Q Scheme Marks AOs. Attempt to multiply out the denominator (for example, 3 terms correct but must be rational or 64 3 seen or implied).

Transcription

1 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 1 Attempt to multiply the numerator and denominator by k(8 3). For example, Attempt to multiply out the numerator (at least 3 terms correct). M1 3rd M1 1.1a Rationalise the denominator of a fraction with a simple surd denominator Attempt to multiply out the denominator (for example, 3 terms correct but must be rational or 64 3 seen or implied). M p and q stated or implied (condone if all over 61) or p, q A1 (4 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 1

2 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions a Statement that discriminant is b 4ac, and/or implied by writing k8 418k 1 Attempt to simplify the expression by multiplying out the brackets. Condone sign errors and one algebraic error (but not missing k term from squaring brackets and must have k, k and constant terms). k k k k o.e. M1 1.1a 4th M1 Understand and use the discriminant; conditions for real, repeated and no real roots k 16k 60 A1 (3) b Knowledge that two equal roots occur when the discriminant is zero. This can be shown by writing b 4ac = 0, or by writing k 16k 60 0 k 10, k 6 A1 M1 5th Solve problems involving the discriminant in context and construct simple proofs involving the discriminant c Correct substitution for k = 8: f( x) x 16x 65 Attempt to complete the square for their expression of f(x). B1 M1.a 3rd Solve quadratic equations by use f( x) x8 1 of formula () Statement (which can be purely algebraic) that f(x) > 0, because, for example, a squared term is always greater than or equal to zero, so one more than a square term must be greater than zero or an appeal to a reasonable sketch of y = f(x). A1.3 (3) (8 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

3 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions a Not all steps have to be present to award full marks. For example, the second method mark can still be awarded if the answer does not include that step. b Award full marks for k = 6, k = 10 seen. Award full marks for valid and complete alternative method (e.g. expanding (x a) comparing coefficients and solving for k). c An alternative method is acceptable. For example, students could differentiate to find that the turning point of the graph of y = f(x) is at (8, 1), and then show that it is a minimum. The minimum can be shown by using properties of quadratic curves or by finding the second differential. Students must explain (a sketch will suffice) that this means that the graph lies above the x-axis and reach the appropriate conclusion. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 3

4 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 3a 3b 3ci 115 (m) is the height of the cliff (as this is the height of the ball when t = 0). Accept answer that states 115 (m) is the height of the cliff plus the height of the person who is ready to throw the stone or similar sensible comment. Attempt to factorise the 4.9 out of the first two (or all) terms. h( t) 4.9 t.5t 115 or t h( t) ( 4.9) or 5 5 h( t) 4.9 t ( 4.9) t h( t) o.e. (N.B = ) h( t) 4.9t t Accept the first term written to 1,, 3 or 4 d.p. or the full answer as shown. Statement that the stone will reach ground level when h(t) = 0, or 4.9t 1.5t is seen. Valid attempt to solve quadratic equation (could be using completed square form from part b, calculator or formula). Clearly states that t = 6.5 s (accept t = 6.3 s) is the answer, or circles that answer and crosses out the other answer, or explains that t must be positive as you cannot have a negative value for time. B1 3.a 4th (1) Understand the concepts of domain and range M1 3.1a 4th M1 3.1a A1 3.1a (3) Solve simple quadratic equations by completing the square M1 3.1a 4th M1 3.1a A1 3.5a (3) Form and solve quadratic equations in context Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 4

5 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 3cii h max = awrt 13 ft A from part b. t = 5 or t = B1ft 3.1a 4th B1ft 3.a Form and solve quadratic equations in context ft C from part b. () 3c (9 marks) Award 4 marks for correct final answer, with some working missing. If not correct B1 for each of A, B and C correct. If the student answered part b by completing the square, award full marks for part c, providing their answer to their part b was fully correct. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 5

6 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 4a Attempt to solve q(x) = 0 by completing the square or by using the formula. x or 10x 0 0 x (1)( 0) x (1) M1 3rd Solve quadratic equations by use of formula x and/or statement that says a = 5 and b = 5 A1 () 4b Figure 1 q(0) = 0, so y = q(x) intersects y-axis at (0, 0) and x-intercepts labelled (accept incorrect values from part a). B1ft 3rd Sketch graphs of quadratic functions y = p(x) intersects y-axis at (0, 3). B1 y = p(x) intersects x-axis at (6, 0). B1 Graphs drawn as shown with all axes intercepts labelled. The two graphs should clearly intersect at two points, one at a negative value of x and one at a positive value of x. These points of intersection do not need to be labelled. B1 (4) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 6

7 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 4c Statement indicating that this is the point where p(x) = q(x) 1 or x 10x 0 3 x seen. Their equation factorised, or attempt to solve their equation by completing the square. x 19x 46 = 0 (x 3)(x + ) = , 4 M1.a 4th M1 A1 Solve more complicated simultaneous equations where one is linear and one is quadratic,4 A1 (4) 4d x < or x 3 o.e. { x: x, x } { x: x, x 11.5} NB: Must see or or (if missing SC1 for just the correct inequalities). B1.a 4th B1.a () Represent solutions to quadratic inequalities using set notation (1 marks) 4a Equation can be solved by completing the square or by using the quadratic formula. Either method is acceptable. 4b Answers with incorrect coordinates lose accuracy marks as appropriate. However, the graph accuracy marks can be awarded for correctly labelling their coordinates, even if their coordinates are incorrect. 4c If the student incorrectly writes the initial equation, award 1 method mark for an attempt to solve the incorrect equation. Solving the correct equation by either factorising or completing the square is acceptable. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 7

8 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 5 Figure Asymptote drawn at x = 6 B1 5th Asymptote drawn at y = 5 B1 Understand and use properties of asymptotes for graphs of the form y = a/x and y = a/x 13 Point 0, 3 labelled. B1 Condone 13 3 clearly on y axis. Point 6,0 5 Condone 6 5 labelled. clearly on x axis. Correctly shaped graph drawn in the correct quadrants formed by the asymptotes. B1 B1 (5) (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 8

9 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 6a Figure 3 Evidence of attempt to show stretch of sf 1 in x direction (e.g. one correct set of coordinates not (0, )). M1 3rd Transform graphs using stretches Fully complete graph with all points labelled. A1 () Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 9

10 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 6b Figure 4 Evidence of attempt to show reflection in y axis (e.g. one correct set of coordinates not (0, )). M1 3rd Transform graphs using translations Fully complete graph with all points labelled. A1 () (4 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 10

11 Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions 7a Figure 5 Graph of y = x + 5 drawn. B1 4th Graph of y + x = 6 drawn. B1 Represent linear and quadratic inequalities on graphs Graph of y = drawn onto the coordinate grid and the triangle correctly shaded. B1.a (3) 7b Attempt to solve y = x + 5 and y + x = 6 simultaneously for y. y = 3.4 A1 Base of triangle = 3.5 B1.a Area of triangle = 1 ( 3.4 ) 3.5 M1.a M1.a 5th Solve problems involving linear and quadratic inequalities in context Area of triangle is.45 (units ). A1 (5) 7b (8 marks) It is possible to find the area of triangle by realising that the two diagonal lines are perpendicular and therefore finding the length of each line using Pythagoras theorem. Award full marks for a correct final answer using this method. In this case award the second and third accuracy marks for finding the lengths.45 and 9.8 Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 11

12 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 1a Use of the gradient formula to begin attempt to find k. k 1 ( ) 3 or 1 (3k 4) ( k 1) 3 (i.e. correct 3k 4 1 substitution into gradient formula and equating to k + 6 = k 1 = 7k 3 ). k = 3* (must show sufficient, convincing and correct working). M1.a 1st A1* Assumed knowledge. 1b Student identifies the coordinates of either A or B. Can be seen or implied, for example, in the subsequent step when student attempts to find the equation of the line. A(5, ) or B(1, 4). Correct substitution of their coordinates into y = mx + b or y y 1 = m(x x 1) o.e. to find the equation of the line. For example, 3 5 b b 3 3 y 4 x1 or y x 5 or 3 11 y x or 3x y11 0 or () B1 nd M1 A1 (3) Find the equation of a straight line given the gradient and a point on the line. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

13 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 1c Midpoint of AB is (3, 1) seen or implied. B1.a 3rd Slope of line perpendicular to AB is 3, seen or implied. B1.a Attempt to find the equation of the line (i.e. substituting their midpoint and gradient into a correct equation). For example, b 3 or y1 x 3 M1 Find the equation of a perpendicular bisector. x3y3 0 or 3y x 3 0. Also accept any multiple of x3y3 0 providing a, b and c are still integers. A1 (4) (9 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

14 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane a 11 ( 7) 18 9 m Correct substitution of (4, 7) or ( 6, 11) and their gradient into y = mx + b or y y 1 = m(x x 1) o.e. to find the equation of the line. For example, b b 5 9 or 5 or y 7 x or y 11 x 6 B1 nd M1 5y + 9x 1 = 0 or 5y 9x + 1 = 0 only A1 (3) Find the equation of a straight line given two points. b 1 1 y0, x so,0 9 A 9. Award mark for 1 x seen. 9 1 x0, y so 5 Area = B 1 0, 5. Award mark for 1 y seen. 5 B1 3rd B1 B1 Solve problems involving length and area in the context of straight line graphs. (3) (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

15 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 3 y = mx seen or implied. M1 4th Substitutes their y = mx into x 6x mx 8( mx ) 4 o.e. Rearranges to a 3 term quadratic in x (condone one arithmetic error). 1 m x (6 1 m) x 16 0 x 6x y 8y 4 M1 3.1a M1 Use the discriminant to determine conditions for the intersection of circles and straight lines. Uses b 4ac 0, 6 1m 41 m 16 0 M1 3.1a Rearranges to 0m 36m 7 0 or any multiple of this. A1 Attempts solution using valid method. For example, M1.a m 9 9 m or m o.e. (NB decimals A0). 10 A1 (7) (7 marks) y Elimination of x follows the same scheme. x leading to m y y 6 y 8y 4 m m This leads to (1 m ) y (4 6m 8 m ) y 4 1m 4m 0 Use of b 4ac 0 gives 4 6m 8m 41 m 4 1m 4m 0 which reduces to 4m 0m 36m 7 0. m cannot equal 0, so this must be discarded as a solution for the final A mark. b 4ac 0could be used implicitly within the quadratic equation formula. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

16 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 4a Student attempts to complete the square twice for the first equation (condone sign errors). x y x y Centre ( 5, 6) A1 3.a Radius = 8 A1 3.a M1.a 4th Find the centre and radius of a circle, given the equation, by completing the square. Student attempts to complete the square twice for the second equation (condone sign errors). x 3 9 y q q 9 x 3 y q 18 q M1.a Centre (3, q) A1 3.a Radius = 18 q A1 3.a (6) 4b Uses distance formula for their centres and 80. For example, q 80 Student simplifies to 3 term quadratic. For example, q 1q 0 0 M1.a 5th M1 Solve coordinate geometry problems involving circles in context. Concludes that the possible values of q are and 10 A1 (3) (9 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

17 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 5a Student completes the square twice. Condone sign errors. x y x y So centre is (4, 5) A1 and radius is 40 M1 4th A1 (3) Find the centre and radius of a circle, given the equation, by completing the square. 5b Substitutes x = 10 into equation (in either form). or y y 10y 1 0 Rearranges to 3 term quadratic in y y 10y 1 0 (could be in completed square form y 5 4) M1.a 5th M1 Solve coordinate geometry problems involving circles in context. Obtains solutions y = 3, y = 7 (must give both). A1 Rejects y = 7 giving suitable reason (e.g. 7 < 5) or it would be below the centre or AQ must slope upwards o.e. B1.3 5c 3 ( 5) 1 m AQ = m 3 (i.e. 1 over their l (4) m AQ ) B1ft.a Substitutes their Q into a correct equation of a line. For example, b or y 3 3 x 10 B1 5th M1 y = 3x + 7 A1 (4) Find the equation of the tangent to a given circle at a specified point. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

18 Pure Mathematics Year 1 (AS) Unit Test : Coordinate geometry in the (x, y) plane 5d uuur 6 AQ o.e. (could just be in coordinate form). M1 3.1a 5th Solve coordinate geometry problems involving circles in context. uuur AP o.e. so student concludes that point P has 6 coordinates (, 1). Substitutes their P and their gradient 1 ( m from 5c) into a 3 correct equation of a line. For example, y x 3 3 b 1 3 or y1 x AQ M1 3.1a M1.a A1 5e PA 40 Uses Pythagoras theorem to find 40 EP. 9 Area of EPA = (could be in two parts). 9 Area = 0 3 (4) B1 3.1a 5th B1.a M1 A1 (4) Solve coordinate geometry problems involving circles in context. (19 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

19 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 1 Correctly shows that either 1 f(3) = 0, f( ) = 0 or f = 0 Draws the conclusion that (x 3), (x + ) or (x + 1) must therefore be a factor. Either makes an attempt at long division by setting up the long division, or makes an attempt to find the remaining factors by matching coefficients. For example, stating or x ax bx c x x x or x rx px q x x x x 1 ux vx w x x 13x 6 For the long division, correctly finds the the first two coefficients. For the matching coefficients method, correctly deduces that a = and c = or correctly deduces that r = and q = 3 or correctly deduces that u = 1 and w = 6 For the long division, correctly completes all steps in the division. For the matching coefficients method, correctly deduces that b = 5 or correctly deduces that p = 5 or correctly deduces that v = 1 States a fully correct, fully factorised final answer: (x 3)(x + 1)(x + ) M1 3.1a 4th M1.a M1 A1.a A1 A1 Divide polynomials by linear expressions with no remainder (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 1

20 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra Other algebraic methods can be used to factorise h(x). For example, if (x 3) is known to be a factor then 3 x x 13x 6 x ( x 3) 5 x( x 3) ( x 3) by balancing (M1) (x 5x )( x 3) by factorising (M1) (x 1)( x )( x 3) by factorising (A1) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

21 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra a States or implies the expansion of a binomial expression to the 8th power, up to and including the x 3 term ( a b) C a C a b C a b C a b... or ( a b) a 8a b 8a b 56 a b... Correctly substitutes 1 and 3x into the formula: (1 3 x) x 81 3x x... M1 1.1a 5th M1 Understand and use the general binomial expansion for positive integer n Makes an attempt to simplify the expression ( correct coefficients (other than 1) or both 9x and 7x 3 ) (1 3 x) 1 4x 89x 56 7 x... M1 dep States a fully correct answer: A1 8 3 (1 3 x) 1 4x 5x 151 x... b States x = 0.01 or implies this by attempting the substitution: Attempts to simplify this expression ( calculated terms correct): (4) M1.a 5th M1 Find approximations using the binomial expansion for positive integer n = (5 s.f.) A1 (3) (7 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 3

22 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 3a States or implies the expansion of a binomial expression to the 9th power, up to and including the x 3 term ( a b) C a C a b C a b C a b... or ( a b) a 9a b 36a b 84 a b... Correctly substitutes and px into the formula. ( px) px 36 px 84 px... Makes an attempt to simplify the expression (at least one power of calculated and one bracket expanded correctly) ( px) px 3618 p x 8464 p x... M1 1.1a 5th M1 M1dep Use the binomial expansion to find arbitrary terms for positive integer n States a fully correct answer: ( px) px 4608 p x 5376 p x... A1 (4) 3bi 3 States that 5376 p 84 Correctly solves for p: M1ft A1ft.a 5th Understand and use the general binomial p p expansion for 64 4 positive integer n 3bii Correctly find the coefficient of the x term: Correctly find the coefficient of the x term: B1ft 5th B1ft (4) Understand and use the general binomial expansion for positive integer n (8 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 4

23 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra ft marks pursues a correct method and obtains a correct answer or answers from their 5376 from part a. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 5

24 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 4a Attempt is made at expanding p q 5. coefficients 1, 5, 10, 10, 5, 1 or seeing p q C p C p q C p q C3 p q C4 pq C5 q o.e. Fully correct answer is stated: Accept seeing the M1 1.1a 5th A1 Understand and use the general binomial expansion for positive integer n p q p 5p q 10p q 10p q 5pq q () 4b States that p, or the probability of rolling a 4, is 1 4 States that q, or the probability of not rolling a 4, is 3 4 States or implies that the sum of the first 3 terms (or 1 the sum of the last 3 terms) is the required probability. For example, B th B1 3.3 M1.a Use the binomial expansion to find arbitrary terms for positive integer n p 5p q 10 p q or (10 p q 5 pq q ) or or M or Either o.e. or awrt A1 (5) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 6

25 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra (7 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 7

26 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 5a Makes an attempt to interpret the meaning of f(5) = 0. For example, writing p + q = 0 5p + q = 150 A1 Makes an attempt to interpret the meaning of f( 3) = 8. For example writing p + q = 8 M1.a 5th M1.a 3p + q = 6 A1 Makes an attempt to solve the simultaneous equations. M1ft 1.1a Solves the simultaneous equations to find that p = A1ft Substitutes their value for p to find that q = 40 A1ft (7) Solve non-linear simultaneous equations in context 5b Draws the conclusion that (x 5) must be a factor. M1.a 5th Either makes an attempt at long division by setting up the long division, or makes an attempt to find the remaining factors by matching coefficients. For example, stating: 3 x 5 ax bx c x x x 40 (ft their or 40) M1ft Divide polynomials by linear expressions with a remainder For the long division, correctly finds the the first two coefficients. For the matching coefficients method, correctly deduces that a = 1 and c = 8 For the long division, correctly completes all steps in the division. For the matching coefficients method, correctly deduces that b = 6 States a fully correct, fully factorised final answer: (x 5)(x + 4)(x + ) A1.a A1 A1 (5) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 8

27 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra (1 marks) Award ft through marks for correct attempt/answers to solving their simultaneous equations. In part b other algebraic methods can be used to factorise: x 5 is a factor (M1) 3 x x x 40 x ( x 5) 6 x( x 5) 8( x 5) by balancing (M1) ( x 6x 8)( x 5) by factorising (M1) ( x 4)( x )( x 5) by factorising (A1 A1) (i.e. A1 for each factor other than (x 5)) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 9

28 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 6 13 Considers the expression x x 16 either on its own or as part of an inequality/equation with 0 on the other side. Makes an attempt to complete the square. For example, stating: x 4 (ignore any (in)equation) States a fully correct answer: M1 3.1a 6th M1 A1 Complete algebraic proofs in unfamiliar contexts using direct or exhaustive methods x 4 16 (ignore any (in)equation) Interprets this solution as proving the inequality for all values 13 of x. Could, for example, state that x 4 squared is always positive or zero, therefore 0 as a number x 0 4. Must be logically connected with the 16 statement to be proved; this could be in the form of an additional statement. So 1 x 6x 18 x (for all x) or by a string of connectives which must be equivalent to if and only if s. A1.1 (4) (4 marks) Any correct and complete method (e.g. finding the discriminant and single value, finding the minimum point by differentiation or completing the square and showing that it is both positive and a minimum, sketching the graph 13 supported with appropriate methodology etc) is acceptable for demonstrating that x x16 0 for all x. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 10

29 Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra 7a Makes an attempt to expand the binomial expression 1 x 3 (must be terms in x 0, x 1, x, x 3 and at least correct). 3 3 x x x x x A < 3x A1 x > 0* as required. A1*.a M1 1.1a 6th Solve problems using the binomial expansion (for positive integer n) in unfamiliar contexts (including the link to binomial probabilities) 7b Picks a number less than or equal to zero, e.g. x = 1, and attempts a substitution into both sides. For example, Correctly deduces for their choice of x that the inequaltity does not hold. For example, 3 0 (4) M1 1.1a 5th A1.a () Use the binomial expansion to find arbitrary terms for positive integer n (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free. 11

30 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 1a A 45 o seen or implied in later working. B1 5th Makes an attempt to use the sine rule, for example, writing M1 Solve problems involving surds in o o context and sin10 sin 45 complete simple 8x3 4x1 proofs involving o States or implies that sin10 surds 3 A1 1. o and sin 45 Makes an attempt to solve the equation for x. Possible steps could include: M1ft 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 Makes an attempt to rationalise the denominator by multiplying top and bottom by the conjugate. Possible steps could include: x M1ft x x 80 States the fully correct simplifed version for x. A1*.1 Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

31 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 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 (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 Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

32 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 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 p cos74 o Makes attempt to simplify, for example, stating p M1ft States the correct final answer. QR = 14.7 km. A1 (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. sin Q sin o M1 3.1a 4th M1ft Solve triangle problems in a range of contexts Solves to find Q = A1ft Makes an attempt to find the bearing, for example, writing bearing = M1ft 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). Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

33 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 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 M1 M1 A1 Solve more complicated trigonometric 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 Finds all other solutions to the equation. A1 (6) (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

34 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 4a 3 or awrt 3.46 B1 4th Determine exact values for trigonometric 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 B1.a B1.a B1 Transform the graphs of trigonometric functions using stretches and translations Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

35 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry (180, 0). (7) 4c o 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) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

36 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 5 Makes an attempt to begin solving the equation. For example, states that sin o cos sin Uses the identity tan to write, cos o 4 1 tan o States or implies use of the inverse tangent. For example, o 1 1 o o 3 0 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 M1 M1 Solve more complicated trigonometric equations in a given interval such as ones requiring use the tan identity (degrees) o o o o , 10, 390,... (ignore any out of range values). Subtracts 0 and divides each answer by 3. M1 o o o ,,, (ignore any out of range values). States the correct final answers to 1 decimal place. 3.3, 63.3, 13.3 cao A1 (6) (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

37 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 6a Any reasonable explanation. For example, the student did not correctly find all values of x 3 which satisfy cosx. 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 trigonometric equations in a given interval (degrees) (1) 6a 6b x = 75 B1.a 4th x = 105 B1.a () Solve simple trigonometric 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. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

38 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 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 trigonometric 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) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

39 Pure Mathematics Year 1 (AS) Unit Test 4: Trigonometry 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. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

40 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 1a Observation or measurement of every member of a population. B1 1. nd (1) Understand the vocabulary of sampling. 1b Two from: B1 1. 3rd takes a long time/costly difficult to ensure whole population surveyed cannot be used if the measurement process destroys the item B1 1. Comment on the advantages and disadvantages of samples and censuses. can be hard to manage and analyse all the data. () 1c The list of unique serial numbers. B1 1. nd Understand the vocabulary of sampling. 1d A circuit board. B1 1. nd (1) (1) Understand the vocabulary of sampling. (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

41 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling a A complete collection of relevant individual people or items. B1 1. nd Understand the vocabulary of sampling. b Opportunity (convenience). B1 1. 3rd (1) Understand quota and opportunity sampling. c Systematic. B1 1. 3rd (1) (1) Understand and carry out systematic sampling. d Two from: not random electoral register may have errors there may not be enough (500) households on the register. B1 B th Select and critique a sampling technique in a given context. () e Either: random sampling it avoids bias. Or: quota sampling no sampling frame required, continue until all quotas filled. Either: Random sampling from people buying kitchen cleaners in a large store, as this would reduce potential bias. Or: Quota sampling from people based on a chosen set of ages and genders who use kitchen cleaners, continuing until all quotas are filled, as this would avoid the need for a sampling frame and allow for a more clearly representative sample. B1.4 5th B1.4 () Select and critique a sampling technique in a given context. (7 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

42 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 3a One of: to obtain a representative sample large number of students compared to staff so would be unfair to take same numbers of both. B1.4 5th (1) Select and critique a sampling technique in a given context. 3b A list of the names of staff and students. B1 1. nd Understand the vocabulary of sampling. 3c A member of staff or a student. B1 1. nd (1) (1) Understand the vocabulary of sampling. 3d Find proportions for different strata out of 60 (either explained or some sensible calculation seen) » 54 students, » 6 staff. A1 M1 3.1b 3rd Understand and carry out stratified sampling. Select at random using a random number generator. B1 (3) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

43 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 3e One of: absence on the day of the survey sampling frame may contain errors. B1.b 5th (1) Select and critique a sampling technique in a given context. (7 marks) 3d Must be whole numbers for A1. 4a All readers of the online newspaper. B1 1. nd Understand the vocabulary of sampling. 4b A list of readers who subscribe to the extra content. B1 1. nd (1) Understand the vocabulary of sampling. 4c The subscribers. B1 1. nd (1) (1) Understand the vocabulary of sampling. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

44 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 4d Advantage: accuracy of the data, unbiased. B1 1. 3rd Disadvantage: difficult to get a 100% response to a survey. B1 1. Comment on the advantages and disadvantages of samples and censuses. 4e Natural variation in a small sample. B1 1. 3rd Bias. B1 1. () () Comment on the advantages and disadvantages of samples and censuses. (7 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

45 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 5a Quota. B1 1. 3rd (1) Understand quota and opportunity sampling. 5b Advantages two from: B1.4 5th easy to get sample size inexpensive fast can be stratified if required. B1.4 Select and critique a sampling technique in a given context. Disadvantages one from: not random could be biased. B1.4 5c Allocate each of the males a number from 1 to 300 B1 3.1b 3rd Use calculator or number generator to generate 50 different random numbers from 1 to 300 inclusive. (3) B1 Select males corresponding to those numbers. B1 Understand and carry out simple random sampling. 5d = 6 B1 3.1b 3rd Use a random number generator to select the first name (or one of the first 6 names on the list) as a starting point and then select every 6th name thereafter to get 50 names. (3) B1 () Understand and carry out simple random sampling. (9 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

46 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 6a There are a very large number of bags. B1.4 3rd Bags are tested to destruction there would be no bags left. B1.4 () Comment on the advantages and disadvantages of samples and censuses. 6b One value is less than 1 kg B1.4 3rd therefore claim is not reliable. B1.3 () Comment on the advantages and disadvantages of samples and censuses. 6c Different samples can lead to different conclusions due to natural variations. Only a small sample taken so unreliable. B1.3 B1.3 3rd Comment on the advantages and disadvantages of samples and censuses. 6d Larger sample. B1.4 3rd () (1) Comment on the advantages and disadvantages of samples and censuses. (7 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

47 Statistics Year 1 (AS) Unit Test 1: Statistical Sampling 7a (Quantitative) continuous. B1 1. 1st (1) Understand the difference between qualitative and quantitative data. 7b A list of the first two digits of the date. B1 1. nd (1) Understand the vocabulary of sampling. 7c Simple random sample B1 3.1b 5th using a random number generator to select five dates. B1 () Select and critique a sampling technique in a given context. 7d Number ordered list of data. B1 3.1b 3rd Use random number generator is choose first selected piece of data. æ 187ö Then take every 6th value è ç 30 ø B1 3.1b B1 Understand and carry out systematic sampling. 7e Some data may be missing or erroneous. B1 3.b 5th (3) (1) Select and critique a sampling technique in a given context. (8 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

48 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 1a All points correctly plotted. B nd Draw and interpret scatter diagrams for bivariate data. 1b The points lie reasonably close to a straight line (o.e.). B1.4 nd () Draw and interpret scatter diagrams for bivariate data. 1c f B1 1. nd (1) (1) Know and understand the language of correlation and regression. 1d Line of best fit plotted for at least. x 8 with D and F above and B and C below. 6 to 31 inclusive (must be correctly read from x = 7 from the line of best fit). M1 1.1a 4th A1 () Make predictions using the regression line within the range of the data. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

49 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 1e 1f It is reliable because it is interpolation (700 km is within the range of values collected). No, it is not sensible since this would be extrapolation (as 180 km is outside the range of distances collected). B1.4 4th (1) Understand the concepts of interpolation and extrapolation. B1.4 4th (1) Understand the concepts of interpolation and extrapolation. (8 marks) 1a First B1 for at least 4 points correct, second B1 for all points correct. 1b Do not accept The points lie reasonably close to a line. Linear or straight need to be noted. 1e Also allow It is reliable because the points lie reasonably close to a straight line. 1f Allow the answer It is sensible since even though it is extrapolation it is not by much provided that the answer contains both ideas (i.e. it IS extrapolation but by a small amount compared to the given range of data). Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

50 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation a = (Accept awrt 6.7 miles) 43 M1 A1 3rd Estimate median values, quartiles and percentiles using linear interpolation. () b x 10 or = o.e. (Accept awrt 9.6 miles) x 10 or x B1 4th M1 1.1a Calculate variance and standard deviation from grouped data and summary statistics. s σ = (Accept awrt 16.6 miles) (or s = = 16.6 miles) A1 c Any sensible reason linked to the shape of the distribution. For example: The distribution is (positively) skewed. A few large distances (values) distort the mean. (3) B1.4 4th (1) Calculate means, medians, quartiles and standard deviation. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

51 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation d Comparison of the two means. B1 4th For example, the mean distance for London is smaller than for Devon. Sensible interpretation comparing a county to a city. For example, distance to work into one city may not be as far as travelling to different cities in a county. B1.b Compare data sets using a range of familiar calculations and diagrams. For example, commuters need to travel further to the cities in Devon for work. Comparison of the two standard deviations: B1 For example, the standard deviation for London is larger than for Devon. Sensible interpretation relating to variability/consistency For example, there is more variability (less consistency) in the commute distances from the Greater London station than from the Devon station. B1.b (4) (10 marks) a Allow consistent use of n + 1 (i.e. for median 60.5th rather than 60th), median = 6.8 c Candidates must compare both the means and standard deviations with interpretations for full marks. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

52 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 3ai 37 (minutes). B1 nd (1) Draw and interpret box plots. 3aii Upper quartile or Q 3 or third quartile or 75 th percentile or P 75 B1 1. nd (1) Understand quartiles and percentiles. 3b Outliers. B1 1. 3rd Sensible interpretation: For example: Observation that are very different from the other observations (and need to be treated with caution). Possible errors. These two children probably walked/took a lot longer. B1.4 Recognise possible outliers in data sets. () 3c = 80 or =0 M1 4th Maximum value =55 < 80 minimum value = 5 > 0 No outliers. A1 B1 Calculate outliers in data sets and clean data. (3) 3d The scale must be the same as for school A. Figure 1 B1 nd Draw and interpret box plots. Box & whiskers 30, 37, 50 B1 5, 55 B1 (3) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

53 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 3e Three comparisons in context. Comment on comparing averages. For example, children from school A took less time on average. Comment comparing consistency of times. For example, there is less variation in the times for school A than school B. Comment on comparing symmetry: For example,both positive skew (or neither symmetrical or median closer to LQ (o.e.) for both). (Most children took a short time with a few taking longer.) Comment on comparing outliers. For example, school A has two children whose times are outliers (or errors) where as school B has no outliers. B3.b 4th (3) Compare data sets using a range of familiar calculations and diagrams. (13 marks) 3c Allow horizontal line through box. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

54 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation y =.335 (seen or implied) 00 x.5y = = (Accept awrt 749) B1 M1 M1 A1 3.1a 5th Calculate the mean and standard deviation of coded data. σ y = = σ x = M1 A1 A1 M1 3.1a = (Accept awrt 15.9) A1 (9) (9 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

55 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 5a Order the data. 15, 160, 169, 171, 175, 186, 10, 43, 50, 58, 390, 40 1 Q 3 = ( ) = 54 M1 nd A1 () Understand quartiles and percentiles. 5b Q (Q 3 Q 1 ) = (54 170) M1 4th = 380 A1 Patients F (40) and B (390) are outliers (so may be suspected by the doctor as smoking more than one packet of cigarettes per day). B1 3.a (3) Calculate outliers in data sets and clean data. (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

56 Statistics Year 1 (AS) Unit Test : Data presentation and interpretation 6 Three comparisons in context: B3.4 4th For example: Very much warmer in Beijing than Perth. Both consistent in the temperatures. Less rainfall in Beijing. Compare data sets using a range of familiar calculations and diagrams. Less likely to have high rainfall in Beijing. Rainfall in Beijing is consistently less than in Perth. B1.4 Evidence of use of a statistic from the boxplots: For example: Medians Measure of a difference in medians Mention of a particular outlier For accurately reading data from boxplots. B1.4 (5) (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

57 Statistics Year 1 (AS) Unit Test 3: Probability 1a + 3 total number of students = 5 30 = 1 M1 or awrt A1 1st Calculate probabilities for single events. () 1b total = or 7 15 or awrt A1 M1 3rd Understand and use Venn diagrams for multiple events. 1c 0 B1 3rd () No student reads both magazine A and magazine C. B1 Understand and use the definition of mutually exclusive in probability calculations. 1d P(C reads at least one magazine) = = 9 0 () B1 3rd (1) Understand and use Venn diagrams for multiple events. 1e P(B) = = 1 3, P(C) = 9 30 = 3 10 P(B and C) = 3 30 = 1 10, and M1.a P(B) P(C) = = 1 = P(B and C) 10 So yes, they are independent. B1.1 4th A1.4 Understand and use the definition of independence in probability calculations. (3) (10 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

58 Statistics Year 1 (AS) Unit Test 3: Probability 1e Allow alternative using formal conditional probability: P(B) = 1 3 (B1). Finding P(B C) = 3 1 and (3 6) 3 comparing with P(B) (M1). Correct conclusion (A1). Or P(C) = 3 10 (B1). Finding P(C B) = 3 3 and comparing with P(C) (M1). Correct conclusion (A1). ( 3 5) 10 a 3rd Draw and use tree diagrams with three branches and/or three levels. For shape and labels: 3 branches followed by 3,, with some R, B and G seen. M1 3.1a First set of branches correct. A1 Second set of branches correct. A1 (3) b P(Blue bead and a green bead) = M1 A rd Draw and use tree diagrams with three branches and/or three levels. () (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

59 Statistics Year 1 (AS) Unit Test 3: Probability a Allow 3 branches followed by 3, 3, 3 if 0 properly placed on redundant branches. R B G labels can be implied on second set but only if order is consistent and probabilities correct. Further sets of branches max M1 A1 A0 (/3). b M1 for or a 3rd Understand and use Venn diagrams for multiple events. Three closed curves and four in centre. Evidence of subtraction (any one of 31, 36, 4, 41, 17 or 11). Any three of 31, 36, 4, 41, 17 or 11 correct. All correct. Labels on sets, 16 and closed curve or box outside. M1 M1 A1 A1 B1 (5) 3.1a 3.3 3bi P(None of the 3 options) = = 4 45 or awrt B rd Understand and use Venn diagrams for multiple events. (1) 3bii P(Networking only) = or awrt B rd Understand and use Venn diagrams for multiple events. (1) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

60 Statistics Year 1 (AS) Unit Test 3: Probability 3c P(Takes all three options takes S and N) = 4 40 = 1 10 or 0.1 M1 A rd Understand and use Venn diagrams for multiple events. () (9 marks) 4a 3rd Draw and use tree diagrams with three branches and/or three levels. Correct tree structure. All labels correct. All probabilities correct. B1 B1 B1 (3) 3.1a 4bi = 1 30 or equivalent. M1 A1 () 3.4 3rd Draw and use tree diagrams with three branches and/or three levels. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

61 Statistics Year 1 (AS) Unit Test 3: Probability 4bii æ Car NL + Bike NL + Foot NL = 1 4 ö è ç 5ø + æ ö è ç 5ø + æ ö M rd è ç 10ø Draw and use tree diagrams with three branches = 4 and/or three 5 or equivalent. A1 levels. () (7 marks) 4bii ft from their tree diagram. Allow one error for M1. æ æ 1 Can also be found from 1-1 ö è ç 5ø + æ 1 6 ö è ç 5ø + æ öö è ç è ç 10ø ø 5a nd Draw and use simple tree diagrams with two branches and two levels. Tree (both sections) and labels 0.85, , 0.97, 0.06, 0.94 B1 B1 B1 3.1a (3) 5b P(Not faulty) = ( ) + ( ) M1 3.4 nd = M1dep A1 Draw and use simple tree diagrams with two branches and two levels. (3) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

62 Statistics Year 1 (AS) Unit Test 3: Probability 5b (6 marks) M1 for either or (ft from their tree diagram) and M1 (dep) for adding two such probabilities (allow one error). 6a Find total frequency = åwidth frequency density = (5 ) + (4 4) + (4 6) + (7 5) + (15 1) = 100 P(Takes longer than 18 mins) = = = 1 or equivalent. M1 A1 M1 A1 3.1a 3.1a nd Calculate probabilities from relative frequency tables and real data. (4) 6b = 5 P(Takes less than 30 mins) = or equivalent = = 9 10 M1 M1 A1.b nd Calculate probabilities from relative frequency tables and real data. (3) (7 marks) 6a M1 for attempt to find total frequency by adding at least three width frequency density terms (which may contain errors).alternative: M1 for " M1 for 1 10 ". A1 for 9 3 " 100" 10 o.e. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

63 Statistics Year 1 (AS) Unit Test 3: Probability 7a Total frequency = 10 B1 3.1a nd P(Less than 17 cm) or equivalent or M1 A1 Calculate probabilities from relative frequency tables and real data. (3) 7b P(Between 1 cm and 18 cm) or awrt Assumption: foot lengths between 17 and 19 are uniformly distributed. M1 A1 B1.b 3.5b nd Calculate probabilities from relative frequency tables and real data. (3) (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

64 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 1a k(16 9) + k(5 9) + k(36 9) (or 7k + 16k + 7k). M1.1 4th = 1 M1 Þ k = 1 50 (answer given). A1* Model simple discrete random variables as probability distributions. (3) 1b x P(X = x) Note: decimal values are 0.14, 0.3, 0.54 respectively. B1 B1.5 4th Calculate probabilities from discrete distributions. () (5 marks) 1b Ignore any extra columns with 0 probability. Otherwise 1 for each. If 4, 5 or 6 missing B0B0. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

65 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions a = = 1 M1 4th = 0.5 A1 Calculate probabilities from discrete distributions. () b P( 1 X < ) = P( 1) + P(0) + P(1) = 0.6 B1 4th Calculate probabilities from discrete distributions. c P(X >.3) = P( ) + P( 1) + P(0) + P(1) + P() = 0.85 B1 4th (1) (1) Calculate probabilities from discrete distributions. (4 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

66 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 3a k + k k = 1 M1.1 4th Þ4k = 1, so k = 0.5 (answer given). A1* Calculate probabilities from discrete distributions. 3b P(X 1 + X = 5) = P(X 1 = 3 and X = ) + P(X 1 = and X = 3) = = 0 (answer given). () B1*.4 4th Calculate probabilities from discrete distributions. (1) 3c x1 + x P(X1 + X) M1 A1 A1.5 4th Calculate probabilities from discrete distributions. (3) 3d P(1.3 X 1 + X 3.) = P(X 1 + X = ) + P(X 1 + X = 3) M th = = or 5 16 A1ft () Calculate probabilities from discrete distributions. (8 marks) 3b Must show that 5 can only be obtained from and 3 or 3 and, and so must use P(X = ) = 0 but condone explanation in words. 3c M1 for correct set of values for X 1 + X. Condone omission of 5 column. A1 for correct probabilities for 0, and 6. A1 for others. Equivalent fractions are ,,,,, Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

67 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 4a Let X be the random variable the number of games Amir loses. X ~ B(9, 0.) P(X = 3) = = to 3 sf from calculator B1 B th Calculate binomial probabilities. () 4b P( X 4) M th = awrt from calculator A1 () Use statistical tables and calculators to find cumulative binomial probabilities. (4 marks) 4a P( X 3) P ( X ) = or 9! 3!6! (0.)3 (0.8) 6 or 9 C or b 0.98 is M1A0 Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

68 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 5a X ~ B(0, 0.05) B1 for binomial B1 for 0 and 0.05 B1 B1 3.1b 3.1b 5th Understand the binomial distribution (and its notation) and its use as a model. () 5b P(X = 0) = (awrt) B1 A1 () 3.4 5th Calculate binomial probabilities. 5c P(X > 4) = 1 P( X 4) = = ( s.f.) (answer given) A1* M th () Use statistical tables and calculators to find cumulative binomial probabilities. (6 marks) 5b P(X = 0) = Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

69 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 6a X ~ B(15, 0.5) B1 3.1b 5th B1 for binomial B1 for 15 and 0.5 B1 3.1b Understand the binomial distribution (and its notation) and its use as a model. () 6bi from calculator P(X = 8) = M1 A1 () 3.4 5th Calculate binomial probabilities. 6bii P(X 4) = 1 P(X 3) = = awrt 0.98 or M th A1 Use statistical tables and calculators to find cumulative binomial probabilities. () (6 marks) 6bi P(X = 8) = P(X 8) P(X 7) = or 15! 8!7! 0.58 (1-0.5) 7 or 15 C or = awrt or Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

70 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 7a 7a Binomial (distribution). B1 1. 5th Each plate is either blue or not blue, independently of each other, with constant probability and there is a fixed number of them. B1.4 () Understand the binomial distribution (and its notation) and its use as a model. 7b X ~ B(10, 0.06) (could be seen in part a) B1.5 5th P(X > ) = 1 P(X = from calculator M1 3.4 = awrt A1 (3) Calculate binomial probabilities. (5 marks) Ignore any parameter values given for the first B1. For second B1 all four points must be made with some context. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

71 Statistics Year 1 (AS) Unit Test 4: Statistical Distributions 8a There is a fixed number of trials. B1 1. 5th Each trial results in 1 of outcomes, success and failure. B1 1. Probability of success on each trial is the same. B1 1. The trials are independent. B1 1. (4) Understand the binomial distribution (and its notation) and its use as a model. 8bi P(X = 5) = 7 ; P(X 5) = 5 7 (either may be implied in either part) Idea of five failures followed by a success (or 0 successes out of five and then a success) seen or implied. B1 3.3 B1 M th Calculate binomial probabilities. P(5 on sixth throw) = M1 = awrt A1 (5) 8bii æ B 8, ö è ç 7ø or B æ 8, 5 ö è ç 7 ø seen or implied. M th Calculate binomial probabilities. P(X = 3) = from calculator M1 A1 (3) (1 marks) 8bii P(exactly 3 fives in first eight throws) = 8! æ ö 5!3! è ç 7ø 3 æ 5ö è ç 7ø 5 o.e. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

72 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 1a Two from: Each bolt is either faulty or not faulty. The probability of a bolt being faulty (or not) may be assumed constant. Whether one bolt is faulty (or not) may be assumed to be independent (or does not affect the probability of) whether another bolt is faulty (or not). B th Understand the binomial distribution (and its notation) and its use as a model. There is a fixed number (50) of bolts. A random sample. () 1b Let X represent the number of faulty bolts. X~B(50, 0.5) P(X 6) = P(X 7) = P(X 19) = P(X 0) = M1 M1dep 3.4 5th Find critical values and critical regions for a binomial distribution. Critical Region is X 6 X 0 A (4) (6 marks) 1a Each comment must be in context for its mark. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

73 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing a The set of values of the test statistic for which the null B 1. 5th hypothesis is rejected in a hypothesis test. 1. Understand the language of hypothesis testing. () b P(X 15) = = P (X 16) = = M1 5th Critical region is 16 X ( 30) A1 Probability of rejection is A1 (3) Find critical values and critical regions for a binomial distribution. c Not in critical region therefore insufficient evidence to reject H 0. There is insufficient evidence at the 1% level to suggest that the value of p is bigger than 0.3. B1.b 6th B1 3.a () Interpret the results of a binomial distribution test in context. (7 marks) c Conclusion must be in context (i.e. use p), mention the significance level and be non-assertive. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

74 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 3a P(X 1) = and P (X ) = M1 5th P(X 10) = = and P(X 11) = = A1 Critical region is X 1 11 X ( 0) A1 Find critical values and critical regions for a binomial distribution. (3) 3b Significance level = = or.47% B1 6th Calculate actual significance levels for a binomial distribution test. 3c Not in critical region therefore insufficient evidence to reject H 0. There is insufficient evidence at the 5% level to suggest that the value of p is not 0.3. (1) B1.b 6th B1 3.a () Interpret the results of a binomial distribution test in context. (6 marks) 3c Conclusion must contain context and non-assertive for first B1. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

75 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 4a X~B(8, 0.37) M th P(X 15) = = and P(X 16) = = M1dep Critical region is X 16 A1 Find critical values and critical regions for a binomial distribution 4a 4b In critical region therefore sufficient evidence to reject H 0 B1.b 6th There is sufficient evidence at the 5% level to suggest that the value of p is bigger than (3) B1 3.a First M1 for correct distribution seen or implied. Second M1 (dependent on first) for evidence that correct probabilities for either critical value examined. 4b Conclusion must contain context and non-assertive for first B1. () Interpret the results of a binomial distribution test in context. (5 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

76 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 5a H 0 : p = 0. H 1 : p > 0. Let X represent the number of times the taxi is late. X~B(5, 0.) seen or implied. B1.5 M th Carry out 1-tail tests for the binomial distribution. Either P(X 3) = 1 P(X ) = = > 0.05 There is insufficient evidence at the 5% significance level that there is an increase in the number of times the taxi/driver is late. M1 A1 B1 B1 3.a Or P(X 3) = 1 P(X ) = P(X 4) = 1 P(X 3) = So critical region is X 4 3 < 4 or 3 is not in the critical region So there is insufficient evidence at the 5% significance level that there is an increase in the number of times the taxi/driver is late. M1 A1 B1 B1 (6) 3.a 5b Two sensible reasons. For example, Different time of the day Linda travels to work. More traffic on different days (e.g. Monday morning, Friday afternoon). Weather conditions. B.b.b 5th Understand the binomial distribution (and its notation) and its use as a model. Road works. () (8 marks) Conclusion must be non-assertive. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

77 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 6a Let X represent the number of bowls with minor defects (seen or implied). XB(5, 0.) P(X l) = P(X = 0) = P(X 8) = P(X 9) = P(X 9) = P(X 10) = may be implied M1 M1dep A1 M th Find critical values and critical regions for a binomial distribution. Critical region is X = 0 X 10 A (6) 6b Significance level = = or.11% B1 1. 6th Calculate actual significance levels for a binomial distribution test. (1) 6c H 0 : p = 0.; H 1 : p < 0. Let Y represent number of bowls with minor defects (Under H 0) Y~B(0, 0.) (may be implied) B1 M th Carry out 1-tail tests for the binomial distribution. Either P(Y ) = > 0.1 (or 10%) Insufficient evidence at the 10% level to suggest that the proportion of defective bowls has decreased. Or P(Y ) = P(Y 1) = so critical region is Y 1 Insufficient evidence at the 10% level to suggest that the proportion of defective bowls has decreased > 0.10 or < 0.9 B1 M1 A1 B1 M1 A1 (5) 3.b 3.a (1 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

78 Statistics Year 1 (AS) Unit Test 5: Statistical Hypothesis Testing 6a M1 for examining probabilities for on both sides for either critical value, A1 for each correct pair. 6c Conclusion must be non-assertive. 7 H 0 : p = 0.5, H 1 : p> 0.5 B1.5 5th Carry out 1-tail Let X represent the number of seeds that germinate. M1 3.4 tests for the binomial (Under H 0) X~B(5, 0.5) distribution. P(X 10) = 1 P(X 9) = M1 > 0.05 A1 10 is not in critical region therefore insufficient evidence to reject H 0. There is insufficient evidence at the 5% level to suggest that the book has underestimated the probability. (o.e.) B1.b B1 3.a (6 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

79 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics 1a States correct answer: 5.3 (m s 1 ) B1.a 4th Understand the difference between a scalar and a vector. 1b States correct answer: 4.8 (m s 1 ) B1.a 4th (1) Understand the difference between a scalar and a vector. 1c States correct answer: 30 (m) B1.a 4th (1) (1) Understand the difference between a scalar and a vector. (3 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

80 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics ai States that x = 0 needs to be substituted or implies it by writing h Correctly substitutes x = 0 to get h = 1.7 (m) A1 M1 3.1b 3rd Understand how mechanics problems can be modelled mathematically. aii b States that x = 7 needs to be substituted or implies it by writing h = (7) 0.01(7) Correctly substitutes x = 7 to get h =.47 (m) Accept awrt.5 (m) Understands that the ball will hit the ground when h = 0 or writes x 0.01x 0 Realises that the quadratic formula is needed to solve the quadratic. For example a = 0.01, b = 0.18, c = 1.7 seen, or makes attempt to use the formula: x Simplifies the b x 0.0 Calculates x = 4.84 (m) Accept awrt 4.8 (m) 4ac part to get or shows Does not need to show that x (m) States that the ball will be called in, or says, for example, yes as 4.84 < 5. () M1 3.1b 3rd A1 () Understand how mechanics problems can be modelled mathematically. M1 3.1b 3rd M1 M1 A1 B1 3.a (5) Understand how mechanics problems can be modelled mathematically. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

81 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics c km 1000 m 1min 1min 1km 60 sec Award 1 method mark for multiplication by 1000 and 1 method mark for division by (m s 1 ) or 33.3 (m s 1 ) M 3rd A1 Understand how mechanics problems can be modelled mathematically. (3) (1 marks) ai Award both marks for a correct final answer. aii Award both marks for a correct final answer. b a 0.01, b 0.18, c 1.7 is also acceptable. b Award the third method mark even if this step is not seen, providing the final answer is correct. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

82 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics 3a Understands that the pole vaulter will land when h = 0 or writes 1 15 x 1 x 0 60 Correctly factorises to get x x 15 Solves to get x (m) 1 Accept awrt 10.4 (m) o.e. M1 M1 3.1b 3rd A1 Understand how mechanics problems can be modelled mathematically. 3b States that the greatest height will occur when x = 5.0 (m) M1 3.1b 3rd Makes an attempt to substitute x = 5.0 into the equation for 1 h. For example, h seen. 60 h = 5.4 (m) Accept awrt 5.4 (m) (3) M1 A1 ft (3) Understand how mechanics problems can be modelled mathematically. 3c States h = 4.9 or states that x x Simplifies this to reach 1x 15x 94 0 o.e. M1 Realises that the quadratic formula is needed to solve the quadratic. For example a 1, b 15, c 94 seen, or makes attempt to use the formula: x M1 3.1b 3rd M1 Understand how mechanics problems can be modelled mathematically. Simplifies the b x 4 x = 6.8 (m) Accept awrt 6.8 (m) x = 3.58 (m) Accept awrt 3.6 (m) 4ac part to get 1513 or shows The pole vaulter can leave the ground between 3.6 m and 6.8 m from the bar. M1 A1 A1 B1 3.a (7) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

83 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics 3di Allows the person to be treated as a single mass and allows the effects of rotational forces to be ignored. B rd (1) Understand assumptions common in mathematical modelling. 3b 3dii The effects of air resistance can be ignored. B rd (1) Understand assumptions common in mathematical modelling. (15 marks) For the first method mark, accept their answer to part a divided by. Continue to award marks for a correct answer using their initial incorrect value. 3c Accept 3.6 x 6.8 Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

84 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics 4a Makes an attempt to find the distance from A to B. For example, 8 80 is seen. Makes an attempt to find the distance from B to C. For example, is seen. Demonstrates an understanding that these two values need to be added. For example, is seen (m) Accept anything which rounds to 16 (m) M1 3.1b 4th M1 3.1b M1 A1 Find the magnitude and direction of a vector quantity. uuur 4b States that AC 10i95j(m) Award one point for each value. States or implies that Finds Accept awrt tan 10 (4) B 3.1b 4th M1 A1 Find the magnitude and direction of a vector quantity. (4) (8 marks) Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

85 Mechanics Year 1 (AS) Unit Test 6: Quantities and Units in Mechanics 5a Makes an attempt to find the absolute value. For example, 14 is seen. Simplifies to 680 M1 Finds speed = 6.07 (ms 1 ) Accept awrt 6.1 (ms 1 ) M1 3.1b 4th A1 (3) Find the magnitude and direction of a vector quantity. 5b States that tan 14 Finds the value of θ, θ = 57.5 A1 Demonstrates that the angle with the unit j vector is Finds 3.47 ( ) Accept awrt 3.5( ) M1 4th M1 A1 Find the magnitude and direction of a vector quantity. 5c Ignore the value of friction between the hockey puck and the ice. (4) B rd (1) Understand assumptions common in mathematical modelling. 5d 1.4 g 1kg 100 cm 100 cm 100 cm 1000 g 1m 1m 1m 1cm 3 Award 1 method mark for division by 1000 and 1 method mark for multiplication by 100 only once and the final method mark for multiplication by 100 three times kg m 3 A1 M3 4th Know derived quantities and SI units. (4) 5b Award all 4 marks for a correct final answer. Award marks for a student stating making a mistake with the inverse or subtracting that answer from 90. (1 marks) 14 tan, and then either Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

86 Mechanics Year 1 (AS) Unit Test 7: Kinematics 1 (constant acceleration) 1a Figure 1 General shape of the graph is correct. i.e. horizontal line, followed by negative gradient, followed by a positive gradient. Vertical axis labelled correctly. Horizontal axis labelled correctly. M th A1 A1 Use and interpret graphs of velocity against time. 1b Makes an attempt to find the area of trapezoidal section where T the car is decelerating. For example, is seen. 4 Makes an attempt to find the area of the trapezoidal section 3T where the car is accelerating. For example, 10 0 is 4 seen. 5T 90T States that 15T (3) M1 4th M1 M1 Calculate and interpret areas under velocity time graphs. Solves to find the value of T: T = 30 (s). A1 (4) (7 marks) 1a Accept the horizontal axis labelled with the correct intervals. 1b Award full marks for correct final answer, even if some work is missing. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.

87 Mechanics Year 1 (AS) Unit Test 7: Kinematics 1 (constant acceleration) a Velocity = acceleration time seen or implied. M1 3.1b 4th Velocity = 11 8 = 88 m s 1 A1 Figure General shape of the graph is correct. i.e. positive gradient, followed by horizontal line, followed by negative gradient not returning to zero. M1 3.3 Use and interpret graphs of velocity against time. Vertical axis labelled correctly. Horizontal axis labelled correctly. A1 A1 b Makes an attempt to find the area of the trapezoidal section. 1 For example, is seen. Demonstrates an understanding that the three areas must total For example, T or 35 88T is seen. (5) M1 4th M1.1 Calculate and interpret areas under velocity time graphs. Correctly solves to find T = 10.5 (s). A1 (3) (8 marks) a Accept the horizontal axis labelled with the correct intervals. b Award full marks for correct final answer, even if some work is missing. Education Ltd 017. Copying permitted for purchasing institution only. This material is not copyright free.