12ap herlzia prelims '07

7/27/2019 12AP Herlzia Prelims '07

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Herzlia High School Additional Maths September 2007

Time: 3hrs Grade 12 Marks: 400Page 1 of 10

Section A: Calculus

1. Limits:

1.1.

>

=


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3. Given the function xxf 2arcsin)( =

3.1. Sketch the graph off(x). (6)

3.2. If241

2

12arcsin.)( xxxxF += , show that

F(x) =f(x) = arcsin 2x (6)

3.3. Now obtain the same result (i.e. thatF(x) is an antiderivative off(x)) by first

using a substitution, and then integration by parts to find the indefinite

integral dxx 2arcsin . (15)

3.4. Use the above answer to find the area betweenf(x) and thex-axis fromx = 0

to21=x . (Leave answer in terms of) (8)

3.5. Shade in this area on your graph. (1)

[36]

4. Consider the graph of xxxy += 34 2

4.1. Show that the graph has a turning point atx = 0,5. (6)

4.2. Explain why this turning point could not have been found using Newtons

Method with a starting value of either 0 or 1. (10)

4.3. Show that there is another turning point between 1 and 2, and hence use

Newtons Method to calculate thex-coordinate of this point, correct to 3

decimal places. (14)

[30]


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5.

O

CA

B

RR

A (and C)

R R

O

h

r

Starting with a circular piece of paper, with fixed radius R, and major arc ABC

which subtends angle at centre O. AO and CO are now joined to form a

canonical cup with circular top of radius rand height h.

5.1. Write down the length of arc ABC in terms of R and . (2)

5.2. Show that

2

Rr= (3)

5.3. Given that the volume of a cone radius r, height h is hr2

.3

1

Show that the volume of the cup is:

22

2

23

424

=

RV

(7)

5.4. Hence, find the value of for maximum volume (14)

[26]

6. When a semi-circle 0,122 =+ yyx , is revolved about thex-axis, the solid

generated is a sphere of radius 1.

t D(1,0)

B

CA(0,1)

E

O

The shaded region is a slice of the sphere of max thickness t (i.e. BD has length

t. EC is parallel to they-axis.

Show that the volume of this slice = )3(332

tt

[16]


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7. Use a right Riemann sum to determine dxxx .)2(2

0

2

you are given:

=

++=

n

i

nnni1

2

6

)12).(1.(

=

+=

n

i

nni1

223

4

)1(

(Hint: remember to check your answer with antiderivative) [20]

8. Evaluate the following integrals:

8.1.

+0

1

623)1( dxxx (14)

8.2. +

4

02

2

sec2

tan.sec

dxx

xx (18)

[32]

Total Section A: 200


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Section B: Algebra

(NB Start on a new page)

1. Use Mathematical Induction to prove that

=

=n

knk

1 2

11

2

1for all positive integers n. [14]

2. Express as partial fractions:1

6

4

23

+

x

xxx[18]

3. Given that 23+= is a root of 21243320122)(2345 +++= xxxxxxf .

3.1. State the Conjugate Surd Theorem. (5)

3.2. Expressf(x) in the formf(x) = a(x). b(x), where a(x) is a quadratic factor in

(x). (10)

3.3. Use Eisensteins Criterion to prove that b(x) is irreducible in (x). (8)

[23]


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4. Given: 68732)( 234 += xxxxxn

243)(23

+++= xxxxd

)()()(

xdxnxf =

4.1. Use the Euclidean Algorithm to prove that an HCF of )(xn and )(xd is:

22)( 2 ++= xxxh (remember, an HCF is any constant multiple of the result

of the algorithm) (9)

4.2. Prove that )(xh has no real roots. (3)

4.3. Factorise both )(xn and )(xd completely in Z(x) (5)

4.4. Use your results in 4.2 and 4.3 to:

4.4.1. express )(xf in its simplest form. (2)

4.4.2. write down its zeros. (2)

4.5. Determine the equations of:

4.5.1. fs vertical asymptote. (2)

4.5.2. fs oblique asymptote. (6)

4.6. Find the co-ordinates offs turning points, correct to 1 decimal place. (10)

4.7. Sketch the graph off, showing intercepts with axes, asymptotes and turning

points. (6)

(45)

Total Section B: 100


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Section C: Statistics

(NB Start on a new page)

1. Consider two normal packs of cards, Pack A and Pack B, each of which contain 4

suits Spades, Hearts, Diamonds, and Clubs with each suit comprising 13

cards Ace, King, Queen, Jack, 10, 9, 8, , 2.

You decide to develop a gambling game based on Packs A and B, Gamblers pay

R1 to draw 2 cards from each pack. You pay out R10 000 if they draw 4 Aces or

R10 if they draw 4 cards of the same suit.

1.1. What is the probability that you will pay out on any R1 bet? (8)

1.2. If one million R1 bets are placed in a year, how much profit could youreasonably expect to make? Show all your working. (11)

[19]

2. How large should a group be if 45 combinations of 2 can be made from it.

(i.e.

452 =n

) [5]

3. Two Archers, A and B, take part in a competition. The probability that each one

will shoot a bulls eye (centre of the target), is 0,7 and 0,5 respectively. Accept

that these two events are independent.

3.1. Represent the situation with a Venn Diagram. (8)

3.2. Use the Venn Diagram, or otherwise, and write down the probability that:

3.2.1. both A and B shoot bulls eyes if each gets one shot only. (2)

3.2.2. neither A nor B shoots a bulls eye if each gets one shot only. (2)

3.3. How many times must B shoot so that his chances of getting at least one

bulls eye is 0.9? (10)


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[22]


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4. An electrical company is selling a large number of solar panels (>1000) to the

Eastern Cape Government for installation in the rural areas as a source of

electricity. Due to the difficulties in the manufacturing process, 5% of these panels

turn out to be defective. Before the consignment is accepted by the Government, a

sample of 15 panels is tested and the number of defective panels counted. If more

than 2 of these 15 panels are defective, the consignment is rejected. What is the

probability of the consignment being rejected? [14]

5. The probability density function for the lifespan of a certain insect species is given

by:

+

=

lsewhere0

0if43

163-

)(

2

e

mxxxf wherex is the age of the insect in years.

Find m, the maximum lifespan of these insects. [10]

6. A certain make of car tyre can be safely used for 30 000km on average before it is

replaced. The makers guarantee to pay compensation to anyone whose tyre doesnot last for 25 000km. They expect 6% of all tyres sold to qualify for

compensation. Assume that the distance in km, X, travelled before a tyre is

replaced has a Normal distribution.

6.1. Draw a sketch illustrating the above information. (6)

6.2. Calculate, to the nearest km, the standard deviation of X. (6)

6.3. Estimate the number of tyres per 1000, which will not need to be replaced

before 32 000km. (6)

[18]

7. It is widely believed that 85% of Add Maths candidates pass the subject in Matric

each year. What size sample will be necessary to get the estimate correct to within

4% with 90% confidence? [12]

Total Section C: 100


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THE END!

12ap herlzia prelims '07

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