Verulam School. C3 Diff ms. 0 min 0 marks

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Adelia Hodges
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1 Verulam School C3 Diff ms 0 min 0 marks 1. (a) Attempt use of product rule * ln x + 1 [or unsimplified equiv] Equate attempt at first derivative to zero and obtain value involving e D e 1 [or exact equiv] 4 (b) Attempt use of quotient rule [ using product rule or...] [] Show that first derivative cannot be zero [AG; derivative but be correct] 3 [7] 2. (a) Attempt use of product rule involving + 2x(x + 1) x 2 (x + 1) 5 ; ignore subsequent attempt at simplification 3 (b) Attempt use of quotient rule or, with adjustment, product rule; allow u / v confusion

2 3 from correct derivative only 3 [6] 3. (i) Differentiate to obtain kx(5 x 2 ) 1 any non-zero constant correct 2x(5 x 2 ) 1 4 for value of derivative Attempt equation of straight line through (2, 0) with numerical value of gradient obtained from attempt at derivative not for attempt at eqn of normal y = 4x (ii) State or imply h = Attempt calculation involving attempts at y values addition with each of coefficients 1, 2, 4 occurring at least once k(ln5 + 4 ln ln4 + 4 ln ln1) perhaps with decimals; any constant k 2.44 allow ± (iii) Attempt difference of two areas allow if area of their triangle < area A and hence 5.56 following their tangent and area of ft 2 A providing answer positive [11] 4. Differentiate to obtain any non zero constant k, perhaps unsimplified for value of first derivative or unsimplified equiv Attempt equation of tangent through (2, 3) using numerical value of first derivative provided

3 derivative is of form k (4x +1) n y = x + or 2x 3y + 5 = 0 involving 3 terms 5 [5] 5. (i) Attempt to express x in terms of y * obtaining two terms x = + 1 State or imply volume involves Attempt to express x 2 in terms of y * dep *M; expanding to produce at least 3 terms any constant k including 1; allow if dy absent Integrate to obtain Use limits 0 and p dep *M *M; evidence of use of 0 needed π(e p + + p 5) AG; necessary detail required 8 (ii) State or imply = 0.2 maybe implied by use of 0.2 in product π(e p + +1) as derivative of V Attempt multiplication of values or expressions for and 0.2π (e 4 +2e 2 + 1) following their expression 44 ft or greater accuracy 5 [13] 6. Attempt use of quotient rule to find derivative allow for numerator wrong way round ; or attempt use of product rule for gradient

4 Attempt eqn of straight line with numerical gradient obtained from their ; tangent not normal 5x + 4y 11 = 0 5 or similar equiv 5 7. (i) derivative of form any constant k correct or (unsimplified) equiv derivative of form ke any constant k different from 6 correct 4 (ii) Either: Form product of two derivatives numerical or algebraic Substitute for t and x in product using t = 4 and calculated value of x 39.7 allow ± 0.1; allow greater accuracy 3 Or: k(4t + 9) n e differentiating y = correct Substitute t = 4 to obtain 39.7 allow ± 0.1; allow greater accuracy (3) [7] 8. (i) as derivative of Attempt product rule allow if sign errors or no chain rule * 8x 7 or (unsimplified) equiv

5 Either: Equate first derivative to zero and attempt solution dep *M; taking at least one step of solution Confirm 2 5 AG Or: 0 Substitute 2 into derivative and show attempt at evaluation AG; necessary correct detail required (5) (ii) Attempt calculation involving attempts at y values with each of 1, 4, 2 present at least once as coefficients Attempt k(y 0 + 4y 1 + 2y 2 + 4y 3 + y 4 ) with attempts at five y values corresponding to correct x values ( ) with at least 3 d.p. or exact values or greater accuracy; allow ± (iii) Attempt 4(y value) 2(part (ii)) 13.3 or greater accuracy; allow ± (i) Attempt use of product rule 3x 2 (x + 1) 5 + 5x 3 (x + 1) 4 [Or: (following complete expansion and 2 differentiation term by term) 8x x x x x 3 + 3x 2 B2 allow if one term incorrect] (ii) derivative of form kx 3 (3x 4 + 1) n any constants k and n derivative of form kx 3 (3x 4 + 1) correct 6x 3 (3x 4 + 1) or (unsimplified) equiv 3 [5]

6 10. (i) Attempt use of quotient rule allow for numerator wrong way round ; Confirm 3 AG; necessary detail required (ii) Identify ln x = State or imply x = e Substitute e k completely in expression for derivative and deal with ln e k term 4 or exact (single term) equiv (iii) State or imply integral of form or k (4lnx + 3) 1 * any constant k Substitute both limits and subtract right way round dep *M or exact equiv 4 [11] 11. (i) derivative of form kh 5 (h 6 +16) n any constant k; any n < ; allow if 4 term retained correct or (unsimplified) equiv; no 4 now Substitute to obtain or greater accuracy or exact equiv (ii) Attempt multn or divn using 8 and answer from (i) Attempt 8 divided by answer from (i) or greater accuracy; allow 0.75 ± 0.01; following their answer from (i) [6]

7 12. (i) Attempt use of product rule for x e 2x obtaining + e 2x +2xe 2x ; maybe within QR attempt Attempt use of quotient rule with or without product rule unsimplified 5 AG; necessary detail required (ii) Attempt use of discriminant 4k 2 8k = 0 and hence k = 2 Attempt solution of 2x 2 + 2kx + k = 0 using their numerical value of k or solving in terms of k using correct formula x = 1 e 2 5 or exact equiv [10] 13. Attempt use of product rule + form Substitute e to obtain 3e for gradient or exact (unsimplified) equiv Attempt eqn of straight line with numerical gradient allowing approx values y e 2 = 3e(x e) ; following their gradient provided obtained by diffn attempt; allow approx values y = 3ex 2e 2 in terms of e now and in requested form [6]

Verulam School. C3 Trig ms. 0 min 0 marks

Verulam School. C3 Trig ms. 0 min 0 marks Verulam School C3 Trig ms 0 min 0 marks 1. (i) R =, or 3.6 or 3.61 or greater accuracy Attempt recognisable process for finding α [allow sine/cosine muddles] α = 33.7 [or greater accuracy] 3 (ii) Attempt