Algebra Examination Questions HL

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

Full Solution

Question 2

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

Find \(x\)

Hint

Full Solution

 

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

Full Solution

Question 4

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

Hint

Full Solution

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

Full Solution

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

Full Solution

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

Full Solution

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

Full Solution

 

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

Full Solution

 

Question 3

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

Find the possible values of x

Hint

Full Solution

 

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

Full Solution

 

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

Full Solution

 

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

Full Solution

 

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

Full Solution

 

 

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

Full Solution

 

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

Full Solution

Question 2

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

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

Hint

Full Solution

Question 3

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

Hint

Full Solution

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

Full Solution

Question 5

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

Hint

Full Solution

 

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

Full Solution

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

Full Solution

Counting Principles

Question 1

A team of five players is chosen from six males and 5 females.

  1. Determine how many different teams can be formed.
  2. Determine how many different teams can be formed consisting of 3 males and 2 females.
  3. Determine how many different teams can be formed if the team consists of more females than males

Hint

Full Solution

Question 2

A five-digit number is formed by using the digits 1-5 exactly once.

  1. How many five-digit numbers are there?
  2. How many of these five-digit numbers are even?

Hint

Full Solution

Question 3

Seven students are placed at random in a line.

  1. How many different arrangements are there?
  2. What is the probability that the two youngest students are separated?

Hint

Full Solution

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

Full Solution

Question 2

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

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

Hint

Full Solution

Question 3

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

Hint

Full Solution

 

Question 4

a) Use the binomial theorem to expand \(\sqrt{4-9x}\) in ascending powers of x up to and including x3

b) State the value of x for which the expansion is valid.

Hint

Full Solution

 

Question 5

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

Find a

Hint

Full Solution

Question 6

In the expansion \((1+bx)^n\), the coefficient of the x term is -6 and the coefficient of the x² term is 27.

Work out b and n.

Hint

Full Solution

 

Question 7

a) Write \(f(x)=\frac{2+x}{1+x-2x^2}\) as the sum of partial fractions

b) Find the binomial expansion of f(x) in ascending powers of x up to and including

c) State the values of x for which the series is valid

Hint

Full Solution

 

Question 8

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

Find a and b

Hint

Full Solution

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

Full Solution

 

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

Full Solution

 

Question 3

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

Hint

Full Solution

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

Full Solution

 

Question 5

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

Hint

Full Solution

 

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

Full Solution

Proof by Contradiction

Question 1

Prove by contradiction that, if n² is even, then n is even

Hint

Full Solution

 

Question 2

a) Use a deductive proof to prove that even x even = even

b) Similarly prove that odd x odd = odd

c) Hence, use proof by contradiction to prove that \(log_25\) is irrational

Hint

Full Solution

Question 3

Prove by contradiction that a rational number + an irrational number = irrational number.

Hint

Full Solution

 

Question 4

Prove that \(\sqrt{3}\) is irrational

Hint

Full Solution

 

Alternate Solution

 

Question 5

Prove that \(\sqrt[3]{5}\) is irrational

Hint

Full Solution

 

Question 6

Prove by contradiction that the length of the hypotenuse of a right-angled triangle is less than the sum of the other two sides.

Hint

Full Solution

Question 7

Prove that \(\sqrt[n]{p}\) is irrational, given that p is prime.

Hint

Full Solution

Question 8

Prove by contradiction that there are no rational roots to the equation \(x^3+x+1=0\)

Hint

Full Solution

 

Proof by Induction

Question 1

Prove that \(12^n+2\times5^{n-1}\) is divisible by 7 , \(n\in\mathbb{Z^+}\)

Hint

Full Solution

Question 2

Prove by induction that \(\sum _{ r=1 }^{ n }{(r\times{ 2 }^{ r-1 })} =(n-1)2^ n+1 \ ,\ n\in\mathbb{Z^+}\)

Hint

Full Solution

Question 3

Let \(y = sinx\)

Prove by induction that \(\frac{d^ny}{dx^n}=sin(x+\frac{n\pi}{2})\)

Hint

Full Solution

Question 4

Prove by induction that \(sinx+sin3x+sin5x+...+sin(2n-1)x=\frac{1-cos2nx}{2sinx} , \quad n\in\mathbb{Z^+} ,\quad sinx\neq 0\)

Hint

Full Solution

Complex Numbers - The Basics

Question 1

Find the possible values of a, given that \(|\frac{z_1}{z_2}|=2\) and \(z_1=a+2i\) and \(z_2=1-2i\)

Hint

Full Solution

 

Question 2

It is given that \(z_1=2+3i\) and \(z_2=4+ai\)

Find a if \(Im(z_1z_2^*)=0\)

Hint

Full Solution

Question 3

Let \(z=a+bi\)

Find a and b if \(z^2=\left|z\right|^2-4\)

Hint

Full Solution

 

Question 4

Show that \((1-\frac{1}{3} e^{2iθ})(1-\frac{1}{3} e^{-2iθ})=\frac{1}{9}(10-6\cos 2θ)\)

Hint

Full Solution

 

Question 5

Given that z is a complex number and \(\frac{3z-4}{5}=\frac{p-2i}{3-i}\), where \(p \in\mathbb{R} \)

a) Express z in the form \(a+bi\), \(a,b\in\mathbb{R}\)

Given that \(arg(z)=-\frac{\pi}{2}\)

b) Find the value of p

Hint

Full Solution

 

Question 6

a) Given that \(z=a+bi\), show that \(z+\frac{1}{z}=2cos\theta\)

b) Hence, show that \(z^n+\frac{1}{z^n}=2cosn\theta\)

c) Use the Binomial expansion to expand \((z+{1\over z})^5\)

d) Hence, show that \(cos^5 θ={1 \over 16}cos⁡5θ+{5 \over 16} cos3θ+{5\over 8} cosθ\)

Hint

Full Solution

Complex Numbers - De Moivre's Theorem

Question 1

Find the roots of the equation \(z^3=8i \)

Express your answers in Cartesian Form

Hint

Full Solution

Question 2

\(z=-2+2\sqrt{3}i\)

a) Find |z| and arg(z)

b) Find \(z^6\) and simplify your answer

c) Given that \(w^4=z^3\) , find the values of \(w\) giving your answers in the form \(a+bi\)

Hint

Full Solution

 

Question 3

Find the values of n such that \((\sqrt{3}-i)^n\) is a real number

Hint

Full Solution

 

Question 4

a) By writing \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\), show that \(sin(\frac{\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)

b) Work out \(cos(\frac{\pi}{12})\)

c) Hence, find the roots of the equation \(z^4=2+2\sqrt{3}i\), giving answers in the form \(z=a+ib\)

Hint

Full Solution

Question 5

a) Find the roots of the equation \(z^4-1=0\)

b) Find the roots of the equation \(z^4+1=0\)

c) Show that roots of \(z^4-1=0\) and \(z^4+1=0\) together make the roots of \(z^8-1=0\)

d) Hence, find all the roots to \(z^6+z^4+z^2+1=0\)

Hint

Full Solution

Complex Numbers - Roots of Polynomials

Question 1

One root of the equation z² + bz + c = 0 is 2+3i where \(b,c\in\mathbb{Z}\).

Find the value of b and the value of c.

Hint

Full Solution

Question 2

\(\frac{2}{1+i}\) is a root to the quadratic equation z² + px + q = 0

a) Find the other root

b) Hence find the values of p and q.

Hint

Full Solution

Question 3

2 - 3i is a root of the equation \(z^3-7z^2+az+b=0\quad ,a, b\in\ \mathbb{R}\)

Work out a and b and the other roots of the equation.

Hint

Full Solution

Question 4

The quartic equation \(z^4+az^3+bz^2+cz+d\) has roots 2 + i and 2i

a) Work out the other roots of the equation

b) Find the values of a , b , c and d

Hint

Full Solution

Question 5

The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).

a. Show that \(\alpha+\beta=\frac{1}{2}.\)

b. Find \(\alpha\beta\).

c. Hence find the two real roots α and β.

d. Find the values of p and q.

Hint

Full Solution

MY PROGRESS

How much of Algebra Examination Questions HL have you understood?