2014 hsc maths ry trial4unitmaths.com/penrith-2u-2014.pdf-2-section i: 10 marks attempt questions...

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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.

Q1. What is 6.04976 correct to 4 significant figures.

(A) 6.049

(B) 6.0497

(C) 6.050

(D) 6.0498

Q2.

What are the solutions of 3 7 3 0?

(A) 7 √856

(B) 7 √856

(C) 7 √136

(D) 7 √136

Q3. is equal to

(A) 4

6

(B) 4

6

(C) 2

3

(D) 4

3

-3-

Q4.

What are the solutions of 2 √3 for 0 2 ?

(A) and

(B) and

(C) and

(D) and

Q5. The line which is perpendicular to 2 1 0 with a intercept of 4 has equation:

(A) 2 4

(B) 2

4

(C) 2 4

(D) 2

4

Q6. What is the derivative of ?

(A) 3

2 3

(B) 12

(C) 4 32 3

(D) 14

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Q7.

Two six-sided dice are thrown. The probability that the sum of the uppermost faces is even is:

(A) 1

(B) even

(C) 14

(D) 12

Q8.

The diagram shows the graph of . The equation of is:

(A) 12

3

(B) |2 3|

(C) 12

3

(D) 12

3

6

3

0

-5-

Q9.

The diagram shows the graph of Which of the following statements is true?

(A) 0 and 0

(B) 0 and 0

(C) 0 and 0

(D) 0 and 0

Q10. A geometric series will have a limiting sum if:

(A) | | 1, where is the common ratio

(B) | | 1, where is the common ratio

(C) 1, where is the common ratio

(D) 1, where is the common ratio

0

-6-

Section II

60 Marks Attempt Questions 11–16 Allow about 2 hours and 45 minutes for this section Answer each question in a SEPARATE booklet. Extra writing booklets are available. In Questions 11–16, your responses should include relevant mathematical reasoning and/or calculations.

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

a) Simplify the expression 3 5 2 2

b) Given that , find when 12, 3 and 2 2

c) Differentiate 2 2 1

d) Factorise 16 2

e) Express √

with a rational denominator 2

f) James invests $1000 at 7% per year compound interest, compounded quarterly. Calculate that value of the investment after 5 years. Give your answer correct to the nearest dollar.

3

g)

Given that 2.75 and 0.25, find the value of:

(i)

(ii)

3

End of Question 11

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a) Differentiate and simplify where necessary.

(i) 3 2

(ii) 3 4 7

2

b) (i) Evaluate 6 1 2

(ii) Find 4 2

c) Find the equation of the tangent to the curve , at the point , 0 2

d)

Sketch 3 for showing all key features. 2

e) A point P(x, y) moves so that it is always twice the distance from the point A(1, 4) as it is to point B(2, -8). Show that the equation of the path traced by P is 3 14 3 72 255 0.

3

End of Question 12

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a) The diagram shows a triangle ABC. The point A 4, 1 lies on given by

the equation 2 6, and the point B 5, 2 lies on the line , given by the equation 2 8.

(i) Show that the point C, which is the point of intersection of and

has coordinates (-2, 4). 1

(ii) Find the gradient of the line joining A and B. 1

(iii) Hence, or otherwise, find the equation of the line AB. 1

(iv) Find the perpendicular distance from the point A to the line . 2

(v) Hence, or otherwise, find the area of the triangle ABC. 2

Question 13 continues on page 9

0 4, 1

5, 2

-9-

b)

Food tins are stacked so there are 49 tins on the bottom row, 45 tins on the next row, 41 tins on the row after and so on until a total of 321 cans are stacked.

(i) Write down a formula for the number of cans in the nth row. 1

(ii) How many rows are in the stack in total? 2

(iii) How many cans are in the top row of this stack? 1

c)

If and are the roots of the quadratic equation 3 8 7 0, find the value of:

(i) 1

(ii) 1

(iii) 1 1

1

(iv) 1

End of Question 13

. . . . .

. . . . .

. . . . .

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a) A factory manufactures light bulbs. Testing showed that 1 out of 20 bulbs

tested was faulty. Three of these bulbs are selected at random and tested. What is the probability that:

(i) All three bulbs tested are faulty? 1

(ii) None of the bulbs are faulty? 1

(iii) Exactly two bulbs are faulty? 2

(iv) At most two bulbs are faulty? 2

b)

is a sector. ∠ BCD = , BA = BC = 9 cm and ⏊ .

(i) Calculate the area of sector . 3

(ii) Calculate the area of the shaded region. Leave your answer in exact

form. 3

c) Find the primitive of:

(i) 2

3 1

(ii) 3

2

End of Question 14

6

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a) Consider the curves and 4 5.

(i) Find any points of intersection. 1

(ii) Sketch the graphs of the two equations on the same set of axes. 2

(iii) Find the area of the region enclosed by these two equations. 2

b) Consider the function 2

(i) Copy and complete the table of values correct to three decimal

places where necessary.

0 4

2

34

1

(ii) Use two applications of Simpson’s rule to calculate the approximate area under the curve between 0 and . Leave your answer correct to 2 decimal places.

3


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c)

A cylinder is made to fit inside a sphere with fixed radius as shown in the diagram.

Let be the distance from the base of the cylinder to the centre of the sphere, as shown in the diagram. Let be the radius of the circular base of the cylinder.

(i) Find an expression for, R, the radius of the base of the cylinder in

terms of and . 1

(ii) Show that the volume, V, of the cylinder is given by

2 2

(iii) Find, in terms of , the maximum volume of the cylinder.

Give your answer in exact form. 3

End of Question 15

Not to scale

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a) Find the equation of the parabola whose axis is parallel to the y-axis, vertex is 2, 1 and has a tangent with equation 2 7.

3

b) A quantity of radium at time in years is given by

where is a constant and is the initial amount of radium at time 0.

(i) Given that when 1530 years, calculate , correct to

three significant figures.

1

(ii) After how many years does only 20% of the initial amount of radium remain, to the nearest whole number.

1

c)

and are two congruent rectangles with sides 3 and 7 units as

shown in the diagram. ( 7, 3

(i) Show that 3

(ii) Find the area of the figure . 2


Not to scale

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d) A truck is to travel 1000 kilometres at a constant speed of km/h.

When travelling at km/h, the truck consumes fuel at the rate of

60 litres per hour.

The truck company pays $1.40 for fuel and pays each of the two drivers $40 per hour whilst the truck is travelling.

(i) Let the total cost of fuel and the drivers’ wages for the trip be dollars. Show that

28164000

3

(ii) The truck must take no longer than 12 hours to complete the trip,

and speed limits require 100. At what speed should the truck travel to minimise the cost ?

2

End of Paper

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Student Number:_________________________________________________

Multiple Choice Answer Sheet

Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely.

Sample:

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer.

If you change your mind and have crossed out what you consider to be the correct answer, then indicate the correct answer by writing the word ‘correct’ and drawing an arrow as follows.

2014 hsc maths ry trial4unitmaths.com/penrith-2u-2014.pdf-2-section i: 10 marks attempt questions...

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