[6] 11 4. , M1 A1 M1 oe or When dependent on the previous M1 see notes or or A1 oe [5] 5 5. (a) M1 oe So, Adds their expanded x (which is in terms of t) to * Correct A1 * [3] (b) Applies to achieve an equation containing only x’s and y’s. M1 A1 [2] 5 6. (i) 1
(b) or B1ft [1] (c) M1 Applies dot product formula between their and their M1 Correct proof A1 cso [3] (d) or , with either or , or a multiple of . M1 Correct vector equation. A1 ft [2] (e) Either or M1 At one set of coordinates are correct. A1 ft Both sets of coordinates are correct. A1 ft [3] (f) M1 or or or awrt 4.9 or equivalent A1 oe dM1 A1 cao [4] 15 8. (f) Helpful Diagram!
A1 A1 (c) (fish) (nearest 100) B1 8. lies on the curve, (a) M1 or so A1 (b) , B1 B1 So, At , M1; A1 cao cso (c) M1 gives A1 So or M1; A1 or dM1 A1 Question Number C4 JUNE 2013 MARK SCHEME 1. (a) , 1st Application: , 2nd Application:
or and Uses a correct method in order to find both possible sets of coordinates of B. M1 Both coordinates are correct. A1 cao (5) [9] Question Number C4 JUNE 2013 (R) MARK SCHEME 1. At one of “A” or “C” are correct. B1 Breaks up partial fraction correctly into three terms and both and . B1 cso Writes down a correct identity and attempts to find the value of either one “A” or “ B” or “C”. M1 Either leading to Correct value for “B” which is found using a correct identity and follows from their partial fraction A1 cso
A1 Substitutes limits of 8 and 0 into an integrated function of the form of either or and subtracts the correct way round. dM1 A1 (6) (d) Difference 1.46 or awrt 1.46 B1 (1) [11] 6. , , (a) A is on l, so B1 Substitutes their value of into M1 A1 cao (3) (b) Finds the difference betweenand . Ignore labelling. M1 M1; A1 ft ddM1; A1 cso cao (5)
1. or B1 see notes M1 A1 See notes below! A1; A1 [5] 5 2. (a) , In the form M1 simplified or A1 simplified or dM1 Correct answer, with/without A1 [5] (b) Applies limits of 2 and 1 to their part (a) answer and subtracts the correct M1
. way round. or A1 [2] 7 3. Method 1: Using one identity their constant term B1 Forming a correct identity. B1 Either or Attempts to find the value of either one of their B or their C from their identity. M1 Correct values for their B and their C, which are found using a correct identity. A1 [4] Method 2: Long Division their constant term B1 So, Forming a correct identity. B1 Either or Attempts to find the value of either one of their B or their C from their identity. M1 equivalent
Correct values for their B and their C, which are found using A1 So, [4] 4 4. (a) 1.0981 B1 cao [1] (b) B1; M1 (3 dp) 2.843 or awrt 2.843 A1 [3] (c) or B1 M1 A1 Expands to give a “four term” in u. Eg: M1 An attempt to divide at least three terms in their cubic by u. See notes. M1 A1 = Applies limits of 3 and 2 in u or 4 and 1 in x and subtracts either way M1 Correct exact answer or equivalent. A1
[8] 12 5. Working parametrically: (a) Applies to obtain a for t. M1 When , Correct for y. A1 [2] (b) Applies to obtain a value for t. (Must be seen in part (b)). M1 When , A1 [2] (c) and either or B1 Attempts divided by their M1 At A, so Applies and M1 or or equivalent. M1 A1 oe cso [5] (d) Complete substitution for both and M1 B1 Either or or M1* A1
6. (a) 0.73508 B1 cao (1) (b) B1 M1 (4 dp) awrt 1.1504 A1 (3) (c) B1 B1 M1 dM1 AG A1 cso (5) (d) Applying limits and either way round. M1 or or awrt 1.2 A1 awrt 0.077 or awrt 6.3(%) A1 cso (3) (12 Scheme Marks 7. (a) M1; A1 [2] (b) or M1 A1ft [2]
Let d be the shortest distance from C to l. (c) M1 Applies dot product formula between and their M1 Correct followed through expression or equation. A1 awrt 109 A1 cso AG [4] (d) M1 So, A1 [2] (e) M1; dM1 A1 [3] (f) or M1 awrt 3.54 A1 [2] (15 marks) Question Number Scheme Marks 8. (a) Can be implied. M1 Either one. A1
giving A1 cao, aef [3] (b) B1 M1* A1ft dM1* eg: Using any of the subtraction (or addition) laws for logarithms CORRECTLY dM1* eg: or eg: Eliminate ln’s correctly. dM1* gives Make P the subject. dM1* or etc. A1 [8] (c) . So population cannot exceed 5000. B1 [1] (12 Question Number C4 JUNE 2011 MARK SCHEME Marks 1. B1 M1 Any two of A, B, C A1
4. (a) , awrt , B1 B1 (2) (b) B1 M1 Accept 1.3 A1 (3) (c) B1 B1 M1 cso A1 (4) (d) M1 A1 M1 A1 = M1 A1 (6) [15] Question Number Scheme 5. … B1 … M1 A1 At , M1