Document related concepts

Horner's method wikipedia , lookup

Transcript
```Markscheme Review Log Exp
1a. [6 marks]
Markscheme
(i)
(A1)
A1 N2
(ii) evidence of doubling (A1)
e.g. 560
setting up equation A1
e.g.
,
(A1)
in the year 2007 A1 N3
[6 marks]
1b. [6 marks]
Markscheme
(i)
(A1)
(A1)
A1 N3
(ii)
A1
not doubled A1 N0
valid reason for their answer R1
e.g.
1
[6 marks]
1c. [5 marks]
Markscheme
(i) correct value A2 N2
e.g.
, 91.4,
(ii) setting up an inequality (accept an equation, or reversed inequality) M1
e.g.
,
finding the value
(A1)
after 10 years A1 N2
[5 marks]
2a. [2 marks]
Markscheme
interchanging x and y (seen anywhere) (M1)
e.g.
(accept any base)
evidence of correct manipulation A1
e.g.
,
,
,
AG N0
[2 marks]
2b. [1 mark]
Markscheme
,
A1 N1
[1 mark]
2c. [4 marks]
2
Markscheme
METHOD 1
finding
(seen anywhere) A1
attempt to substitute (M1)
e.g.
evidence of using log or index rule (A1)
e.g.
,
A1 N1
METHOD 2
attempt to form composite (in any order) (M1)
e.g.
evidence of using log or index rule (A1)
e.g.
,
A1
A1 N1
[4 marks]
3. [7 marks]
Markscheme
recognizing
e.g.
(seen anywhere) (A1)
,
recognizing
(A1)
e.g.
correct simplification A1
3
e.g.
,
evidence of correct approach to solve (M1)
correct working A1
e.g.
,
A2 N3
[7 marks]
4a. [3 marks]
Markscheme
METHOD 1
recognizing that
(M1)
e.g.
recognizing that
(A1)
e.g.
A1 N2
METHOD 2
attempt to find the inverse of
e.g.
(M1)
,
substituting 1 and 8 (M1)
e.g.
,
A1 N2
[3 marks]
4b. [4 marks]
4
Markscheme
METHOD 1
recognizing that
(M1)
e.g.
(A1)
(accept
) A2 N3
METHOD 2
attempt to find inverse of
(M1)
e.g. interchanging x and y , substituting
into
correct inverse (A1)
e.g.
,
A2 N3
[4 marks]
5a. [3 marks]
Markscheme
(i) interchanging x and y (seen anywhere) M1
e.g.
correct manipulation A1
e.g.
,
AG N0
(ii)
A1 N1
[3 marks]
5
5b. [4 marks]
Markscheme
collecting like terms; using laws of logs (A1)(A1)
e.g.
,
,
,
simplify (A1)
e.g.
,
A1 N2
[4 marks]
6a. [1 mark]
Markscheme
5 A1 N1
[1 mark]
6b. [4 marks]
Markscheme
METHOD 1
(A1)
(A1)
(A1)
,
(accept
) A1 N3
METHOD 2
(A1)
6
(A1)
(A1)
,
(accept
) A1 N3
[4 marks]
7a. [2 marks]
Markscheme
substituting (0, 13) into function M1
e.g.
A1
AG N0
[2 marks]
7b. [3 marks]
Markscheme
substituting into
e.g.
A1
,
evidence of solving equation (M1)
e.g. sketch, using
(accept
) A1 N2
[3 marks]
7c. [5 marks]
Markscheme
(i)
7
A1A1A1 N3
Note: Award A1 for
, A1 for
, A1 for the derivative of 3 is zero.
(ii) valid reason with reference to derivative R1 N1
e.g.
, derivative always negative
(iii)
A1 N1
[5 marks]
7d. [6 marks]
Markscheme
finding limits
,
(seen anywhere) A1A1
evidence of integrating and subtracting functions (M1)
correct expression A1
e.g.
,
area
A2 N4
[6 marks]
8a. [2 marks]
Markscheme
(A1)
A1 N2
[2 marks]
8b. [2 marks]
Markscheme
evidence of using the derivative (M1)
A1 N2
8
[2 marks]
8c. [4 marks]
Markscheme
METHOD 1
setting up inequality (accept equation or reverse inequality) A1
e.g.
evidence of appropriate approach M1
e.g. sketch, finding derivative
(A1)
least value of k is 36 A1 N2
METHOD 2
, and
A2
least value of k is 36 A2 N2
[4 marks]
9a. [1 mark]
Markscheme
A1 N1
[1 mark]
9b. [2 marks]
Markscheme
substitution into formula (A1)
e.g.
,
A1 N2
[2 marks]
9
9c. [5 marks]
Markscheme
set up equation (M1)
e.g.
attempting to solve (M1)
e.g. graph, use of logs
correct working (A1)
e.g. sketch of intersection,
A1
correct time 18:33 or 18:34 (accept 6:33 or 6:34 but nothing else) A1 N3
[5 marks]
10a. [2 marks]
Markscheme
combining 2 terms (A1)
e.g.
,
e.g.
,
AG N0
[2 marks]
10b. [3 marks]
Markscheme
attempt to substitute either value into f (M1)
e.g.
,
10
,
A1A1 N3
[3 marks]
10c. [6 marks]
Markscheme
(i)
,
A1A1 N1N1
(ii)
A1A1A1 N3
Note: Award A1 for sketch approximately through
, A1 for approximately correct
shape, A1 for sketch asymptotic to the y-axis.
11
(iii)
(must be an equation) A1 N1
[6 marks]
10d. [1 mark]
Markscheme
A1 N1
[1 mark]
10e. [4 marks]
Markscheme
12
A1A1A1A1 N4
Note: Award A1 for sketch approximately through
shape of the graph reflected over
, A1 for approximately correct
, A1 for sketch asymptotic to x-axis, A1 for point
clearly marked and on curve.
[4 marks]
11a. [4 marks]
Markscheme
attempt to apply rules of logarithms (M1)
13
e.g.
,
correct application of
(seen anywhere) A1
e.g.
correct application of
(seen anywhere) A1
e.g.
so
(accept
) A1 N1
[4 marks]
11b. [3 marks]
Markscheme
transformation with correct name, direction, and value A3
e.g. translation by
, shift up by
, vertical translation of
[3 marks]
12a. [4 marks]
Markscheme
evidence of substituting the point A (M1)
e.g.
manipulating logs A1
e.g.
A2 N2
[4 marks]
12b. [5 marks]
Markscheme
14
(i)
(accept
A1 N1
(ii)
A1A1A1A1 N4
Note: Award A1 for asymptote at
, A1 for an increasing function that is concave up, A1 for a
positive x-intercept and a negative y-intercept, A1 for passing through the point
[5 marks]
12c. [4 marks]
Markscheme
METHOD 1
recognizing that
(R1)
15
.
evidence of valid approach (M1)
e.g. switching x and y (seen anywhere), solving for x
correct manipulation (A1)
e.g.
A1 N3
METHOD 2
recognizing that
(R1)
identifying vertical translation (A1)
e.g. graph shifted down 3 units,
evidence of valid approach (M1)
e.g. substituting point to identify the base
A1 N3
[4 marks]
Printed for American Community School at Beirut