) the value of the function at Q. (7) (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the -coordinate of this point. (4) (d) Determine the area of the shaded region. 8. In the diagram below, the points O(0, 0) and A(8, 6) are fied. The angle varies as the point P(, 0) moves along the horizontal line = 0. OPˆ A () (Total 5 marks) P(, 0) =0 A(8, 6) O(0, 0) (i) Show that AP = diagram to scale (ii) Write down a similar epression for OP in terms of. () Hence, show that cos OPˆA = {( ) (, + 00)} () (c) Find, in degrees, the angle OPˆ A when = 8. (d) Find the positive value of such that O PˆA = 60. () (4) 0

11 REPASO Let the function f be defined b f ( ) = cos OPˆA =, 0 5. {( ) ( + 00)} (e) Consider the equation f () =. (i) Eplain, in terms of the position of the points O, A, and P, wh this equation has a solution. (ii) Find the eact solution to the equation. 9. The diagram below shows the graph of = sin, for 0 < m, and 0 < n, where is in radians and m and n are integers. n (5) n 0 m m Find the value of m; n.

12 REPASO 0. The diagram shows parts of the graphs of = and = 5 ( 4). 8 = 6 = 5 ( 4) The graph of = ma be transformed into the graph of = 5 ( 4) b these transformations. A reflection in the line = 0 a vertical stretch with scale factor k a horizontal translation of p units a vertical translation of q units. followed b followed b followed b Write down the value of k; p; (c) q.. Factorise 0. Solve the equation 0 = 0.

13 REPASO. The diagram represents the graph of the function f : a ( p) ( q). C Write down the values of p and q. The function has a minimum value at the point C. Find the -coordinate of C. Total 4 marks). The diagrams show how the graph of = is transformed to the graph of = f() in three steps. For each diagram give the equation of the curve. 0 = 0 (c)

14 4 REPASO 4. On the following diagram, sketch the graphs of = e and = cos for. 0 The equation e = cos has a solution between and. Find this solution. 5. The function f is defined b f : a,. Evaluate f (5). 6. The following diagram shows the graph of = f (). It has minimum and maimum points at 4

15 5 REPASO (0, 0) and (, )..5 On the same diagram, draw the graph of = f ( ) +. What are the coordinates of the minimum and maimum points of = f ( ) +? 7. Michele invested 500 francs at an annual rate of interest of 5.5 percent, compounded annuall. Find the value of Michele s investment after ears. Give our answer to the nearest franc. () 5

16 6 REPASO How man complete ears will it take for Michele s initial investment to double in value? () (c) What should the interest rate be if Michele s initial investment were to double in value in 0 ears? (4) (Total 0 marks) 8. Note: Radians are used throughout this question. Let f () = sin ( + sin ). (i) Sketch the graph of = f (), for 0 6. (ii) Write down the -coordinates of all minimum and maimum points of f, for 0 6. Give our answers correct to four significant figures. (9) Let S be the region in the first quadrant completel enclosed b the graph of f and both coordinate aes. (i) Shade S on our diagram. (ii) Write down the integral which represents the area of S. (iii) Evaluate the area of S to four significant figures. (5) (c) Give reasons wh f () 0 for all values of. () 6

17 7 REPASO 9. f() = 4 sin + π. For what values of k will the equation f() = k have no solutions? 0. A group of ten leopards is introduced into a game park. After t ears the number of leopards, N, is modelled b N = 0 e 0.4t. How man leopards are there after ears? How long will it take for the number of leopards to reach 00? Give our answers to an appropriate degree of accurac. Give our answers to an appropriate degree of accurac.. Consider the function f : a +, Determine the inverse function f. What is the domain of f?. The diagram shows the graph of = f(), with the -ais as an asmptote. B(5, 4) A( 5, 4) On the same aes, draw the graph of =f( + ), indicating the coordinates of the 7

18 8 REPASO images of the points A and B. Write down the equation of the asmptote to the graph of = f( + ).. Sketch, on the given aes, the graphs of = and sin for Find the positive solution of the equation = sin, giving our answer correct to 6 significant figures. 8

19 9 REPASO 4. A ball is thrown verticall upwards into the air. The height, h metres, of the ball above the ground after t seconds is given b h = + 0t 5t, t 0 (c) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). Show that the height of the ball after one second is 7 metres. At a later time the ball is again at a height of 7 metres. () () (i) (ii) Write down an equation that t must satisf when the ball is at a height of 7 metres. Solve the equation algebraicall. (4) (d) (i) Find d h. dt (ii) (iii) Find the initial velocit of the ball (that is, its velocit at the instant when it is released). Find when the ball reaches its maimum height. (iv) Find the maimum height of the ball. (7) (Total 5 marks) 5. The diagram shows part of the graph with equation = + p + q. The graph cuts the -ais at 9

20 0 REPASO and Find the value of p; q. 6. Each ear for the past five ears the population of a certain countr has increased at a stead rate of.7% per annum. The present population is 5. million. What was the population one ear ago? What was the population five ears ago? 7. The diagram shows the graph of the function = a + b + c. 0

21 REPASO Complete the table below to show whether each epression is positive, negative or zero. Epression positive negative zero a c b 4ac b 8. Initiall a tank contains litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given b V = (0.9 t ). (c) Find the value of V after 5 minutes. Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out of the tank. The tank is regarded as effectivel empt when 95% of the liquid has flowed out. Show that it takes almost three-quarters of an hour for this to happen. () () () (d) (i) Find the value of V when t = 0.00 minutes. (ii) Hence or otherwise, estimate the initial flow rate of the liquid. Give our answer in litres per minute, correct to two significant figures. () (Total 0 marks) 9. Epress f() = in the form f() = ( h) + k, where h and k are to be determined.

22 REPASO Hence, or otherwise, write down the coordinates of the verte of the parabola with equation The diagram shows three graphs. B A C A is part of the graph of =. B is part of the graph of =. C is the reflection of graph B in line A. Write down the equation of C in the form =f(); the coordinates of the point where C cuts the -ais. 4. The quadratic equation 4 + 4k + 9 = 0, k > 0 has eactl one solution for. Find the value of k. 4. Two functions f, g are defined as follows: f : + 5

23 REPASO Find g : ( ) f (); (g o f)( 4). 4. Two functions f and g are defined as follows: f() = cos, 0 = π; g() = +,. Solve the equation (g o f) () = The function f is given b F() = +,,. (i) Show that = is an asmptote of the graph of = f(). () (ii) (iii) Find the vertical asmptote of the graph. Write down the coordinates of the point P at which the asmptotes intersect. () () (c) Find the points of intersection of the graph and the aes. Hence sketch the graph of = f(), showing the asmptotes b dotted lines. (4) (4) (d) Show that f () = 7 ( ) the point S where = 4. and hence find the equation of the tangent at (6) (e) The tangent at the point T on the graph is parallel to the tangent at S. Find the coordinates of T. (5)

24 4 REPASO (f) Show that P is the midpoint of [ST]. (l) 45. The diagram shows the parabola = (7 )(l + ). The points A and C are the -intercepts and the point B is the maimum point. B A 0 C Find the coordinates of A, B and C. 46. The function f is given b f() = n ( ). Find the domain of the function. 47. Let f() =, and g() =. Solve the equation (f o g)() = A population of bacteria is growing at the rate of. % per minute. How long will it take for the size of the population to double? Give our answer to the nearest minute. 49. Three of the following diagrams I, II, III, IV represent the graphs of = + cos = cos( + ) 4

25 5 REPASO (c) = cos +. Identif which diagram represents which graph. I II 4 π π π π π π π π π π III IV 5 4 π π π π π π π π π π 50. $000 is invested at 5% per annum interest, compounded monthl. Calculate the minimum number of months required for the value of the investment to eceed $000. 5