2, find c in terms of k. x

Size: px

Start display at page:

Download "2, find c in terms of k. x"

Gwendoline Knight
5 years ago
Views:

1 1. (a) Work out (i) (ii) (iii) (iv) (4) (b) Given that x = 2 k and 4 c 2, find c in terms of k. x c =. (1) (Total 5 marks)

2 2. Solve the equation x 2 x 1 (Total 7 marks)

3 3. Prove algebraicall that the sum of the squares of an two consecutive even integers is never a multiple of 8. (Total 4 marks)

4 4. Solve 7r + 2 = 5(r 4) r =... (Total 2 marks) 5. Simplif full (i) (p 3 ) 3... (ii) 4 3q 2q q (Total 3 marks) 6. Simplif full (a) 2(3x + 4) 3(4x 5)... (b) (2x 3 ) 5...

5 (c) 2 n 1 n 1 n (3) (Total 7 marks) 7. This is a sketch of the curve with equation =f(x). It passes through the origin O. = f( x) O x A (2, 4) The onl vertex of the curve is at A (2, 4) (a) Write down the coordinates of the vertex of the curve with equation (i) = f(x 3), (...,...) (ii) = f(x) 5, (...,...) (iii) = f(x), (...,...) (iv) = f(2x). (...,...) (4)

6 The curve with equation = x 2 has been translated to give the curve = f(x). (b) Find f(x) in terms of x. f(x) =... (4) (Total 8 marks) 8. Nassim thinks of a number. When he multiplies his number b 5 and subtracts 16 from the result, he gets the same answer as when he adds 10 to his number and multiplies that result b 3. Find the number Nassim is thinking of. (Total 4 marks)

7 9. (a) Simplif (i) p 2 p 7 (ii) x 8 x 3 (iii) (3) (b) Expand t(3t 2 + 4) (Total 5 marks) 10. The fraction, p, of an adult s dose of medicine which should be given to a child who weighs w kg is given b the formula 3 20 p w 200 (a) Use the formula adult s dose p w to find the weight of a child whose dose is the same as an 200 kg (3)

8 (b) Make w the subject of the formula 3 20 p w 200 w = (3) 3w = A A 12 (c) Express A in terms of w. A = (4) (Total 10 marks)

9 11. A straight line, L, passes through the point with coordinates (4, 7) and is perpendicular to the line with equation = 2x + 3. Find an equation of the straight line L. (Total 3 marks) 12. The expression x 2 6x + 14 can be written in the form (x p) 2 + q, for all values of x. (a) Find the value of (i) p, (ii) q. (i) p = (ii) q = (3) The equation of a curve is = f(x), where f(x) = x 2 6x Here is a sketch of the graph of = f(x). = f( x) M O x (b) Write down the coordinates of the minimum point, M, of the curve. (1)

10 Here is a sketch of the graph of = f(x) k, where k is a positive constant. The graph touches the x-axis. = f( x) k O x (c) Find the value of k. k = (1) (d) For the graph of = f(x 1), (i) write down the coordinates of the minimum point, (ii) find the coordinates of the point where the curve crosses the -axis. (3) (Total 8 marks) 13. (a) Simplif k 5 k 2... (1)

11 (b) Expand and simplif (i) 4(x + 5) + 3(x 7)... (ii) (x + 3)(x + 2)... (4) (c) Factorise (p + q) 2 + 5(p + q)... (1) (d) Simplif (m 4 ) 2... (1) (e) Simplif 2t 2 3r 3 t 4... (Total 9 marks)

12 14. 6 C B Diagram NOT accuratel drawn D 1 A O x ABCD is a rectangle. A is the point (0, 1). C is the point (0, 6). The equation of the straight line through A and B is = 2x + 1 (a) Find the equation of the straight line through D and C.... (b) Find the equation of the straight line through B and C.... (c) It is alwas possible to draw a circle which passes through all four vertices of a rectangle. Explain wh (1) (Total 5 marks)

13 15. (a) (ii) Factorise 2x 2 35x (ii) Solve the equation 2x 2 35x + 98 = 0... (3) A bag contains (n + 7) tennis balls. n of the balls are ellow. The other 7 balls are white. John will take at random a ball from the bag. He will look at its colour and then put it back in the bag. (b) (i) Write down an expression, in terms of n, for the probabilit that John will take a white ball. Bill states that the probabilit that John will take a white ball is (ii) Prove that Bill s statement cannot be correct. (3)

14 After John has put the ball back into the bag, Mar will then take at random a ball from the bag. She will note its colour. (c) Given that the probabilit that John and Mar will take balls with different colours is 9 4, prove that 2n 2 35n + 98 = 0 (5) (d) Using our answer to part (a) (ii) or otherwise, calculate the probabilit that John and Mar will both take white balls.... (Total 13 marks)

15 16. A sketch of the curve = sin x for 0 x 360 is shown below. 2 1 O x 1 2 (a) Using the sketch above, or otherwise, find the equation of each of the following two curves. (i) 2 1 O x 1 2 (i) Equation =...

16 (ii) 2 1 O x 1 2 (ii) Equation =... (b) Describe full the sequence of two transformations that maps the graph of = sin x onto the graph of = 3 sin 2x (3) (Total 5 marks) 17. (a) Expand the brackets p(q p 2 )... (1)

17 (b) Expand and simplif 5(3p + 2) 2(5p 3)... (Total 3 marks) 18. ABC is an isosceles triangle. x A Diagram NOT accuratel drawn B C AB = AC Angle A = x (a) Find an expression, in terms of x, for the size of angle B....

18 (b) Solve the simultaneous equations. 3p + q = 11 p + q = 3 p =... q =... (3) (Total 5 marks) 19. (a) Simplif 6 x (i) 2 x... (ii) ( 4 ) 3... (b) Expand and simplif (t + 4)(t 2)...

19 (c) Write down the integer values of x that satisf the inequalit 2 x < 4... (d) Find the value of 1 (i) (ii) (Total 8 marks) 20. Make u the subject of the formula D = ut + kt 2 u =... (Total 2 marks)

20 21. The length of a rectangle is twice the width of the rectangle. The length of a diagonal of the rectangle is 25 cm. x 2x Work out the area of the rectangle. Give our answer as an integer.... cm 2 (Total 3 marks) = a sin x Diagram NOT accuratel drawn = cos x + b x The diagram shows part of two graphs. The equation of one graph is The equation of the other graph is = a sin x = cos x + b (a) Use the graphs to find the value of a and the value of b. a =... b =...

21 (b) Use the graphs to find the values of x in the range 0 x 720 when a sin x = cos x + b. x =... (c) Use the graphs to find the value of a sin x (cos x + b) when x = (Total 6 marks) 23. Solve 2 x x 1 = 2 x 5 1 x =... (Total 4 marks)

22 24. (a) Expand and simplif ( x 7)( x 4). (b) Expand ( 3 2). (c) Factorise p p. (d) Factorise completel 6x 2 9x. (Total 8 marks)

23 25. Here is a sketch of the graph of 2 ( x 8) 25 for 0 x 12 4 P and Q are points on the graph. P is the point at which the graph meets the -axis. Q is the point at which has its maximum value. (a) Find the coordinates of (i) P, (, ) (ii) Q. (, ) (3) (b) Show that ( x 8) (2 x)(18 x) 4 (3) (Total 6 marks)

24 26. Expand and simplif 3 (5x 2) 2 (2x 5).. (Total 2 marks) 27. (a) Factorise x 2 2x 15 (b) Hence, or otherwise, solve x 2 2x 15 = 0 x = or x =.. (1) (Total 3 marks) 28. (a) Solve 6x + 2 = 4(x 7) x =..

25 15 2x (b) Solve 4 3 x =.. (3) (Total 5 marks) 29. (a) Factorise m 2 m (1) (b) Solve 7(p 2) = 3p + 4 p = (3) (Total 4 marks) 30. (a) Solve the inequalit 5x + 12 > 2

26 (b) Expand and simplif (x 6)(x + 4) (Total 4 marks) 31. (a) Simplif full 2x 2 4x 2 3x 9 (Total 3 marks) 32. (a) Complete the table of values for = x 2 3x 1. x (b) On the grid below, draw the graph of = x 2 3x 1.

27 (c) Use our graph to find an estimate for the minimum value of. (1) O x (Total 5 marks) 33. (a) Simplif x 5 x 2 (1)

28 (b) Simplif 2w 4 3w 3 2 (Total 3 marks) 34. Solve x 2 3x 18 = 0 (Total 3 marks) (b) Simplif 2w 4 3w 3 2 (Total 3 marks) 35. (a) Solve 4 = 22 p p =...

29 (b) Solve 7r + 2 = 5(r 4) r =... (Total 4 marks) 36. (a) Find the equation of the straight line which passes through the point (0, 3) and is perpendicular to the straight line with equation = 2x....

30 The graphs of = 2x 2 and = mx 2 intersect at the points A and B. The point B has coordinates (2, 8). = mx 2 = 2x 2 B (2, 8) O A x (b) Find the coordinates of the point A. (...,...) (Total 4 marks) 37. Solve 2(5x + 3) = 3x 22 x =.. (Total 3 marks)

31 38. n is a whole number such that 7 3n 15 List all the possible values of n. (Total 3 marks) 39. A straight line has equation 2 6x = 5 (a) Find the gradient of the line... The point (k, 6) lies on the line. (b) Find the value of k. k =.. (Total 4 marks) 40. (a) = k Find the value of k. k =

32 15a b (b) Simplif 2 3 3a b (Total 4 marks) 41. P Q R S O x The diagram shows 4 straight lines, labelled P, Q, R and S. The equations of the straight lines are A: = 2x B: = 3 2x C: = 2x + 3 D: = 3 Match each straight line, P, Q, R and S to its equation. Complete the table. Equation A B C D Straight line (Total 2 marks)

33 42. A D (115 + x)º C B Diagram NOT accuratel drawn AB = AC. BCD is a straight line. Angle ACD = (115 + x). Find, in terms of x, the size of angle BAC. Give our answer in its simplest form. Angle BAC = (Total 3 marks) 43. A straight line has equation = 5 3x (a) Write down the gradient of the line... (1) (b) Write down the coordinates of the point where the line crosses the axis. (, ) (1) (Total 2 marks)

34 44. Solve 2x + 7 = 6(x + 3) x =... (Total 3 marks) C B D 1 A O x Diagram NOT accuratel drawn ABCD is a rectangle. A is the point (0, 1). C is the point (0, 6). The equation of the straight line through A and B is = 2x + 1 (a) Find the equation of the straight line through D and C....

35 (b) Find the equation of the straight line through B and C.... (Total 4 marks) 46. A straight line passes through the points (0, 5) and (3, 17). Find the equation of the straight line.... (Total 3 marks) 47. Solve x 2 + 6x = 4 Give our answers in the form p q, where p and q are integers.... (Total 3 marks)

36 48. Solve 4 x = 2(3x 1) x =... (Total 3 marks) 49. Make a the subject of the formula s = 4 a + 8u a =... (Total 2 marks)

37 50. A straight line has equation 4 5x = 2 Work out the gradient of this line.... (Total 2 marks) 51. = 4 x 8 = 2 x+ 3 A O x The diagram shows two straight lines intersecting at point A. The equations of the lines are = 4x 8 = 2x + 3 Work out the coordinates of A. Diagram NOT accuratel drawn (...,...) (Total 3 marks)

38 52. Factorise p 2 + 6p (Total 2 marks) 53. Diagram NOT accuratel drawn The diagram shows three points A ( 1, 5), B (2, 1) and C (0, 5). The line L is parallel to AB and passes through C. (a) Find the equation of the line L... (4)

39 The line M is perpendicular to AB and passes through (0, 0). (b) Find the equation of the line M. (Total 6 marks)

Solve Quadratics Using the Formula

Clip 6 Solve Quadratics Using the Formula a + b + c = 0, = b± b 4 ac a ) Solve the equation + 4 + = 0 Give our answers correct to decimal places. ) Solve the equation + 8 + 6 = 0 ) Solve the equation =