PSf Further Calculus Past Papers Unit 3 utcome 2 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Differentiate 3 cos ( 2 π ) 6 with respect to. A. 3 sin(2) B. 3 sin(2 π 6 ) C. 6 sin(2 π 6 ) D. 6 sin(2 π 6 ) Ke utcome Grade Facilit Disc. Calculator Content Source C 3.2 C 0.68 0.23 NC C20 HSN 096 PSf Written Questions [END F MULTIPLE CHICE QUESTINS] 2. Differentiate sin 2 + 2 with respect to. 4 hsn.uk.net Page 1 Questions marked c SQA
PSf 3. Given that f () = (5 4) 2 1, evaluate f (4). 3 Part Marks Level Calc. Content Answer U3 C2 5 1 C CN C21 8 2000 P2 Q8 2 A/B CN C21 1 2 3 pd: differentiate power pd: differentiate 2nd function pd: evaluate f () 1 1 2 (5 4) 1 2 2 5 3 f (4) = 5 8 4. Given f () = cos 2 sin 2, find f (). 3 5. Given that f () = 5(7 2) 3, find the value of f (4). 4 hsn.uk.net Page 2 Questions marked c SQA
PSf 6. Differentiate 2 2 3 + sin 2 with respect to. 4 7. Find the derivative, with respect to, of 1 + cos 3. 4 3 8. If f () = cos 2 2 3 2, find f (). 4 hsn.uk.net Page 3 Questions marked c SQA
PSf 9. Differentiate 4 + 3 cos 2 with respect to. 4 10. Find d d given that = 1 + cos. 3 11. Given f () = (sin + 1) 2, find the eact value of f ( π 6 ). 3 hsn.uk.net Page 4 Questions marked c SQA
PSf 12. Find the equation of the tangent to the curve = 2 sin( π 6 ) at the point where = π 3. 4 Part Marks Level Calc. Content Answer U3 C2 4 C CN C5, C20 = 3 + 1 π 3 2002 P2 Q6 pd: find derivative 2 ss: know derivative at =... represents grad. 3 pd: find corresponding -coordinate 4 ic: state equation of tangent 1 1 d d = 2 cos( π 6 ) 2 m = 3 3 = π 3 = 1 4 1 = 3( π 3 ) 13. Find 1 1 + 3 d and hence find the eact value of 1 + 3 d. 4 0 14. Differentiate sin 3 with respect to. Hence find sin 2 cos d. 4 hsn.uk.net Page 5 Questions marked c SQA
PSf 15. Find 1 (7 3) 2 d. 2 Part Marks Level Calc. Content Answer U3 C2 2 A/B CN C22, C14 1 + c 3(7 3) 2000 P2 Q10 1 2 pd: integrate function pd: deal with function of function 1 1 1 (7 3) 1 2 1 3 16. Evaluate 0 3 ( 2 + 3) 2 d. 4 17. (a) Evaluate π 2 0 cos 2 d. 3 (b) Draw a sketch and eplain our answer. 2 hsn.uk.net Page 6 Questions marked c SQA
PSf 18. (a) Show that (cos + sin ) 2 = 1 + sin 2. 1 (b) Hence find (cos + sin ) 2 d. 3 19. Find ( 6 2 + cos ) d. 4 hsn.uk.net Page 7 Questions marked c SQA
PSf 20. (a) B writing sin 3 as sin(2 + ), show that sin 3 = 3 sin 4 sin 3. 4 (b) Hence find sin 3 d. 4 21. (a) Find the derivative of the function f () = (8 3 ) 2 1, < 2. 2 2 (b) Hence write down d. 1 (8 3 ) 2 1 Part Marks Level Calc. Content Answer U3 C2 (a) 2 A/B CN C21 3 2 2 (8 3 ) 1 2 2002 P1 Q10 (b) 1 A/B CN C24 2 3 (8 3 ) 1 2 + c 1 2 pd: process differentiation pd: use the chain rule 3 ic: interpret answer from (a) 1 1 2 (8 3 ) 1 2 2... 3 2 3 2 3 f () or 2 3 (8 3 ) 1 2 hsn.uk.net Page 8 Questions marked c SQA
PSf 22. The curve = f () passes through the point ( 12 π, 1) and f () = cos 2. Find f (). 3 23. The graph of = f () passes through the point ( π 9, 1 ). If f () = sin(3) epress in terms of. 4 Part Marks Level Calc. Content Answer U3 C2 4 A/B NC C18, C23 = 1 3 cos(3) + 7 6 2000 P1 Q8 1 ss: know to integrate 2 pd: integrate 3 ic: interpret ( π 9, 1) 4 pd: process 1 = sin(3) d stated or implied b 2 3 2 1 3 cos(3) 1 = 1 3 cos( 3π 9 ) + c or equiv. 4 c = 7 6 24. A curve for which d ( d = 3 sin(2) passes through the point 5π 12, ) 3. Find in terms of. 4 Part Marks Level Calc. Content Answer U3 C2 4 A/B CN C18, C23 = 3 2 cos(2) + 1 4 3 2001 P2 Q10 pd: integrate trig function pd: integrate composite function 3 ss: use given point to find c 4 pd: evaluate c 1 2 1 3 sin(2) d stated or implied b 2 2 3 2 cos(2) 3 3 = 3 2 cos(2 5 12 π) + c 4 c = 1 4 3 ( 0 4) hsn.uk.net Page 9 Questions marked c SQA
PSf 25. A point moves in a straight line such that its acceleration a is given b a = 2(4 t) 1 2, 0 t 4. If it starts at rest, find an epression for the velocit v where a = dv dt. 4 Part Marks Level Calc. Content Answer U3 C2 4 C NC C18, C22 V = 4 3 (4 t) 3 2 + 32 3 2002 P2 Q8 1 ss: know to integrate acceleration 2 pd: integrate 3 ic: use initial conditions with const. of int. 4 pd: process solution 1 V = (2(4 t) 1 2 ) dt stated or implied b 2 2 2 1 (4 t) 3 2 3 2 3 0 = 2 1 (4 0) 3 2 3 2 + c 4 c = 10 2 3 26. hsn.uk.net Page 10 Questions marked c SQA
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PSf 28. hsn.uk.net Page 12 Questions marked c SQA
PSf 29. [END F WRITTEN QUESTINS] hsn.uk.net Page 13 Questions marked c SQA