Cambridge A Level Mathematics 9709 — 2024 May/June Paper 1 · Variant 3

1 Find the coefficient of x2 in the expansion of ( 2 - 5x) ( 1 + 3x) 10 . [4] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... ....................................................................................................................................................................

Mark scheme: 1 Correct second term 30x in expansion of 10 (1 3 ) x B1 WWW, may be implied later. Correct third term 2 405  x B1 Ignore subsequent terms, may be implied later. Multiply   2 5 x by their 2 30 405  x x to obtain two 2 x terms only M1 Expect 2 2 150 , 810  x x . Coefficient is 660 A1 Must be clearly identified. Allow final answer 2 660x . 4

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2 (a) y A x O B The diagram shows the curve y = k cos ( x - 1 r) where k is a positive constant and x is measured 6 in radians. The curve crosses the x-axis at point A and B is a minimum point. Find the coordinates of A and B. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the exact value of t that satisfies the equation 3 sin -1 ( 3t) + 2 cos -1 b 1 2l = r . [2] 2 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 2(a) State 5 3( π, 0) for point A Allow 5 π 3 x  or exact equivalent. 19 π 6 x  for point B B1 Or exact equivalent. May be implied in coordinate or vector form.  y k for point B B1 May be implied in coordinate or vector form. 3 2(b) Solve at least as far as 1 sin 3 π   t k with correct value for   1 1 2 cos 2  M1 Allow use of π 3.14...  . Allow 1 sin 3 30   t . 1 1 6 sin 3 π   t and hence 1 6  t A1 Or exact equivalent. Can use degrees if consistent. 2

More questions on Trigonometry

3 i rad C r cm r cm A B The diagram shows a sector of a circle with centre C. The radii CA and CB each have length r cm and the size of the reflex angle ACB is i radians. The sector, shaded in the diagram, has a perimeter of 65 cm and an area of 225 cm 2. (a) Find the values of r and i. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the area of triangle ACB. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 3(a) State 2 65 r r   and 2 1 225 2 r  Form a 3-term quadratic or cubic in r or  or r from correct arc and sector formula *M1 Condone sign errors. Solve their 3 term quadratic or cubic to obtain values of r or  DM1 Expect    2 2 65 450 2 45 10      r r r r or    2 18 97 72 9 8 2 9          . 10  r and 4.5  ignore 8 22.5 and 9 r    , do not ignore 0  r A1 B1 SC if no quadratic or cubic solution. If 0  r included A0 or B0 SC. 4 3(b) Use correct formula for area of triangle with clear use of angle being 2π their   M1 Expect 1.783 or 102.2o, their  must be reflex. 48.9 A1 AWRT, WWW or a second answer. Or greater accuracy; condone absence of units. 2

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4 (a) Show that the equation cos i ( 7 tan i - 5 cos i) = 1 can be written in the form a sin 2i + b sin i + c = 0 , where a, b and c are integers to be found. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Hence solve the equation cos 2 x ( 7 tan 2x - 5 cos 2 x) = 1 for 0° 1 x 1 180° . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 4(a) Use identity sin tan cos    M1 Use identity 2 2 cos 1 sin θ  M1 ± ( 2 5sin θ 7sinθ 6 0    ) A1 3 Question Answer Marks Guidance 4(b) Attempt solution of their 3 term equation and correct process to find at least 1 value of sin x or sin 2x or sin  M1 Expect    5 3 2 0, 3 / 5. s s s     18.4  x A1 Or greater accuracy. B1 SC if no solution to the quadratic. 71.6  x or (90 – their 18.4) or greater accuracy; and no other solutions for 0 180    x A1FT WWW B1 SC FT if no solution to the quadratic. B1 SC both correct in radians, 0.322, 1.25. 3

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2 1 5 The equation of a curve is y = 2x - + 3 . 2x (a) Find the coordinates of the stationary point. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Determine the nature of the stationary point. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) For positive values of x, determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 5(a) Differentiate to obtain 2 1 2 4   x x B1 OE Condone ‘+c’. Equate first derivative to zero and solve 2 4 0   K x x as far as 3 , and x k K  k non- zero M1 Not given if ‘+c’ used. 1 2  x and 9 2  y A1 OE B1 SC if no visible solution of the cubic. 3 Question Answer Marks Guidance 5(b) Differentiate their first derivative, substitute their x value. Substitution may be implied by a correct inequality or correct value, M1 Must differentiate one term correctly. Expect 3 1 4 12 at 2 x x      Alternative: substitute values of x into d d y x . One value 1 2  x and one value 1 0. 2 x    conclude minimum A1 Following correct work only 2 5(c) State increasing … B1 … with clear reference to first derivative always being positive [for 0] x  B1 Dependent on first derivative being correct. It is not sufficient to substitute values of x. 2

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4 dy - 20 6 A curve passes through the point b , - 3l and is such that = . 5 dx ( 5 x - 3 ) 2 (a) Find the equation of the curve. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) The curve is transformed by a stretch in the x-direction with scale factor 1 followed by a translation 2 2 of e o. 10 Find the equation of the new curve. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 6(a) Integrate to obtain form 1 (5 3)  k x *M1 OE 1 4(5 3)  x A1 Or unsimplified equivalent. Condone absence of ...c so far. Substitute 4 5  x and 3  y to attempt value of c DM1 DM0 for substituting 4 3, 5        . 1 4(5 3) 7     y x allow f(x) 1 4(5 3) 7     or f x A1 OE Condone c = –7 as the final answer providing    4 or 5 3     y f x c x OE is seen earlier. Attempts to write equation in   y mx c form scores A0. Do not ISW. Gains max 3/4. 4 6(b) Carry out stretch by replacing x by 2x in their equation M1 Award if given as the second transformation. Do not ignore sign errors. Carry out translation by replacing x by 2  x and y by 10  y M1 OE Award if given as the first transformation. Do not ignore sign errors. 4 3 10 23    y x A1 Or similarly simplified equivalent, WWW. 3

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7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 . (a) Find the common difference. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 7(a) B1 OE 2.5  d B1 2 7(b) Attempt to find either the first term or the last term in the set by considering 1.5 2.5( 1) 25    n or 1.5 2.5( 1) 100    n or equivalent equations M1 Using their d. May be implied by correct answers. State or imply that 11th term or 26.5 is the first in the set A1 State or imply that 40th term or 99 is the last in the set A1 Either use 40 10  S S Or use 1 2 ( )  n a l with correct results for their d Or use 1 2 [2 ( 1) ]   n a n d with correct results for their d DM1 Their 40 and 10 from correct working with their d. Correct values 30, 26.5 and 99 respectively. Correct values 30, 26.5 and 2.5 respectively. Obtain 1882.5 A1 OE 5

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8 A circle with equation x 2 + y 2 - 6x + 2y - 15 = 0 meets the y-axis at the points A and B. The tangents to the circle at A and B meet at the point P. Find the coordinates of P. [8] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... 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Mark scheme: 8 Substitute and attempt solution of 3-term quadratic equation in y M1 If 0 y used can score a maximum of M0 A0 B1 M1 A0 A1FT DM1 A0, i.e. 4/8. –5 and 3 A1 B1 SC if no working to solve the quadratic. State or imply centre of circle is (3, 1)  B1 Condone errors which don’t affect finding centre. May be implied by the correct final y coordinate. Attempt gradient of AC or BC *M1 4 3  or 4 3 A1 State or imply gradient of tangent is 3 4 or 3 4  A1FT Following their gradient of radius. Only FT when previous 2 marks are M1 A0. Either solve simultaneous equations (of 2 tangent equations) to find x- coordinate Or Substitute y-value of centre into either tangent equation DM1 16 3 , 1   x y A1 Alternative Method 1: for the 4th and 5th marks Rearrange and differentiate the circle equation or differentiate implicitly (M1) Replaces the second M1. d 3 d 1 y x x y    or     1 2 2 d 3 d 25 3 y x x x     (A1) Replaces the second A1. Question Answer Marks Guidance 8 Alternative Method 2: for the last 5 marks    ACP MAP 1 4 tan 3   or identifying similar triangles PMA and AMC (M1A1) C is the circle centre, P is intersection of the two tangents, M is intersection of PC and the y-axis. 4 16 tan , , 4 3 4 3    PM PM MAP PM or use of similar triangles (M1A1) P is 16, 1 3         (A1) Alternative Method 3: for the last 5 marks Pythagoras on triangle PAC, 2 2 2,   PC PA AC (M1) Identifies the required 3 sides and sets up formula.   2 2 2 2 2 3 , 4 , radius 5       PC PM PA PM AC (A1) Finds each side with two in terms of PM OE.   2 2 2 2 3 4 5     PM PM leads to 16 6 32, 3   PM PM (M1A1) Sets up and solves equation. P is 16, 1 3         (A1) 8

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9 y x O 1 3 The diagram shows the curve with equation y = 2x 3 + 10 . (a) Find the equation of the tangent to the curve at the point where x = 3 . Give your answer in the form ax + by + c = 0 where a, b and c are integers. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) The region shaded in the diagram is enclosed by the curve and the straight lines x = 1, x = 3 and y = 0 . Find the volume of the solid obtained when the shaded region is rotated through 360° about the x-axis. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 9(a) Differentiate to obtain form 1 2 2 3 (2 10)   kx x M1 OE 1 2 2 3 3 (2 10)   x x A1 Or unsimplified equivalent. Substitute 3  x in first derivative and evaluate to find gradient *M1 Expect 27 8 . Allow if first derivative of forms 1 3 2 (2x 10)   k , 1 3 2 (2x 10)   kx or 1 2 3 2 (2x 10)   kx . Attempt equation of tangent at   3, 8 with numerical gradient DM1 Use of gradient of the normal is DM0. [±]( 27 8 17) 0    x y or integer multiples A1 5 9(b) State or imply volume is 3 π (2 10) d   x x B1 Implied if π appears only at the end. Do not allow an unsimplified:     2 1/2 3 π 2 10 x   . Integrate to obtain 4 1 2  k x k x and evaluate using limits 1 and 3 M1 Where 1 2 0  k k . 60π A1 OE Allow from a correct integral and sight of limits. Allow numerical answers in the range 188-189. 3

More questions on Differentiation

10 The geometric progression a , a , a , … has first term 2 and common ratio r where r 2 0 . 1 2 3 It is given that 9 a + 7a = 8 . 2 5 3 (a) Find the value of r. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the sum of the first 20 terms of the geometric progression. Give your answer correct to 4 significant figures. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the sum to infinity of the progression a , a , a , … . [3] 2 5 8 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................

Mark scheme: 10(a) Substitute to obtain equation 4 2 9 14 8 0   r r B1 OE Attempt solution of quadratic equation in 2 r to obtain at least one value of r or 2 r M1 Expect    2 2 9 4 2   r r . 2 3  r only A1 SC B1 answer without working. 3 10(b) Substitute 2  a and their r in correct formula and attempt to evaluate M1 Expect 20 2 2 1 3 2 1 3                       or 20 2 2 1 3 . 2 1 3                       5.998 A1 AWRT and no other value. 2 10(c) Identify 2 4 3  a and common ratio as 8 27 . B1 FT Following their r provided 1. r  May be implied in the sum to infinity. Allow 3 . 2 3       Substitute their new a and r in correct formula for sum to infinity and evaluate M1 1  r otherwise M0. 36 19 A1 OE Accept 1.89 or better from 1.894736….. 3

More questions on Series

11 The function f is defined by f ( x) = 10 + 6x - x 2 for x d R . (a) By completing the square, find the range of f. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The function g is defined by g( )x = 4 x + k for x d R where k is a constant. (b) It is given that the graph of y = g -1 f ( x) meets the graph of y = g ( x) at a single point P. Determine the coordinates of P. 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Mark scheme: 11(a) Express f ( )x as: 2 ( 3)   a x or 2 3   a x where 19 or 1 a   If the form   2 6 10     f x x x is used the form must be returned to  f x Completed square form must give 2 x . Answers must come from completion of the square (not calculus or graphs). 2 19 (3 )  x or 2 19 ( 3)   x A1 OE  19 f x  or 19 y  with ⩽, not < or –∞ < f(x) 19  or –∞ ⩽ f(x) 19  or (–∞, 19] or [–∞, 19] A1 FT Using their constant following the award of M1. SC B1 answer only or answer from a method not involving completion of the square. 3 Question Answer Marks Guidance 11(b) 1 1 4 g ( ) ( )    x x k B1    1 2 1 10 6 4 4        g f x x x k x k M1 OE May use their completed square form for f(x). Simplify the quadratic equation obtained from 1 g f ( ) g( )   x x provided k is present and apply 2 4 0   b ac to this quadratic equation *M1 Expect 2 10 10 5 0. x x k     Obtain   100 4 5 10 0    k and hence 7  k A1 Use their k to form and solve a quadratic in x DM1 Allow if their quadratic has two solutions.   5, 13   only A1 SC B1 if no method seen. Alternative Method for first 4 marks State f ( ) gg( )  x x (B1) gg( ) 16 5   x x k (M1) Apply 2 4 0   b ac to quadratic equation obtained from f ( ) gg( )  x x (*M1) Provided k is present. 100 4(5 10) 0    k and hence 7  k (A1) 6

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