TRINITY GRAMMAR SCHOOL Mathematics Department

(NESA Student Number | Year 12 only)

2019 Year 12

Mathematics

HSC ASSESSMENT TASK 1

Date of Assessment Task: Thursday, 15 November 2018

General Instructions

• Reading time – not applicable to this Task • Working time – 45 minutes • Write using black pen • NESA approved calculators may be used • A formula and data sheet is provided • In Questions 6 – 7, show relevant mathematical reasoning

and/or calculations • Write your NESA Student Number (Year 12 HSC) or Name

(Year 11 or 10) and your Class teacher on the question paper and on any answer sheets or writing booklets used to write your responses to the questions submitted

• If you do not attempt a question you must submit an answer sheet or writing booklet for that question clearly indicating N/A and your NESA Student Number or Name

Total marks: 35

Section I – 5 marks (pages 2 – 4) • Attempt Questions 1 – 5 • Allow about 5 minutes for this section Section II – 30 marks (pages 5 – 7) • Attempt Questions 6 – 7 • Allow about 40 minutes for this section

• HSC Assessment Weighting: 20%

E

Suggested solutionsi

− 2 −

Section I

5 marks Attempt Questions 1 – 5 Use the multiple-choice answer sheet for Questions 1 – 5.

1 If 2 5( 3)y x= + then dydx

=

A. 2x

B. 2 45( 3)x +

C. 2 42 ( 3)x x +

D. 2 410 ( 3)x x +

2 Ken decided to use differentiation from first principles to differentiate the expression 22x x− . A correct expression, that Ken ought to have used in order to differentiate this

expression is

A.

2

0

2( ) ( )limh

x h x hh→

+ − +

B.

2 2

0

2 ( ) 2( )limh

x x x h x hh→

− − + + +

C.

2 2

0

2 ( ) 2( )limh

x x x h x hh→

− − + − −

D.

2 2

0

2 ( ) 2( )limh

x x x h x hh→

− + + − +

18.448K55

ret's C2nIon Cntt3

0

let fCn _2n R2

fad thirffCathfffat

hhjfcuthfeth52u.IT

− 3 −

Turn over

3 The diagram below is of a parabola with vertex at ( 2, 3)− and it has y-intercepts at 1,−

and 7 respectively.

The Cartesian equation of the parabola is given by

A. 2( 2) 4( 3)x y+ = −

B. 2( 2) 8( 3)x y+ = −

C. 2( 3) 4( 2)y x− = +

D. 2( 3) 8( 2)y x− = +

4 The quadratic equation 22 4 1 0kx kx− + = has two equal roots. The value(s) of k is

A.

12

only

B.

10 and2

C. 2 only

D. 0 and 2

y k5 Kaen h

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a 2

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for equal rootsC 4kt 4Gk CD o

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8kC2k D o

k k only

− 4 −

5 A quadratic equation with integer coefficients for which the sum and product of its roots

is 3− and 12

− respectively, is

A. 22 3 1 0x x− − =

B. 22 3 1 0x x− − =

C. 22 6 1 0x x+ − =

D. 22 6 1 0x x− − =

let Ltp 3

xp 42r2 Gtp r 213 0

H Csad o

set t3a tz D

2 those 1 0

− 5 −

Turn over

Section II

30 marks Attempt Questions 6 – 7 Allow about 40 minutes for this section

Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.

In Questions 6 – 7, your responses should include relevant mathematical reasoning and/or calculations.

Question 6 (15 marks) Use a SEPARATE writing booklet.

(a) Find:

(i) ( )33 2d x

dx+ ; 2

(ii) 3 5

2d xdx x

+−

using the quotient rule; 2

(iii) ( )1d x xdx

+ using the product rule. 2

(b) Evaluate 3

4

64lim4x

xx→

−−

. 3

(c) Find the value of x for which 2( ) 2 1f x x x= − + has zero gradient. 2

Question 6 continues on page 6

622

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simplifywith

tangy 4GetterHD musthavecorregadforisation

42 443 16 48 ask 74

f Cn 2x 2want 2.72270 Herogradient

can use N bha there are severalothermethods

eg ffnj C.seD2r R is aZeroD o aswell440 is thevertex

− 6 −

Question 6 (continued)

(d) Consider the diagram below where the graph of ( )y g x= is a straight line and tangent to the graph of ( )y f x= at the point where x = 1. The graph of ( )y g x= intersects the

y -axis at 10,2

and passes through the point (2, 4.5).

(i) Find the value of '(1)f . 2 (ii) Evaluate (1).f 2

End of Question 6

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gcn set's

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There may bemoetwthods

− 7 −

Turn over

Question 7 (15 marks) Use a SEPARATE writing booklet.

(a) The roots of a quadratic equation 2 3 7 0x x− − = are and .

Without finding the actual roots,

(i) Write down the value of + . 1

(ii) Write down the value of . 1

(iii) Evaluate 1 1

+ . 1

(iv) Evaluate 2 2a + . 1

(v) Evaluate 2 2+ . 1

(b) Find the value(s) of k for which the expression 2 3 2kx x k+ + − is positive definite. 3

(c) Solve for x if 26 37(6 ) 36 0x x− + = . 2

(d) A parabola has its focus at (0, 5) and its vertex at (0, 8). Find the focal length and the equation of the directrix of this parabola.

2

(e) Find the equation of the locus of a point P(x, y) such that AP and PB are perpendicular, given A(3, 2) and B(4, −1). Describe the locus of P.

3

End of Question 7 End of Task

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− 8 −

Blank page

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