c3 january 2006 mark scheme

January 2006 6665 Core Mathematics C3 Mark Scheme

Question

NumberScheme Marks

1. (a) y Shape unchanged B1 (2, 7) Point B1 (2)

O x

(b) y (2, 4) Shape B1 Point B1 (2)

O x

(c)

y Shape B1 (2, 4) B1 B1 (3)

[7]

O x

1


Question

NumberScheme Marks

2. At any stage B1

B1

M1

A1

M1 A1

A1 (7)

[7]

Alternative method

At any stage B1

appearing as a factor of the numerator at any stage B1

M1

can be implied A1

M1

Any one linear factor × quadratic

Complete factors A1

A1 (7)

2


Question

NumberScheme Marks

3. accept M1 A1

At x = 3, Use of M1

M1

Accept A1 (5) [5]

leading to is a maximum of M1 A0 M1 M1 A0 =3/5

4. (a) (i) At any stage B1

Or equivalent M1 A1+A1

(4)

(ii) At any stage M1 A1

M1 A1 (4)

Alternatively using the product rule for second M1 A1

Accept equivalent unsimplified forms

(b) M1

M1 A1

M1 A1 (5)

[13]

3


Question

NumberScheme Marks

5. (a) Dividing equation by x M1

Obtaining … M1

cso A1 (3)

(b) B1, B1, B1 If answers given to more than 2 dp, penalise first time then accept awrt above. (3)

(c) Choosing or a tighter interval M1

Both, awrt A1

Change of sign (and continuity)

to 3 decimal places cso A1 (3) [9]

6. (a)

Accept if just written down, awrt 12.6 M1 A1

awrt M1, A1(4)

(b) M1

x + their awrt 56° A1 = … , their principal value M1 Ignore solutions out of range A1, A1 (5)If answers given to more than 1 dp, penalise first time then accept awrt above.

(c)(i) minimum value is ft their R B1ft

(ii) M1

cao A1 (3) [12]

4


Question

NumberScheme Marks

7. (a) (i) Use of in an attempt to prove the identity. M1

cso A1 (2)

(ii) Use of in an attempt to prove the identity. M1 Use of in an attempt to prove the identity. M1

cso A1 (3)

(b) Using (a)(i) M1

Using (a)(ii) M1

A1 (3)

(c) M1

any one correct value of A1

Obtaining at least 2 solutions in range M1

The 4 correct solutions A1 (4)

If decimals or degrees are [12]

given, but all 4 solutions are found, penalise one A mark only.Ignore solutions out of range.

5


Question

NumberScheme Marks

8. (a) M1

M1 M1 Give mark at this point, cso A1 (4)

(b) y

Shape and point B1 (1)

4

O x

(c) Range is Accept B1 (1)

(d)

M1 A1

M1

A1 (4) [10]

6

c3 january 2006 mark scheme

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