Algebra Examination Questions SL

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  3. Algebra Examination Questions SL
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On this page you can find examination questions from the topic of algebra  

Arithmetic Sequences

Question 1

In an arithmetic sequence, the first term is 4 and the third term is 16.

a) Find the common difference

b) Find the 8th term

c) Find the sum of the first 8 terms

Hint

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Question 2

Three consecutive terms of an arithmetic sequence are \(x-3 \ , \ 12 \ ,\ 3x-5\)

Find \(x\)

Hint

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Question 3

The 2nd term of an arithmetic sequence is 19 and the 5th term is 37.

a) Find the 10th term

b) The sum of the first n terms of this sequence exceeds 1000. Find the least value of n

Hint

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Question 4

Find the sum of all the integers between 100 and 1000 that are divisible by 9

Hint

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Question 5

An arithmetic sequence has first term U1 and common difference d. The sum of the first 17 terms is 136.

a) Show that \(U_1+8d=8\)

The sum of the 2nd and the 3rd terms is 42.

b) Find d.

The nth term of the sequence is Un.

c) Find the value of \(\sum _{ r=4 }^{ 17 }{ { U }_{ n } } \)

Hint

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Question 6

In an arithmetic sequence, the 9th term is 4 times the 5th term. The sum of the first 2 terms is -13.

Find the 10th term

Hint

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Question 7

The first terms of a sequence are log3 x , log3 x2 , log3 x3 , ...

Find x if the sum of the first 9 terms is 135

Hint

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Geometric Sequences

Question 1

The nth term of a geometric sequence is Un , where Un =\(48(\frac{1}{4})^n\)

a) Find U1

b) Find the sum to infinity of the series

Hint

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Question 2

The first term of a geometric series is 10. The sum to infinity is 50.

a) Find the common ratio

The nth term is Un

b) Find the value of \(\sum _{ n=1 }^{ 20 }{ { U }_{ n } } \)

Hint

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Question 3

Three terms of a geometric sequence are \(x+6 \ , \ 12 \ ,\ x-1\)

Find the possible values of x

Hint

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Question 4

Consider the geometric sequence where the first term is 45 and the second term is 36.

a) Find the least value of n such that the nth term of the sequence is less than 1

b) Find the least value of n such that the sum of the first n terms of the sequence is more than 200.

c) Find the sum to infinity.

Hint

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Question 5

The sum to infinity of a geometric series is 27.

The sum of the first 3 terms is 19.

Find the common ratio

Hint

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Question 6

The 2nd, 3rd and 6th terms of an arithmetic sequence with common difference \(d, \ d\neq 0\) form the first three terms of a geometric sequence, with common ratio, r.

The 1st term of the arithmetic sequence is -2.

a) Find d.

The sum of the first n terms of the geometric sequence exceeds the sum of the first n terms of the arithmetic sequence by at least 1000.

b) Find the least value of n for which this occurs.

Hint

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Question 7

U1 = cosx ,U2 = sin2x are the first two terms of a geometric sequence, \(-\frac{\pi}{2}<x<\frac{\pi}{2}\)

a) Find U3 in terms of cosx

 

 

b) Find the set of values of x for which the geometric series converges

Hint

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Question 8

a) Jessica takes out a loan of $200 000 to buy an appartment. The interest rate is 4% and is calculated at the end of each year. Calculate to the nearest dollar the amount Jessica would owe the bank after 15 years.

b) In order to pay of the loan, she pays $P into a bank at the end of each year. She receives an interest rate of 2.5% per year. Find the amount saved after 15 years.

c) What must be the value of P so that she has saved enough money to pay off the loan.

Hint

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Indices and Logarithms

Question 1

Find the value of each of the following, giving your answer as an integer

a. \({ log }_{ 4 }16\)

b. \({ log }_{ 4 }2+{ log }_{ 4 }32\)

c. \({ log }_{ 4 }8-{ log }_{ 4 }32\)

Hint

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Question 2

a. Given that 3a =27 , write down the value of a

b. Hence of otherwise solve 27x+4 = 93x+1

Hint

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Question 3

Solve the equation log2(x - 3) = 1 - log2(x - 4)

Hint

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Question 4

Solve the equation \(3^{ x-1 }=\frac { 2 }{ { 4 }^{ x+1 } } \) , giving your answer in the form \(x=\frac{lna}{lnb}\) , where a and b are rational numbers.

Hint

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Question 5

Solve \(ln⁡(sinx)−ln⁡(cosx)=e\) for \(0<x<2\pi\)

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Question 6

\(a=\log _{ 2 }{ 2+\log _{ 2 }{ \frac { 3 }{ 2 } + } } \log _{ 2 }{ \frac { 4 }{ 3 } + } \quad ...\quad +\log _{ 2 }{ \frac { 32 }{ 31 } } \)

Given that \(a\in \mathbb{Z}\) , find the value of a

Hint

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Question 7

The first three terms of a geometric sequence are \(\log _{ 3 }{x\quad ,\quad \log _{ 9 }{ x\quad ,\quad \log _{ 81 }{ x } } } \)

Find the value of x if the sum to infinity is 8.

Hint

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Binomial Theorem

Question 1

The values of the third row of Pascal's triangle are given below

a) Write down the values in the fourth row of Pascal's triangle

b) Hence or otherwise, find the term in x² in the expansion of \((3x+2)^4\)

Hint

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Question 2

a) Expand (x - 3)4 and simplify your result

b) Hence find the x3 term in (x - 2)(x - 3)4 .

Hint

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Question 3

Find the term independent of x in the expansion \((2x- \frac{3}{x^2})^6\)

Hint

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Question 4

The x term in the expansion \((4+2x)^3(2+ax)^4\) is -4608x

Find a

Hint

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Question 5

​\(\large(\textbf{a}+2x)^3(4-x)^4 = 6912 + \textbf{b}x +...\)​

Find a and b

Hint

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Deductive Proofs

Question 1

a) Verify that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2=4x^3 \) for x = 3

b) Prove that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2\equiv4x^3 \) for all x

Hint

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Question 2

a) Verify that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=\frac{2x+1}{x^2\left(x+1\right)^2} \) for x = -2

b) Prove that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\equiv\frac{2x+1}{x^2\left(x+1\right)^2} \) for all x

Hint

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Question 3

Prove that the sum of three consecutive integers is divisible by 3

Hint

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Question 4

a) Verify that x² - 4x + 5 is positive for x = -1

b) Prove that x² - 4x + 5 is positive for all x

Hint

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Question 5

Prove that the difference between the square of any two consecutive odd integers is divisible by 8

Hint

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Question 6

a) Verify that \(^3 C_1\ +\ ^3 C_2\ =\ ^4C_2\)

b) Prove that \(^{n-1} C_{r-1}\ +\ ^{n-1} C_r\ =\ ^nC_r\)

Hint

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