Maths Extension 1 Question Booklet 2016.pdf

HSC Trial Examination 2016

Mathematics Extension 1

General Instructions

Total marks – 70

Reading time – 5 minutes Working time – 2 hours

Section I Pages 2–5 10 marks

Write using black pen

Attempt Questions 1–10

Board-approved calculators may be used

Allow about 15 minutes for this section

A reference sheet containing a range of formulae from the Mathematics and Mathematics Extension 1 courses is provided at the back of this paper.

Section II Pages 6–10 60 marks

In Questions 11–14, show relevant mathematical reasoning and/or calculations

Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section

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TENME1_QA_16.FM

HSC Mathematics Extension 1 Trial Examination

Section I – 10 marks Attempt Questions 1–10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1–10.

1.

sin 4 x Finding lim ------------- gives x → 0 x (A)

1 --4

(B)

4

(C)

2π

(D)

π --8

2

2.

3.

2

log a 16 x Simplifying ---------------------- , where a > 0, x > 0, gives log a 4 x (A)

2

(B)

4 x

(C)

log a 4 x

(D)

log a ( 16 x – 4 x )

2

Differentiating sin ( ln x ) gives (A)

1 sin -- x

(B)

cos ( ln x )

(C)

1 --- cos ( ln x ) x

(D)

1 ln x  cos ---   x 

TENME1_QA_16.FM

Copyright © 2016 Neap


4.

5.

Finding



–5 --------------------- d x gives 2 36 + x 2

(A)

– 10 x ln ( 36 + x ) + C

(B)

– 5 sin

(C)

– 1 x 5 ---cos --- + C 6 6

(D)

–1 x 5 – ---tan --- + C 6 6

–1 x

--- + C 6

5 Solving ----------- < 3 gives x – 2 (A) x

0

1

2

3

4

0

1

2

3 11 4 3

0

1

2

3 11 4 3

0

1

2

3 11 4 3

(B) x

(C) x

(D)

6.

x

If cosec x – cot x = 10, then cosec x + cot x equals (A)

0.1

(B)

no solution

(C)

–0.1

(D)

–10


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7.

8.

1 –1 The domain and range of the function y = ---cos 2 x are 2 (A)

D : { – 2 ≤ x ≤ 2 } and R : { 0 ≤ y ≤ 2π }

(B)

 1 1 D :  – --- ≤ x ≤ ---  and R : { 0 ≤ y ≤ 2π } 2  2

(C)

D : { – 2 ≤ x ≤ 2 } and R :  0 ≤ y ≤ ---  2 

(D)

 1  1 π  D :  – --- ≤ x ≤ ---  and R :  0 ≤ y ≤ ---  2 2  2 



π 

2

2

The diagram shows graphs with equations y = 16 – x and y = 2 x + 4. y

2

y = 2 x + 4

4

2

y = 16 – x

2

x

–4

–2

2

4

The shaded area equals (A)

(B)

(C)

(D)

4

   

2 2

20 – 3 x d x –2 2 2

12 – 3 x d x –2 16 2

20 – 3 x d x 4 16 2

3 x – 12 d x 4

TENME1_QA_16.FM



9.

2

The diagram shows the graph with equation y = k ( x + c ) ( x – 6 ) . y

12

2

4

x

6

What are the values of c and k ?

10.

(A)

1 k = ---, c = 2 2

(B)

1 k = ---, c = – 2 2

(C)

1 k = – --- , c = – 2 2

(D)

1 k = – --- , c = 2 2

In the diagram, AB is a diameter produced at Z . XY is a chord meeting the diameter AB externally at Z . ∠ BAY = 15 ° and ∠ XBY = 40 ° .

B

O

A

α

Z

Y X

What is the value of α ? (A)

15°

(B)

20°

(C)

35°

(D)

55°


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Section II – 60 marks Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11–14, your responses should include relevant mathematical reasoning and/or calculations.

Marks

Question 11 (15 marks) Use a SEPARATE writing booklet. (a)

2

Show that ( x + 4) is a factor of ( 2 x + 3 x – 23 x – 12 ) .

(ii)

Factorise and solve 2 x + 3 x – 23 x – 12 = 0.

(iii)

(b)

3

(i)

3

1

2

2 3

2

Graph the curve represented by the function f ( x ) = 2 x + 3 x – 23 x – 12, showing all the x- and y-intercepts.

2

3

3

If α and β are the roots of the equation 2 x + 3 x + 4 = 0, find the value of α + β .

2

2

∞

(c)

Find x if 1 + x =



n

x , x < 1.

2

n=2

x + 1

and y = 3

1 – x

(d)

Find the point of intersection of the curves y = 2 decimal places.

correct to two

(e)

In the diagram two circles with radii 8 cm and 6 cm, respectively, are drawn with their centres 12 cm apart. P is a point of intersection of the two circles. A straight line, ST , is drawn through P. The centres of the circles, C and K , form a straight line with T , and SP = PT . AC and BK are perpendicular to ST .

2

S A P

8 cm

B C

12 c m

(i) (ii)

6

K 6 c m

T

If BT = x and BK = y, find AC in terms of y.

1

Find the length of ST .

3

TENME1_QA_16.FM


HSC Mathematics Extension 1 Trial Examination Marks


(b)

2

Show 2 – 6 cos 2 x = 8sin x – 4cos x .

(ii)

Hence, find



2

2

2

2sin x – cos x d x .

n

Consider the equation A n = 8 + 3 (i)

(c)

2

(i)

n–2

1

, n ≥ 2.

Show that A 2 = 65.

1

(ii)

Prove that A3 is divisible by 5.

1

(iii)

Prove by induction that 8 + 3 an integer.

n

n–2

is divisible by 5 for all n ≥ 2, where n is

3

2

For the parabola, x = 16 y . y

P

S Q O

x

R D

T

(i)

2

If P is a point on the curve x = 16 y , where x = 8 p, show that the value of the

1

2

y-coordinate in terms of p is 4 p .

(ii) (iii)

2

2

Find the equation of the tangent to x = 16 y at (8 p, 4 p ). 2

The focal chord from P meets x = 16 y again at Q.

1 1

Find the coordinates of Q in terms of p. (iv)

(v)

Show that the point of intersection, D, of the tangents from P and Q is on the directrix. 2

For x = 16 y , what is the equation of the locus of points generated by the tangents at each end of a focal chord?


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1

7


Marks


The tips of the blades of a wind turbine are turning at a constant speed. The height of a blade tip at time t seconds is given by h ( t ) = 12 sin 3 t + 5 cos 3 t + 20.

(b)

π

Express 12 sin 3 t + 5 cos 3 t in the form k cos ( 3 t – α ) , where k > 0 and 0 < α < --- . 2

3

(ii)

Hence or otherwise, find the first two values of t for which the blade tip is 30 metres above the ground. Give your answer correct to two decimal places.

3

(i)

Show that the x -intercepts for f ( x ) = sec x – 2, – --- < x < --- , are x = ± --- . 3 3 4

(ii)

Graph the function represented by f ( x ) = sec x – 2, – --- < x < --- . 3 3

(i)

(iii)

π

2

π

π

2

π

1

π

2

2

Find the area enclosed by the curve f ( x ) = sec x – 2, the x -axis and the line y = 2.

(c)

Use one step of Newton’s method to find a closer approximation for the root of the equation cos ( 2 x ) – x = 0 if the initial approximation is x 1 = 1. Give your answer correct to two decimal places.

(d)

Using the substitution u = 9 – x , find the upper limit of integration b for the definite

3

integral

8



2

2

2

b

2

x

3 7 9 – x d x = 5 --- . 9

0

TENME1_QA_16.FM


HSC Mathematics Extension 1 Trial Examination Marks


A gardener randomly plants 2 casuarina, 3 bottle brush and 4 red gum trees in a row. (i) (ii)

(b)

What is the total number of different arrangements of red gum trees in this row?

1

What is the probability that none of the red gum trees are planted next to each other?

2

A soccer player notices that the opposing goalkeeper is out of position and takes a shot at goal. The ball is 50 cm off the ground when it is projected by his foot toward the centre of the goal with an initial velocity of 8 3 m s

–1

and an angle of projection of 30°. The goal

posts are 7.32 m wide. x = Vt cosθ , y = Vt sinθ , g = 9.8 m s

–2

7.32 m

V = 8 3 m s

NOT TO SCALE

–1

goal

line

30° 50 cm (i)

(ii)

If the ball clears the goal line at the same height that it was projected, how far is the goal line from the point of projection, and what is the time taken? Give y our answers correct to two decimal places.

4

The diagram below shows the path of projection of the ball.

2

NOT TO SCALE

ball point of projection

H

6m

goal

centre

50 cm What is the height, H , of the ball above the ground when it is 6 m from the centre of the goal? Question 14 continues on page 10


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Marks

Question 14 (continued) (c)

(i)

Prove that the ratio of the ( k + 1 ) th term and the k th term in the expansion of 14 15 – k ( 3 + tan 2θ ) is -------------- tan 2 θ . 3 k

(ii)

Hence find the greatest coefficient in the expansion of ( 3 + tan 2θ )

(iii)

π

If θ = --- , calculate the greatest term. 6

14

.

3

2 1

End of paper

10

TENME1_QA_16.FM



Mathematics Factorisation 2

Distance between two points

2

a – b = (a + b)(a – b)

d=

3

3

2

2

3

3

2

2

a + b = ( a + b ) ( a – ab + b )

2

( x 2 – x 1 ) + ( y 2 – y 1 )

2

Perpendicular distance of a point from a line

a – b = ( a – b ) ( a + ab + b )

ax 1 + by 1 + c d = ----------------------------------2 2 a +b

Angle sum of a polygon S = ( n – 2 ) × 180 °

Slope (gradient) of a line

Equation of a circle

( x – h ) + ( y – k ) = r

y 2 – y 1 m = --------------- x 2 – x 1

Trigonometric ratios and identities

Point-gradient form of the equation of a line

opposite side sinθ = ------------------------------hypotenuse

y – y 1 = m ( x – x 1 )

2

2

2

1 cosecθ = ---------sinθ

n th

term of an arithmetic series

adjacent side cosθ = ------------------------------hypotenuse

1 secθ = ----------cosθ

T n = a + ( n – 1 ) d

opposite side tanθ = ------------------------------adjacent side

sinθ tanθ = ----------cosθ

Sum to n terms of an arithmetic series n n S n = --- [ 2 a + ( n – 1 ) d ] or S n = --- ( a + l ) 2 2

cosθ cotθ = ----------sinθ 2

2

sin θ + cos θ = 1

n th

term of a geometric series n–1

T n = ar

Sum to n terms of a geometric series

Exact ratios

n

n

a ( r – 1 ) a ( 1 – r ) S n = ---------------------- or S n = ---------------------1 – r r – 1

30°

2 45°

√2

√3

1

45°

60° 1

1

Sine rule

Limiting sum of a geometric series a S = ----------1 – r Compound interest r n A n = P  1 + ---------   100 

a b c ----------- = ----------- = ----------sin A sin B sin C Cosine rule 2

2

2

c = a + b – 2 ab cos C Area of a triangle 1 Area = --- ab sin C 2


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Mathematics (continued) Differentiation from first principles

Integrals

f ( x + h ) – f ( x ) f ′ ( x ) = lim ---------------------------------h h→0

     

Derivatives n n–1 dy If y = x , then ------ = nx dx

dy dv du If y = uv , then ------ = u ------ + v -----dx dx dx du dv v ------ – u -----u dy dx dx If y = ---, then ------ = ------------------------2 v dx v dy du If y = F ( u ), then ------ = F ′ ( u ) -----dx dx

If y = e

f ( x )

f ( x ) dy , then ------ = f ′ ( x ) e dx

dy f ′ ( x ) If y = log e f ( x ) = ln f ( x ), then ------ = -----------dx f ( x ) dy If y = sin f ( x ), then ------ = f ′ ( x ) cos f ( x ) dx dy If y = cos f ( x ), then ------ = – f ′ ( x ) sin f ( x ) dx 2 dy If y = tan f ( x ), then ------ = f ′ ( x ) sec f ( x ) dx

n+1

( ax + b ) ( ax + b ) d x = ------------------------------ + C a(n + 1) n

e

ax + b

1 ax + b + C d x = --- e a

f ′ ( x ) ------------ d x = ln f ( x ) + C f ( x )

1 sin ( ax + b ) d x = – --- cos ( ax + b ) + C a 1 cos ( ax + b ) d x = --- sin ( ax + b ) + C a 2 1 sec ( ax + b ) d x = --- tan ( ax + b ) + C a

Trapezoidal rule (one application)



b

b–a f ( x ) d x ≈ ------------ [ f ( a ) + f ( b ) ] 2

a

Simpson’s rule (one application)



b

b–a a+b f ( x ) d x ≈ ------------ f ( a ) + 4 f  ------------  + f ( b )  2  6

a

Solution of a quadratic equation

log b x log a x = ------------log b a

2

– b ± b – 4 ac x = --------------------------------------2a Sum and product of roots of a quadratic equation b a

α + β = – ---

c a

αβ = ---

Equation of a parabola

Logarithms – change of base

Angle measure 180° = π radians Length of an arc l = r θ

2

( x – h ) = ± 4 a ( y – k )

12

Area of a sector 1 2 Area = --- r θ 2

TENME1_QA_16.FM



Mathematics Extension 1 Angle sum identities

Acceleration

sin (θ + φ ) = sinθ cos φ + cosθ sin φ

d x dv dv d 1 2 --------- = ------ = v ------ = ------  --- v  2 dt dx dx  2  dt

2

cos (θ + φ ) = cosθ cos φ + sinθ sin φ

Simple harmonic motion

tanθ + tan φ tan (θ + φ ) = -------------------------------1 – tanθ tan φ

x = b + a cos ( nt + a )

t formulae

2 ·· x = – n ( x – b )

θ

If t = tan --- , then 2

Further integrals

2 t sinθ = ------------2 1 + t 2

1 – t cosθ = ------------2 1 + t



1 –1 x --------------------- d x = sin --- + C a 2 2 a – x



1 1 – 1 x ---------------- d x = --- tan --- + C 2 2 a a a – x

Sum and product of roots of a cubic equation

2 t tanθ = ------------2 1 – t

b a

α + β + γ = – ---

General solution of trigonometric equations

c a

αβ + αγ + βγ = ---

–1

n

sinθ = a,

θ = n π + ( – 1 ) sin a

cosθ = a,

θ = 2 n π ± cos a

tanθ = a,

θ = n π + tan a

–1

d a

αβγ = – ---

–1

Estimation of roots of a polynomial equation

Division of an interval in a given ratio mx 2 + nx 1 my 2 + ny 1  -------------------------  m + n , -----------------------m+n 

Newton’s method f ( x 1 ) x 2 = x 1 – ------------- f ′ ( x 1 )

Parametric representation of a parabola 2

For x = 4 ay ,

Binomial theorem 2

x = 2 at , y = at

n

n

(a + b) =

2

At ( 2 at, at ) ,

 k = 0

2

n

 n  ak b n – k =  k 



 n  an – k b k  k 

k = 0

tangent: y = tx – at

3

normal: x + ty = at + 2 at At ( x 1, y 1 ) , tangent: xx 1 = 2 a ( y + y 1 ) 2a normal: y – y 1 = – ------ ( x – x 1 ) x 1 Chord of contact from ( x 0, y 0 ) : xx 0 = 2 a ( y + y 0 )


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Maths Extension 1 Question Booklet 2016.pdf

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