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Date May 2017 Marks available 8 Reference code 17M.1.SL.TZ1.S_9
Level Standard Level Paper Paper 1 Time zone Time zone 1
Command term Find Question number S_9 Adapted from N/A

Question

A quadratic function f can be written in the form f ( x ) = a ( x p ) ( x 3 ) . The graph of f has axis of symmetry x = 2.5 and y -intercept at ( 0 ,   6 )

Find the value of p .

[3]
a.

Find the value of a .

[3]
b.

The line y = k x 5 is a tangent to the curve of f . Find the values of k .

[8]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1 (using x-intercept)

determining that 3 is an x -intercept     (M1)

eg x 3 = 0 , M17/5/MATME/SP1/ENG/TZ1/09.a/M

valid approach     (M1)

eg 3 2.5 ,   p + 3 2 = 2.5

p = 2     A1     N2

METHOD 2 (expanding f (x)) 

correct expansion (accept absence of a )     (A1)

eg a x 2 a ( 3 + p ) x + 3 a p ,   x 2 ( 3 + p ) x + 3 p

valid approach involving equation of axis of symmetry     (M1)

eg b 2 a = 2.5 ,   a ( 3 + p ) 2 a = 5 2 ,   3 + p 2 = 5 2

p = 2     A1     N2

METHOD 3 (using derivative)

correct derivative (accept absence of a )     (A1)

eg a ( 2 x 3 p ) ,   2 x 3 p

valid approach     (M1)

eg f ( 2.5 ) = 0

p = 2     A1     N2

[3 marks]

a.

attempt to substitute ( 0 ,   6 )     (M1)

eg 6 = a ( 0 2 ) ( 0 3 ) ,   0 = a ( 8 ) ( 9 ) ,   a ( 0 ) 2 5 a ( 0 ) + 6 a = 6

correct working     (A1)

eg 6 = 6 a

a = 1     A1     N2

[3 marks]

b.

METHOD 1 (using discriminant)

recognizing tangent intersects curve once     (M1)

recognizing one solution when discriminant = 0     M1

attempt to set up equation     (M1)

eg g = f ,   k x 5 = x 2 + 5 x 6

rearranging their equation to equal zero     (M1)

eg x 2 5 x + k x + 1 = 0

correct discriminant (if seen explicitly, not just in quadratic formula)     A1

eg ( k 5 ) 2 4 ,   25 10 k + k 2 4

correct working     (A1)

eg k 5 = ± 2 ,   ( k 3 ) ( k 7 ) = 0 ,   10 ± 100 4 × 21 2

k = 3 ,   7     A1A1     N0

METHOD 2 (using derivatives)

attempt to set up equation     (M1)

eg g = f ,   k x 5 = x 2 + 5 x 6

recognizing derivative/slope are equal     (M1)

eg f = m T ,   f = k

correct derivative of f     (A1)

eg 2 x + 5

attempt to set up equation in terms of either x or k     M1

eg ( 2 x + 5 ) x 5 = x 2 + 5 x 6 ,   k ( 5 k 2 ) 5 = ( 5 k 2 ) 2 + 5 ( 5 k 2 ) 6

rearranging their equation to equal zero     (M1)

eg x 2 1 = 0 ,   k 2 10 k + 21 = 0

correct working     (A1)

eg x = ± 1 ,   ( k 3 ) ( k 7 ) = 0 ,   10 ± 100 4 × 21 2

k = 3 ,   7     A1A1     N0

[8 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
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c.

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques
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