BHASVIC MαTHS. (a) The table shows the number of eggs a bird lays per brood cycle. The mean number of eggs is 1.5. Find the value of p

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1 1 (a) The table shows the number of eggs a bird lays per brood cycle. The mean number of eggs is 1.5. Find the value of p Eggs Frequency 7 p 2 (b) From the large data set, the daily mean visibility, v metres, for Leeming in May and June 2015 was recorded each day. The data is summarised as follows: May: n = 30, v = 724,000 June: n = 31, v = 632,000 Calculate the mean visibility for the total recording period.

2 1 (c) The table shows the number of CDs owned by the students questioned in a survey. CDs Frequency % of the students owned more than p CDs 15% of the students owned fewer than q CDs Find the value of p and q

3 2 This is a SUVAT table. Only model a situation with a SUVAT table if acceleration is constant. s = displacement in metres u = initial velocity in m/s v = final velocity in m/s a = acceleration in m/s 2 t = time in seconds The five SUVAT equations are given below: s = ut at2 v = u + at s = u+v 2 Write these out five times to memorise them. t s = vt 1 2 at2 v 2 = u 2 + 2as

4 2 For each of the situations below, make a SUVAT table, write down the equation needed, and then rearrange to solve the problem. Make sure you use displacement and not distance. (a) A particle moves in a straight line. When t = 0 its velocity is 3m/s. When t = 4 its velocity is 12m/s. Find its acceleration, assumed to be constant. (b) A car is approaching traffic lights at 15m/s when the driver applies the brake and comes to a stop in 45m. Find the deceleration, assumed constant, and the time taken to stop. (c) A particle has constant acceleration 6m/s 2 whilst travelling in a straight line between points A and B. It passes A at 2m/s and B at 5m/s. Calculate the distance AB. (d) A person on the top of a tower of height 45m holds their arm out over the side of the building and drops a stone vertically downwards. The stone takes 3.03 s to reach the ground. Use this information to prove that the value of acceleration due to gravity is 9.8 to 2s.f.

5 3 The diagram below is a speed-time graph representing the motion of a cyclist along a straight road. At time t = 0 seconds, the cyclist is moving with speed u m/s. The speed is maintained until time t = 15 seconds, when the cyclist slows down with constant deceleration, coming to rest when t = 23 seconds. The total distance they travel is 152m in 23 seconds. Find the value of u Hint: From GCSE Physics, you may know that the area under a speed-time graph is the same as the distance, since distance = speed x time.

6 4 Draw a labelled mathematical force diagram to model each of the following situations on the next slide. Help Notes Objects are normally modelled as particles, which are drawn as a small circle. We used particles because we assume that all of the object s mass is centred into a single point. Forces are drawn as arrows pointing in the direction in which they are acting, and are normally drawn coming out of the particle. Forces are usually written in capital letters: W for weight (which is always mass x g). Weight always goes vertically downwards R for Normal Reaction. Normal Reactions always go perpendicular to the surface a particle is in contact with. T for Tension F for Friction An unknown force of indeterminate cause is often called P or X

7 4 Draw a labelled mathematical force diagram to model each of the following situations. (a) A book of mass 400g resting on a horizontal table (b) A fish of mass M kg dangling from a vertical fishing line (c) An ice hockey puck gliding across the ice at a constant velocity (d) A box of mass 75g resting on a rough table which is sloping at an angle of 30 o to the horizontal (e) A dog being dragged along by its lead (at an angle of 45 o to the horizontal) from rest (f) The head of a mop being pushed across the floor at a constant speed by its handle (which is at an angle of 30 o to the horizontal) (g) A wet jumper hanging by a smooth hanger on a washing line

8 5 (a) The diagram shows a rectangle with a square cut out. The rectangle has length 3x y + 4 and width x + 7. The square has length x 2. Find an expanded and simplified expression for the shaded area. (b) Given that 2x + 5y 3x y 2x + y = ax 3 + bx 2 y + cxy 2 + dy 3, where a, b, c, and d are constants. Find the values of a, b, c, and d.

9 Solve the following equations by factorising 6 (a) 10x = 0 (b) 6m 7 3 m = 0 (c) 2r 2 15r + 13 = r 2 (d) x 2 + 4a 2 = 4ax (e) This shape has an area of 44m 2. Find the value of x. x x (x + 3) 2x

10 7 (a) Find the coordinates where the line y = x + 1 intersects the curve x 2 2y + 3 = 0. What can you conclude from the number of solutions that you have found? Draw a sketch to illustrate this (no need for coord axes) (b) Find the coordinates where the line y = 3x crosses the circle x 2 + y 2 = 120. Draw a sketch to illustrate this (no need for coord axes)

11 8 These graph transformation rules are in the GCSE Higher Tier. Shifts: f x + b is f(x) moved up by b units. f x b is f x moved down by b f(x + b) means shift f(x) left by b. f(x b) means shift f(x) right by b (a) If f x = x 2, sketch the following transformations on separate sets of axes, stating the exact coordinates of the x and y intercepts and equations of any asymptotes. i) f x + 1 ii) f x 3 iii) f(x + 1) iv) f(x 2) (b) If f x = 1 x2, sketch the following transformations on separate sets of axes, stating the exact coordinates of the x and y intercepts and equations of any asymptotes. i) f x + 1 ii) f x 3 iii) f(x + 1) iv) f(x 2)

12 9 (a) The straight line l passes through 0,6 and has gradient 2. It intersects the line with equation 5x 8y 15 = 0 at point P. Find the coordinates of P. (b) The straight line l 1 with equation y = 3x 1 intersects the straight line l 2 with equation ax + 4y 17 = 0 at the point P( 3, b). i) Find the value of b. ii) Find the value of a.

13 10 (a) The line segment AB is a diameter of a circle, where A and B are 4,7 and 8,3 respectively. Find the equation of the circle. (b) By completing the square for the terms in x and for the terms in y, put the circle equation below into the form x a 2 + y b 2 = r 2, and hence find the centre and radius of the circle: x 2 14x + y y 12 = 0 Reminder: The equation of a circle with centre a, b and radius r is x a 2 + y b 2 = r 2. You can find the mid-point of a line segment by averaging the x and y coordinates of its endpoints i.e. if you have coordinates A x 1, y 1 and B x 2, y 2 the mid-point between those points would be x 1 +x 2 2, y 1+y 2 2. From Pythagoras, you can find the distance AB between A x 1, y 1 and B x 2, y 2 by the formula AB = x 1 x y 1 y 2 2

14 1 - Answers (a) p = 1 (b) 22,230 m (c) p = 37, q = 36

15 2 - Answers (a) 2.25m/s 2 (b) 2.5m/s 2 and 6 seconds to stop (c) 1.75 m (d) Proof

16 3 - Answers u = 8m/s

17 4 - Answers Watch this video M1 force diagrams 1 made by Reef Maths on YouTube (or use this QR code) for the answer. It is important that you try these yourselves before you look at the video

18 (a) 2x 2 xy + 29x 7y + 24 (b) a = 12, b = 32, c = 3, d = Answers

19 6 - Answers (a) x = ± 5 (b) m = 1 3, m = 3 2 (c) r = 3 2, r = 5 2 (d) x = 2a (e) x = 4

20 7 - Answers (a) (1, 2). One repeated real root, therefore the line is a tangent to the curve (b) 2 3, 6 3 and 2 3, 6 3

21 8 - Answers We want you to get into the habit of checking these graphs early. Only check though once you ve had a good go at drawing them by hand yourself. On your Graphical Calculator, SmartPhone, Tablet, Laptop, or Library Computer, you can plot all of these graphs using Desmos. Type Desmos into Google and you ll see it.

22 9 - Answers (a) P 3,0 (b) i) b = 10 ii) a = 19 S

23 (a) x y 5 2 = 40 (b) C 7, 8 and r = Answers