BHASVIC MαTHS. The mean marks for a stats exam were worked out for 3 classes.

Transcription

1 The mean marks for a stats exam were worked out for 3 classes. Class 1 had 12 students with a mean of 78% and standard deviation of 8 Class 2 had 16 students with a mean of 84% and standard deviation of 10 Class 3 had 18 students with a mean of 54% and standard deviation of 21 (a) Find the mean mark for all 46 students, give your ans to the nearest whole number A new student joins class 1 who has a mark of 83%. (b) Explain with reasons whether the mean for this class would increase, decrease or remain the same when the new student is added to the class, and whether the standard deviation would increase, decrease or remain the same when the new student is added to the class. (c) The teacher of class 3 claims that her class results are not actually worse than class 1 or class 2. The head of department looks at her results and agrees. What could explain the lower average score?

2 2 Rupal throws a ball upwards at 2 ms -1 from a window which is 4 m above ground level. (a) Draw an st, a vt and an at graph for this motion. Explain the links between them. (b) Use suvat to find an equation for the height h m of the ball above the ground after t seconds (while it is still in the air) (c) Use your answer to part (b) to find the time the ball hits the ground (d) How fast is the ball moving just before it hits the ground? (e) In what way would you expect your answers to (c) and (d) to change if you were to take air resistance into account?

3 3 A car travelling on a straight road slows down with constant deceleration. The car passes a road sign with speed 40km h -1 and then a post box with speed 24 km -1. The distance between the road sign and the post box is 240m. (a) Draw an st, a vt and an at graph for this motion. Explain the links between them. Find, in ms -2, the deceleration of the car (b) using your vt graph (c) using suvat equations

4 4 (a) A car is joined to a caravan with a light rigid towbar and is driving along a level road. The car has mass 1200 kg and the caravan has mass 800 kg. Resistance forces of 600 N and 800 N act on the car and caravan respectively. The acceleration of the car and caravan is 0.2 ms -2. Draw a force diagram to model this situation. (i) How have you used the fact that the towbar is light in your model? (ii) How have you used the fact that the towbar is rigid in your model? (iii) It would also be possible to model the car-caravan combination as one particle. Draw this new force-acceleration diagram.

5 4 (b) Particles A and B each of mass 20kg are joined by a light inextensible string which passes over a smooth pulley so that the string hangs vertically on both sides Draw a force diagram to model this situation. (i) How have you used the fact that the string is light in your model? (ii) How have you used the fact that the string is inextensible in your model? (iii) How have you used the fact that the pulley is smooth in your model? (iv) It is not possible to model the mass A - mass B combination as one particle (as we did in part a). Explain why not.

6 5 a) Sketch the following curves of y = f (x), stating the equations of the asymptotes and the coordinates of any axis intercepts: (i) f x = 2 1 x (ii) f x = x 3 (b) Sketch the following curves of y = f (x), stating the coordinates of any axis intercepts (i) f x = 2(x 4) (ii) f x = 5 x 2x + 1 (iii) f x = x + 2 x 3 2x 5 (iv) f x = (x + 2) 2 x 3 (v) f x = x + 2 x x 3x 1

7 6 (a) Given that 2 x+y = 1 and 10 3x y = 100, find x and y (b) Given that y = 1 8 x3, express each of the following in the form kx n where k and n are constants. (i) y 1 3 (ii) 1 2 y 2 (c) Given that 4x3 + x 5 x can be written as 4x a + x b, find a and b (d) Solve the equation 8 + x 12 = 8x 3

8 7 Use desmos to look at the graph of f x = 4x 2 4kx k (when you type the 4kx, click add slider ). (a) Using the slider, (i) find the values of k for which the quadratic f x has a repeated real root (ii) Write in set notation and in interval notation, the set of values of k for which the quadratic f x has no real roots (iii) Write in set notation and in interval notation, the set of values of k for which the quadratic f x has two distinct real roots (b) By completing the square, find in terms of the constant k the roots of the equation f x = 4x 2 4kx k (c) Explain the connection between what you have found in part (b) and what you found in part (a)

9 8 The points A (-1, -2), B (7, 2) and C (k, 4), where k is a constant, are the vertices of the triangle ABC. Angle ABC is a right angle. Find the area of the triangle ABC *lines/circles/triangles are geometry questions so always draw a diagram*

10 The line l 1 has equation 2x y + 4 = 0 The line l 2 has equation 6x 3y 9 = 0 9 (a) Prove that l 1 and l 2 are parallel (b) Prove that the point A (3, 10) lies on l 1 The l 3 line is perpendicular to l 1 and passes through the point A (c) Find the point of intersection of l 3 and l 2 (d) Find the shortest distance between l 1 and l 2 *lines/circles/triangles are geometry questions so always draw a diagram*

11 10 The diagram shows a section of a suspension bridge carrying a road over water. The height of the cables above water level in metres can be modelled by the function h x = x , where x is the displacement in metres from the centre of the bridge. (a) Interpret the meaning of the constant term 200 in the model (b) Use the model to find the two values of x at which the height is 346m. (c) Given that the towers at each end are 346m tall, use your answer to part b to calculate the length of the bridge to the nearest metre.

12 1 - Answers (a) 71% (b) Answer not given (c) Answer not given

13 2 - Answers (a) Answer not given (b) h = 4 + 2t 4.9t 2 (c) 1.13 s (d) 9.08 ms -1 (e) Answer not given

14 3 - Answers (a) Answer not given (b) Answer not given (c) ms -2

15 4 - Answers Answers not given

16 5 - Answers In the library computers you can sketch the graphs on autograph. On your phone you could use the free app desmos or use wolfram alpha. Or, use your graphical calculator to check. It is important you try these yourself first, don t go straight to the answers!

17 6 - Answers (a) Show your method! x = 1 2, y = 1 2 (b) Show your method! (i) 1 2 x (ii) 32x 6 (c) Show your method! a = 5 2, b = 2 (d) x = 4 3

18 7 - Answers (ai) k = 0 (aii) Answer not given (aiii) Answer not given (b) x = 2k ± 4k 2 + k (c) Answer not given

19 8 - Answers (a) 6 (b) 2x + y 16 = 0 (c) 10

20 9 - Answers (a) make sure to conclude your proof with a statement of what you have found. (b) don t sub in both 3 and 10 then say 0 = 0, this isn t a proof! Sub in x = 3 and find that y = 10 or vice versa. Conclude your proof with a statement of what you have proved. (c) 29, (d) 7 5 5

21 10 - Answers (a) Ans not given (b) x = ±1103 (c) 2206 m