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QB (Hard)

The document contains 5 multi-part math questions involving geometric sequences, polynomials, and loans. Question 1 involves finding the common ratio of a geometric sequence and solving a series. Question 2 deals with geometric sequences, finding expressions for the common ratio and sum. Question 3 calculates amounts owed on a loan over 10 years and finds the quarterly deposit needed to pay it off. Question 4 involves finding common ratios and terms of geometric sequences given information about their sums. Question 5 explores relationships between the coefficients and roots of a cubic polynomial, determining values of the coefficients given information about the roots.

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722 views3 pages

QB (Hard)

The document contains 5 multi-part math questions involving geometric sequences, polynomials, and loans. Question 1 involves finding the common ratio of a geometric sequence and solving a series. Question 2 deals with geometric sequences, finding expressions for the common ratio and sum. Question 3 calculates amounts owed on a loan over 10 years and finds the quarterly deposit needed to pay it off. Question 4 involves finding common ratios and terms of geometric sequences given information about their sums. Question 5 explores relationships between the coefficients and roots of a cubic polynomial, determining values of the coefficients given information about the roots.

Uploaded by

nqwrgnbzhq
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Q1.

[Maximum Mark:8] (Non-Calculator)

9 3
The first three terms of a geometric sequence are 𝑙𝑛⁡𝑥 , 𝑙𝑛⁡𝑥 , 𝑙𝑛⁡𝑥, for 𝑥 > 0

(a) Find the common ratio. [3]

∞ 3−𝑘
(b) Solve ∑𝑘=1 3 𝑙𝑛⁡𝑥 = 27. [5]

Q2.[Maximum Mark:15] (Non-Calculator)

2
The first two terms of an infinite geometric sequence are 𝑢1 = 20 and 𝑢2 = 16𝑠𝑖𝑛 ⁡θ,
where 0 < θ < 2π, and θ ≠ π

(a)

(i) Find an expression for 𝑟 in terms of θ.

(ii) Find the possible values of 𝑟. [5]

100
(b) Show that the sum of the infinite sequence is 3+2𝑐𝑜𝑠⁡2θ
. [4]

(c) Find the values of θ which give the greatest value of the sum. [6]

Q3.[Maximum Mark:15] (Calculator)

Bill takes out a bank loan of $100000 to buy a premium electric car, at an annual
interest rate of 5. 49%. The interest is calculated at the end of each year and added
to the amount outstanding.

(a) Find the amount of money Bill would owe the bank after 10 years. Give your
answer to the nearest dollar. [3]

To pay off the loan, Bill makes quarterly deposits of $𝑃 at the end of every quarter in
a savings account, paying a nominal annual interest rate of 3. 2%. He makes his first
deposit at the end of the first quarter after taking out the loan.
40
1.008 −1
(b) Show that the total value of Bill’s savings after 10 years is 𝑃⎡⎢ 1.008−1
⎤.
⎥ [3]
⎣ ⎦
(c) Given that Bill’s aim is to own the electric car after 10 years, find the value for 𝑃 to
the nearest dollar. [3]

Melinda visits a different bank and makes a single deposit of $𝑄, the annual interest
rate being 3. 5%

(d)

(i) Melinda wishes to withdraw $8000 at the end of each year for a period of 𝑛 years.
Show that an expression for the minimum value of 𝑄 is
8000 8000 8000 8000
1.035
+ 2 + 3 +⋯ + 𝑛
1.035 1.035 1.035

(ii) Hence, or otherwise, find the minimum value of 𝑄 that would permit Melinda to
withdraw annual amounts of $8000 indefinitely. Give your answer to the nearest
dollar. [6]

Q4.[Maximum Mark:7] (Non-Calculator)

The sum of the first two terms of a geometric series is 7 and the sum of the first six
terms is 91.
2
(a) Show that the common ratio 𝑟 satisfies 𝑟 = 3. [4]

(b) Given 𝑟 = 3,

(i) find the first term.

(ii) find the sum of the first eight terms. [3]

Q5.[Maximum Mark:14] (Non-Calculator)

3 2
The cubic polynomial equation 𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑑 = 0 has three roots 𝑥1, 𝑥2 and 𝑥3.
( )(
By expanding the product 𝑥 − 𝑥1 𝑥 − 𝑥2 𝑥 − 𝑥3 , show that )( )

(a)

(
(i) 𝑏 =− 𝑥1 + 𝑥2 + 𝑥3 ; )
(ii) 𝑐 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥2𝑥3;
(iii) 𝑑 =− 𝑥1𝑥2𝑥3. [3]

It is given that 𝑏 =− 9 and 𝑐 = 45 for parts (b) and © below.

(b)

(i) In the case that the three roots 𝑥1, 𝑥2 and 𝑥3 form an arithmetic sequence, show
that one of the roots is 3.

(ii) Hence determine the value of 𝑑. [5]

(c) In another case the three roots form a geometric sequence. Determine the value
of 𝑑. [6]

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