AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013

Transcription

1 AS Mathematics Assignment 8 Due Date: Friday 15 th February 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit them with this cover sheet. (Work submitted without a cover sheet complete with name will not be marked). All work is to be submitted either to your module teacher or the Faculty Office by 4.15pm on the due date above. TEACHER ASSESSED QUESTIONS SUBJECT SPECIFIC SKILLS 1. Presentation : written work presented legibly using standard mathematical notation Grade Evidence/Comment : 2. Communication: written solutions presented in a logical and coherent manner ( showing workings clearly) Grade Evidence/Comment : 3. Problem solving: identifying and using appropriate mathematical techniques Grade Evidence/Comment : 4. Competence in using algebraic techniques Grade Evidence/Comment : 5. Competence in using graphical calculator Grade Evidence/Comment : NOTE: GRADE 1 = BRIILIANT /SORTED GRADE 5 = YET TO DEMONSTRATE THIS PARTICULAR SKILL REFERENCES Edexcel AS Core 2 text book: Factor & Remainder Theorems: pg 6-17 Circle Geometry: pg (See your subject for the login details, click topic to access mymaths lesson) Factor theorem Remainder Theorem Circle Geometry Students Feedback: Please comment on how you think you have performed on this assignment and state the questions that you found difficult.

2 FACTOR AND REMAINDER THEOREM 1. f(x) = 2x 3 x 2 + px + 6, where p is a constant. Given that (x 1) is a factor of f(x), find the value of p, the remainder when f(x) is divided by (2x + 1). (Total 4 marks) 2. f(x) = px 3 + 6x x + q. Given that the remainder when f(x) is divided by (x 1) is equal to the remainder when f(x) is divided by (2x + 1), find the value of p. Given also that q = 3, and p has the value found in part, find the value of the remainder. (1) (Total 5 marks) 3. Use the factor theorem to show that (x + 4) is a factor of 2x 3 + x 2 25x Factorise 2x 3 + x 2 25x + 12 completely. 4. f(x) = (3x 2)(x k) 8 where k is a constant. Write down the value of f (k). (1) When f (x) is divided by (x 2) the remainder is 4 Find the value of k. Factorise f (x) completely. Haringey Sixth Form Centre Mathematics Department 2

3 5. Show that (x + 2) is a factor of f(x). Factorise f(x) completely. f(x) = x 3 19x f(n) = n 3 + pn n + 9, where p is a constant. Given that f(n) has a remainder of 3 when it is divided by (n + 2), prove that p = 6. Show that f(n) can be written in the form (n + 2)(n + q)(n + r) + 3, where q and r are integers to be found. Hence show that f(n) is divisible by 3 for all positive integer values of n. (Total 7 marks) 7. f(x) = 2x 3 x 2 + 2x 16. Use the factor theorem to show that (x 2) is a factor of f(x). Given that f(x) = (x 2)(2x 2 + bx + c), find the values of b and c. Hence prove that f(x) = 0 has only one real solution. (Total 8 marks) 8. f(x) = 2x 3 + ax 2 + bx 6 where a and b are constants. When f(x) is divided by (2x 1) the remainder is 5. When f(x) is divided by (x + 2) there is no remainder. Find the value of a and the value of b. (6) Factorise f(x) completely. (Total 9 marks) Haringey Sixth Form Centre Mathematics Department 3

4 9. f(x) = x 3 + (p + 1)x 2 18x + q, where p and q are integers. Given that (x 4) is a factor of f(x), show that 16p + q + 8 = 0. Given that (x + p) is also a factor of f(x), and that p > 0, show that p p + q = 0. Hence find the value of p and the corresponding value of q. (5) (d) Factorise f(x) completely. CIRCLE GEOMETRY (Total 13 marks) 10. The point A has coordinates (2, 5) and the point B has coordinates ( 2, 8). Find, in cartesian form, an equation of the circle with diameter AB. (Total 4 marks) 11. The points A and B have coordinates (5, 1) and (13, 11) respectively. Find the coordinates of the mid-point of AB. Given that AB is a diameter of the circle C, find an equation for C. QUESTION CONTINUED ON NEXT PAGE Haringey Sixth Form Centre Mathematics Department 4

5 12. y ( a, b) 5 O 4 x The circle C, with centre (a, b) and radius 5, touches the x-axis at (4, 0), as shown in the diagram above. Write down the value of a and the value of b. Find a cartesian equation of C. (1) A tangent to the circle, drawn from the point P(8, 17), touches the circle at T. Find, to 3 significant figures, the length of PT. 13. The line joining the points ( 1, 4) and (3, 6) is a diameter of the circle C. Find an equation for C. 14. A circle C has radius 5 and has its centre at the point with coordinates (4, 3). Prove that an equation of the circle C is x 2 + y 2 8x 6y + 20 = 0. The line l, with equation y = 2x, is a tangent to the circle C. Find the coordinates of the point where the line l touches C. (Total 7 marks) Haringey Sixth Form Centre Mathematics Department 5

6 15. The circle C, with centre A, has equation x 2 + y 2 6x + 4y 12 = 0. Find the coordinates of A. Show that the radius of C is 5. The points P, Q and R lie on C. The length of PQ is 10 and the length of PR is 3. Find the length of QR, giving your answer to 1 decimal place. (Total 7 marks) 16. y B M (3, 1) O A (1, 2) P x l The points A and B lie on a circle with centre P, as shown in the diagram above. The point A has coordinates (1, 2) and the mid-point M of AB has coordinates (3, 1). The line l passes through the points M and P. Find an equation for l. Given that the x-coordinate of P is 6, use your answer to part to show that the y-coordinate of P is 1, (1) find an equation for the circle. (Total 9 marks) Haringey Sixth Form Centre Mathematics Department 6

7 17. Two circles C 1 and C 2 have equations respectively. (x 2) 2 + y 2 = 9 and (x 5) 2 + y 2 = 9 For each of these circles state the radius and the coordinates of the centre. Sketch the circles C 1 and C 2 on the same diagram. Find the exact distance between the points of intersection of C 1 and C 2. (Total 9 marks) 18. The circle C, with centre at the point A, has equation x 2 + y 2 10x + 9 = 0. Find the coordinates of A, the radius of C, the coordinates of the points at which C crosses the x-axis. 7 Given that the line l with gradient is a tangent to C, and that l touches C at the point T, 2 (d) find an equation of the line which passes through A and T. (Total 9 marks) 19. A circle C 1 has equation x 2 + y 2 12x + 4y + 20 = 0. Find the coordinates of the centre of C 1. Find the radius of C 1. The circle C 1 cuts the x-axis at the points A and B. Find an equation of the circle C 2 with diameter AB. (6) (Total 10 marks) Haringey Sixth Form Centre Mathematics Department 7

8 20. The circle C has centre A(2,1) and passes through the point B(10, 7). Find an equation for C. The line l 1 is the tangent to C at the point B. Find an equation for l 1. The line l 2 is parallel to l 1 and passes through the mid-point of AB. Given that l 2 intersects C at the points P and Q, find the length of PQ, giving your answer in its simplest surd form. (Total 11 marks) END OF ASSIGNMENT Haringey Sixth Form Centre Mathematics Department 8