January 2006 - 6673 Pure P3 - Mark scheme
Comparison of key skills specifications 2000/2002 with 2004 standardsX015461July 2004Issue 1
| | | |
|Question Number|Scheme |Marks |
| | [pic]+….) | |
|1 (a) | | |
| |=[pic] | |
| | |M1 (corr bin coeffs) |
| |May use McLaurin f(0)=1 and [pic] to obtain 1st two terms 1 + x |M1 (powers of –2x) |
| |Differentiates two further times and uses formula with correct factorials to give | |
| |[pic] |A1, A1 |
| | |(4) |
|Alternative |[pic]. So series is [pic] | |
| | |M1 A1 |
| | |M1 |
| |Uses f(2) = 0 to give 16 – 4 + 2a + b = 0 | |
| | |A1 |
| |Uses f(-1) = 6 to give -2 – 1 +-a + b = 6 |(4) |
| | | |
|(b) |Solves simultaneous equations to give a = -7, and b = 2 |M1A1 ft |
| | |(2) |
| | | |
|2 |Uses circle equation | |
| |[pic] |M1 A1 |
| |Multiplies out to give [pic]and thus [pic] (*) | |
| | |M1 A1 |
| |Or states equation of circle is [pic]has centre (-g, -f) and so | |
| |g = - 4 and f = - 3 |M1 A1 A1 |
| |Uses [pic]to give [pic], i.e. c = 20 |(7) |
|3 (a) |[pic] | |
| | |M1 |
| |y = 2x meets the circle when [pic][pic] |A1 |
| |[pic] | |
| |Solves and substitutes to obtain x = 2 and y = 4. Coordinates are (2, 4) | |
| |Or Implicit differentiation attempt ,[pic] |A1 |
| |Uses y = 2x and [pic]=2 to give 10x – 20 = 0. |(3) |
|Alternative |Thus x =2 and y =4 | |
| | | |
| | | |
| | |M1 A1 |
| | | |
| | |A1 |
| | |(3) |
|(b) | |M1 |
| | | |
| | |A1 |
| | |M1 A1 |
| | |(4) |
| | | |
| | |M1 A1 |
| | | |
| | | |
| | |M1 A1 |
| | |(4) |
| | | |
|Question Number|Scheme |Marks |
| | | |
|4.(a) |[pic] |M1 A1 |
| |[pic] | |
| | |M1 A1 |
| | |(4) |
| |[pic] | |
|(b) |= [pic] | |
| |= [pic] |M1 A1 |
| |= [pic] | |
| | | |
| | | |
| | | |
| |[pic], so A = - 1 |A1 |
| | | |
| |Uses 18 = B ( 3– 2x) + C ( 3+2x ) and attempts to find B and C |M1 A1 |
| | |(5) |
| |B = 3 and C = 3 | |
| |Or | |
| |Uses [pic] = A([pic])+ B ( 3– 2x) + C ( 3+2x ) and attempts to find A, B and C | |
|5. (a) | | |
| |A = -1, B = 3 and C = 3 |B1 |
| | | |
| | | |
| | |M1 |
| |Obtains [pic] | |
| |Substitutes limits and subtracts to give 2[pic] |A1 A1 |
| | |(4) |
| |= -2 +3ln5 or –2 +ln125 | |
| | |M1 |
| | | |
| | |A1, A1, A1 |
| | |(4) |
| | | |
| | | |
|(b) | | |
| | |M1 A1 |
| | | |
| | | |
| | |M1 A1ft |
| | | |
| | | |
| | |A1 |
| | |(5) |
| | | |
| | | |
| | | |
| | | |
|Question Number|Scheme |Marks |
| | | |
| | | |
|6 (a) |[pic]; rate of decrease/negative sign; k constant of proportionality/positive constant | |
| | |B1 |
| | |(1) |
| |[pic] | |
| |[pic] | |
| |[pic] | |
|(b) | |M1 |
| | | |
| |At t = 0 [pic], [pic] |M1 |
| |and at t = 4 [pic], [pic], | |
| |[pic]and [pic] , [pic] |A1 |
| | |(3) |
|(c) | | |
| | | |
| |Solves [pic]to give [pic] so p = 3 |B1 |
| |Solves [pic]to give [pic] so q = 5 | |
| | |M1 |
| |[pic]so unit vector is [pic]6i +6j +3k) | |
| | |M1, A1 |
| |[pic] | |
|7 (a) | |(4) |
| |[pic] | |
| | |M1 A1 |
|(b) |Write down two of [pic] |M1 A1 |
| |Solve to obtain [pic]or [pic] |(4) |
| |Obtain coordinates ( 5, 3, 5) | |
|(c) | |M1 A1 |
| | |(2) |
| | | |
| | |M1 A1 |
| | | |
| | |A1 |
|(d) | |(3) |
| | | |
| | |B1 B1 |
| | | |
| | |M1 A1 |
| | | |
| | |A1 |
| | |(5) |
| | | |
|Question Number|Scheme |Marks |
| | | |
| | | |
|8(a) |[pic] therefore [pic] | |
| |When x = 0, t =[pic] |M1 A1 |
| |Gradient is [pic] | |
| |Line equation is [pic] |B1 |
| | | |
| |Area beneath curve is [pic] | |
| |=[pic] |M1 |
| |[pic] | |
| |Uses limits 0 and [pic] to give [pic] |M1 A1 |
| |Area of triangle beneath tangent is [pic]= [pic] |(6) |
| |Thus required area is [pic]- [pic]= [pic] | |
|(b) | | |
| | |M1 |
| | | |
| |The integration of the product of two sines is worth 3 marks (lines 2 and 3 of scheme to part (b)) |M1 |
| |If they use parts | |
| |[pic] | |
| |8I = cost sin3t – 3 cos3t sint |M1 A1 |
| | | |
| | | |
| | |M1 A1 |
| | | |
| | | |
| | |M1 A1 |
| | | |
| | | |
| | |A1 |
| | |(9) |
| | | |
|N.B. | | |
| | | |
| | | |
| | | |
| | | |
| | |M1 |
| | | |
| | | |
| | |M1 A1 |
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- net scheme hk
- monthly income scheme in bank
- monthly income scheme calculator
- sbi monthly income scheme calculator
- monthly income scheme in india
- monthly income scheme 2019 best
- monthly income scheme post office
- data classification scheme examples
- 0580 scheme of work 2020
- rhyme scheme finder
- mis scheme of post office
- p3 orion