Year 12. Core 1 and 2. Easter Revision Exam Questions

Transcription

1 Year 12 Core 1 and 2 Easter Revision 2014 Exam Questions The Redhill Academy 1

2 Session 1: Transforming Graphs 1. y (0, 7) y = f( x) O (7, 0) x The diagram above shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation (a) y = f(x) + 3, y = f(2x). On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis. (Total 5 marks) 1 2. Given that f(x) =, x 0, x (a) sketch the graph of y = f(x) + 3 and state the equations of the asymptotes. Find the coordinates of the point where y = f(x) + 3 crosses a coordinate axis. (Total 6 marks) The Redhill Academy 2

3 3. (a) Factorise completely x 3 4x. Sketch the curve with equation y = x 3 4x, showing the coordinates of the points where the curve crosses the x-axis. (c) On a separate diagram, sketch the curve with equation y = (x 1) 3 4(x 1), showing the coordinates of the points where the curve crosses the x-axis. (Total 9 marks) 4. (a) Factorise completely x 3 6x 2 + 9x Sketch the curve with equation y = x 3 6x 2 + 9x showing the coordinates of the points at which the curve meets the x-axis. Using your answer to part, or otherwise, (c) sketch, on a separate diagram, the curve with equation y = (x 2) 3 6(x 2) 2 + 9(x 2) showing the coordinates of the points at which the curve meets the x-axis. (Total 9 marks) The Redhill Academy 3

4 Session 2: Sequences and Series 1. The rth term of an arithmetic series is (2r 5). (a) Write down the first three terms of this series. State the value of the common difference. n (c) Show that (2r 5) = n(n 4). r 1 (Total 6 marks) 2. A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term a and common difference d. Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find (a) the value of d, the value of a, (c) the total number of houses built in Oldtown over the 40-year period. (Total 8 marks) The Redhill Academy 4

5 3. A sequence a 1, a 2, a 3..., is defined by a 1 = k, a n+1 = 3a n + 5, n 1, where k is a positive integer. (a) Write down an expression for a 2 in terms of k. Show that a 3 = 9k a r r 1 (c) (i) Find in terms of k. 4 a r r 1 (ii) Show that is divisible by 10. (Total 7 marks) The Redhill Academy 5

6 4. The adult population of a town is at the end of Year 1. A model predicts that the adult population of the town will increase by 3% each year, forming a geometric sequence. (a) Show that the predicted adult population at the end of Year 2 is Write down the common ratio of the geometric sequence. The model predicts that Year N will be the first year in which the adult population of the town exceeds (c) Show that (N 1) log1.03 > log1.6 (d) Find the value of N. At the end of each year, each member of the adult population of the town will give 1 to a charity fund. Assuming the population model, (e) find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest (Total 10 marks) The Redhill Academy 6

7 5. A trading company made a profit of in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio r, r > 1. The model therefore predicts that in 2007 (Year 2) a profit of r will be made. (a) Write down an expression for the predicted profit in Year n. The model predicts that in Year n, the profit made will exceed log 4 Show that n > 1. log r Using the model with r = 1.09, (c) find the year in which the profit made will first exceed , (d) find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest (Total 9 marks) The Redhill Academy 7

8 Session 3: Calculus 1. y C P R O A x The diagram above shows part of the curve C with equation 3 1 y = 2 x 2 4 x 3. The curve C touches the x-axis at the origin and passes through the point A(p, 0). (a) Show that p = 6. Find an equation of the tangent to C at A. The curve C has a maximum at the point P. (c) Find the x-coordinate of P. The shaded region R, in the diagram above, is bounded by C and the x-axis. (d) Find the area of R. (Total 11 marks) The Redhill Academy 8

9 2. y A (1, 5) C R O B D x The diagram above shows part of the curve C with equation 2 y = 9 2x, x > 0. x The point A(1, 5) lies on C and the curve crosses the x-axis at B(b, 0), where b is a constant and b > 0. (a) Verify that b = 4. The tangent to C at the point A cuts the x-axis at the point D, as shown in the diagram above. Show that an equation of the tangent to C at A is y + x = 6. (c) Find the coordinates of the point D. The shaded region R, shown in the diagram above, is bounded by C, the line AD and the x-axis. (d) Use integration to find the area of R. (6) (Total 12 marks) The Redhill Academy 9

10 3. 2 x metres y metres The diagram above shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is 2x metres and the width is y metres. The diameter of the semicircular part is 2x metres. The perimeter of the stage is 80 m. (a) Show that the area, A m 2, of the stage is given by A = 80x 2 x 2. 2 Use calculus to find the value of x at which A has a stationary value. (c) Prove that the value of x you found in part gives the maximum value of A. (d) Calculate, to the nearest m 2, the maximum area of the stage. (Total 12 marks) The Redhill Academy 10

11 4. y A C R P O Q x The diagram above shows part of the curve C with equation y = x 2 6x The curve meets the y-axis at the point A and has a minimum at the point P. (a) Express x 2 6x + 18 in the form (x a) 2 + b, where a and b are integers. Find the coordinates of P. (c) Find an equation of the tangent to C at A. The tangent to C at A meets the x-axis at the point Q. (d) Verify that PQ is parallel to the y-axis. The shaded region R in the diagram is enclosed by C, the tangent at A and the line PQ. (e) Use calculus to find the area of R. (5) (Total 15 marks) The Redhill Academy 11