Q Scheme Marks AOs Pearson Progression Step and Progress descriptor. and sin or x 6 16x 6 or x o.e

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1 1a A 45 seen or implied in later working. B1 1.1b 5th Makes an attempt to use the sine rule, for example, writing sin10 sin 45 8x3 4x1 States or implies that sin10 3 and sin 45 A1 1. Solve problems involving surds in context and complete simple proofs involving surds Makes an attempt to solve the equation for x. Possible steps could include: M1ft 1.1b or 16x6 8x 16x6 4x1 or x6 8x 8 3 x 3 16 x 6 or 4 x x 6 6 x or x x x 6 16x 6 or 4 6 x 6 16x 6 or 6 3 x or 6 6 x or x o.e A1ft 1.1b Makes an attempt to rationalise the denominator by multiplying top and bottom by the conjugate. Possible steps could include: x M1ft 1.1b x x 80 States the fully correct simplifed version for x. A1*.1 Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

2 9 6 x * 0 1b States or implies that the formula for the area of a triangle is 1 sin ab C or 1 sin ac B or 1 sin bc A sin15 or awrt awrt awrt awrt. or sin15 or 0.59 (7) M1 1.1a 3rd M1 3.1a Understand and use the general formula for the area of a triangle. Finds the correct answer to decimal places. 0.6 A1 1.1b (3) (10 marks) 1a Award ft marks for correct work following incorrect values for sin 10 and sin 45 1b 1 Exact value of area is If 0.6 not given, award M1M1A0 if exact value seen. 00 Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

3 a States or implies that the angle at P is 74 B1.a 4th States or implies the use of the cosine rule. For example, p q r qr cos P M1 1.1a Solve triangle problems in a range of contexts Makes substitution into the cosine rule. M1ft 1.1b p cos 74 Makes attempt to simplify, for example, stating p M1ft 1.1b States the correct final answer. QR = 14.7 km. A1 1.1b (5) b States or implies use of the sine rule, for example, writing sinq sin P q p Makes an attempt to substitute into the sine rule. M1ft 1.1b sin Q sin M1 3.1a 4th Solve triangle problems in a range of contexts Solves to find Q = A1ft 1.1b Makes an attempt to find the bearing, for example, writing bearing = M1ft 1.1b States the correct 3 figure bearing as 068 A1ft 3.a (5) (10 marks) a Award ft marks for correct use of cosine rule using an incorrect initial angle. b Award ft marks for a correct solution using their answer to part (a). Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

4 3 States sin xcos x 1 or implies this by making a substitution. 8 7cos x 6 1 cos x Simplifies the equation to form a quadratic in cos x. 6cos x 7cos x 0 Correctly factorises this equation. x x 3cos cos 1 0 or uses equivalent method for solving quadratic (can be implied by correct solutions). Correct solution. cos x or 1 3 M1.1 5th A1 1.1b Solve more complicated equations in a given interval such as ones requiring use the tan identity (degrees) Finds one correct solution for x. (48.,60, or 300 ). A1 1.1b Finds all other solutions to the equation. A1 1.1b (6) (6 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

5 4a 3 or awrt 3.46 B1 1.1b 4th Determine exact values for functions in all four quadrants (1) 4b Figure 1 Sine curve with max and min Sine curve translated 60 to the right. Sin curve cuts x-axis at ( 10, 0) and (60, 0) and the y-axis (0, 3 ). Asymptotes for tan curve at x = 90 and x = 90 Tangent curve is flipped. Uses the value of tan ( 10 ) to deduce no intersection in 3rd quadrant (can be implied). Tangent curve cuts x-axis at ( 180, 0), (0, 0) and B1.a 4th B1.a B1.a B1 1.1b B1.a B1.a B1 1.1b Transform the graphs of functions using stretches and translations Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

6 (180, 0). (7) 4c States that solutions to the equation sin( x 60 ) tan x 0 will occur where the two curves intersect. B1ft 3.1a 4th Use intersection points of graphs to solve equations 4b 4d States that there are two solutions in the given interval. A1.a 4th Ignore any portion of curve(s) outside 180 x 180 4c Award both marks for correctly stating that there are two solutions even if explanation is missing. () Use intersection points of graphs to solve equations (10 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

7 5 Makes an attempt to begin solving the equation. For example, states that sin cos sin Uses the identity tan to write, cos 4 1 tan States or implies use of the inverse tangent. For example, tan or Shows understanding that there will be further solutions in the given range, by adding 180 to 30 at least once. M1.1 5th M1.1 Solve more complicated equations in a given interval such as ones requiring use the tan identity (degrees) , 10, 390,... (ignore any out of range values). Subtracts 0 and divides each answer by ,,,...(ignore any out of range values) States the correct final answers to 1 decimal place. 3.3, 63.3, 13.3 cao A1 1.1b (6) (6 marks) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

8 6a Any reasonable explanation. For example, the student did not correctly find all values of x 3 which satisfy cos x. Student should have subtracted 150 from 360 first, and then divided by. N.B. If insufficient detail is given but location of error is correct then mark can be awarded from working in part (b). B1.3 4th Solve simple equations in a given interval (degrees) (1) 6a 6b x = 75 B1.a 4th x = 105 B1.a () Solve simple equations in a given interval (degrees) (3 marks) Award the mark for a different explanation that is mathematically correct, provided that the explanation is clear and not ambiguous. Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

9 7a Figure Correct shape of sine curve through (0, 0). Sine curve has max value of 1 and min 1 value of Sine curve has a period of (can be implied by 5 complete cycles) and passes through (1,0), (,0),..., (10,0). B1 3.1a 4th B1 3.1a B1 3.1a Transform the graphs of functions using stretches and translations 7b Student states that the buoy will be 0.4 m above the still water level 10 times. (3) B1 3.a 7th (1) Use functions in modelling (including critiquing) Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

10 7c Sensible and correct reason. For example: A buoy would not move up and down at exactly the same rate during each oscillation. The period of oscillation is likely to change each oscillation. The maximum (or minimum) height is likely to change with time. Waves in the sea are not uniform. B1 3.b 7th (1) Use functions in modelling (including critiquing) (5 marks) 7c Award the mark for a different explanation that is mathematically correct. For example, stating that the buoy would not move exactly vertically each time. Pearson Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

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