Term 4 Revision

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Term 4 Revision

Quiz L

• Sequences & Series

• Calculus

• Vectors

• Binomial Distribution

85 mins

56

Term 4 Consolidation Quiz Page 2

1. [SAJC/Prelim]

(i)

Show that ( )

2

1 2e 1

e 2e ee

r r r

r

− − + − +−

− + = . [1]

(ii)

Hence find ( )

2

11

e 1

e

N

rr

+=

− in terms of N.

[4]


(iii)

Using your result in part (ii), find ( )

21

19

e 1

e

N

rr

+

+=

− in terms of e.

[2]


2. [NYJC/Prelim]

Sarah carried out a series of experiments which involved using decreasing amounts of a

chemical. In the first experiment, she used 4 grams of the chemical and the amount of chemical

used formed a geometric progression. In the 25th experiment, she used 1 gram of the chemical.

(i) Find the total amount of chemical she used in the first 25 experiments. [4]

(ii) Show that the theoretical maximum total amount of chemical she would use will not

exceed 71.3 grams. [1]


Robert carried out the same series of experiments. He also used decreasing amounts of the

same chemical but the amount of chemical used formed an arithmetic progression with

common difference d. If the total amount of chemical that both Sarah and Robert used for

the first 25 experiments were the same, and the amount of chemical Robert used for the

25th experiment was still 1 gram, find the value of d and the amount of chemical he used for

the first experiment. [4]


3. [TPJC/Prelim]

A right pyramid toy is placed inside a spherical container of fixed radius r. The toy has a

square base RSTU and vertical height VM of length ( )r h+ where 0 h r . M is the point

where the diagonals SU and RT of the square meet. The vertices R, S, T, U and V of the toy

just touch the interior of the container with the vertical height VM passing through the centre O

of the container.

(i) Show that the length of the side of the square base RSTU is ( )2 22 r h− .

[2]


(ii) Hence, find the maximum volume of the toy in terms of r.

[It is given that the volume of a pyramid is 1

3 x base area x height.]

[5]



4. [TJC/Prelim]

(a) Find ( )ln 1 dx x+ . [3]

Hence find, in terms of k, the area bounded by the curve ( )ln 1y x= + , the x-axis and

the lines x k= − and x k= , where 0 1k . [3]


(b) (i)

By using the substitution tanx = , find the exact value of ( )

1

20 2

1d

1x

x + .

[6]


(ii)

The figure above shows the graph of ( )

22

1

1y

x=

+ for 0x . Rectangles, each

of width 1

n, are drawn under the curve. Show that the total area of all n

rectangles can be written as ( )

3

22 2

1

n

r

n

r n= + .

[2]

(iii)

Hence state the exact value of ( )

3

22 2

1

limn

nr

n

r n→

=

+

.

[1]


6. [MJC/Prelim]

The points A and B have position vectors 2 7 3− − +i j k and +j k respectively. The plane p

has equation 5x y− = .

(i) Find a vector equation of line l passing through points A and B. [2]

(ii) Find the acute angle between l and p. [2]

(iii) Verify that the point A lies on the plane p. Given that the point C is the reflection of the

point B in the plane p, describe the shape formed by the points A, B and C. [2]


(iv) Find a vector equation of the line which is a reflection of the line l in the plane p. [4]


6. [DHS/Prelim]

A test consists of 15 multiple choice questions, where each question has n possible options, of

which only one is correct. A student took the test by randomly choosing an answer to each

question. It is known that the probability of answering exactly 3 questions correctly is the same

as the probability of answering exactly 4 questions correctly.

(i) By forming an equation in terms of n, find the value of n. [3]


Each correct answer is awarded 3 marks and each incorrect answer carries a penalty of 1 mark.

The score is the total marks awarded based on the number of correct and incorrect answers.

(ii) Find the expected score, s, obtained by the student. [3]

(iii) Find the probability that the score obtained by the student is within 4 marks of s. [2]

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Term 4 Revision