• Syllabus

Topic 1 - Core: Algebra

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Sub sections and their related questions

1.1

• 12M.1.hl.TZ1.1: Find the value of k if \({\sum\limits_{r = 1}^\infty{k\left( {\frac{1}{3}} \right)}^r} = 7\).
• 12M.2.hl.TZ2.1a: Find the first term and the common difference.
• 12M.2.hl.TZ2.1b: Find the smallest value of n such that the sum of the first n terms is greater than 600.
• 12M.2.hl.TZ2.8a: What height does the ball reach after its fourth bounce?
• 12M.2.hl.TZ2.8b: How many times does the ball bounce before it no longer reaches a height of 1 metre?
• 12M.2.hl.TZ2.8c: What is the total distance travelled by the ball?
• 12N.2.hl.TZ0.1: Find the sum of all the multiples of 3 between 100 and 500.
• 12N.2.hl.TZ0.5: A metal rod 1 metre long is cut into 10 pieces, the lengths of which form a geometric sequence....
• 08M.1.hl.TZ1.7: The common ratio of the terms in a geometric series is \({2^x}\) . (a)     State the set of...
• 08M.1.hl.TZ2.12a: Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... .
• 08N.1.hl.TZ0.4: An 81 metre rope is cut into n pieces of increasing lengths that form an arithmetic sequence with...
• 08N.2.hl.TZ0.2: A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the...
• 11M.1.hl.TZ2.10a: Show that \(a = - \frac{3}{2}d\) .
• 11M.1.hl.TZ2.10b: Show that the \({{\text{4}}^{{\text{th}}}}\) term of the geometric sequence is the...
• 11M.2.hl.TZ2.2: In the arithmetic series with \({n^{{\text{th}}}}\) term \({u_n}\) , it is given that...
• 11M.3ca.hl.TZ0.4c: Hence determine the exact value of \(\int_0^\infty  {{{\text{e}}^{ - x}}|\sin x|{\text{d}}x} \) .
• 09N.1.hl.TZ0.11: (a)     The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven...
• SPNone.1.hl.TZ0.2b: The roots of this equation are three consecutive terms of an arithmetic sequence. Taking the...
• SPNone.2.hl.TZ0.2: The first term and the common ratio of a geometric series are denoted, respectively, by a and r...
• SPNone.2.hl.TZ0.11a: (i)     Find an expression for \({S_1}\) and show...
• SPNone.2.hl.TZ0.11b: Sue borrows $5000 at a monthly interest rate of 1 % and plans to repay the loan in 5 years (i.e....
• 13M.1.hl.TZ1.8: The first terms of an arithmetic sequence are...
• 10M.2.hl.TZ1.6: Find the sum of all three-digit natural numbers that are not exactly divisible by 3.
• 10M.2.hl.TZ2.1: Consider the arithmetic sequence 8, 26, 44, \( \ldots \) . (a)     Find an expression for the...
• 10M.2.hl.TZ2.13: The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas...
• 10N.1.hl.TZ0.5: The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty...
• 10N.1.hl.TZ0.6: The sum, \({S_n}\), of the first n terms of a geometric sequence, whose \({n^{{\text{th}}}}\)...
• 13M.1.hl.TZ2.6: A geometric sequence has first term a, common ratio r and sum to infinity 76. A second geometric...
• 13M.2.hl.TZ2.5a: Find an expression for \({u_n} - {v_n}\) in terms of n.
• 13M.2.hl.TZ2.11a: (i)     Express the sum of the first n positive odd integers using sigma notation. (ii)     Show...
• 11N.2.hl.TZ0.7a: Find the set of values of x for which the series...
• 11N.2.hl.TZ0.7b: Hence find the sum in terms of x.
• 11N.2.hl.TZ0.12a: In an arithmetic sequence the first term is 8 and the common difference is \(\frac{1}{4}\). If...
• 11N.2.hl.TZ0.12b: If \({a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }} \ldots \) are terms of a geometric sequence...
• 11N.2.hl.TZ0.14b: Consider the geometric series...
• 11N.2.hl.TZ0.14c: Find an expression for the sum to infinity of this series.
• 11M.1.hl.TZ1.3b: An arithmetic sequence \({v_1}\) , \({v_2}\) , \({v_3}\) , \(...\) is such that \({v_2} = {u_2}\)...
• 11M.1.hl.TZ1.3a: Find the common ratio of the geometric sequence.
• 14M.1.hl.TZ1.13: A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by...
• 14M.1.hl.TZ2.9: The first three terms of a geometric sequence are \(\sin x,{\text{ }}\sin 2x\) and...
• 14M.2.hl.TZ2.1: (a)     (i)     Find the sum of all integers, between 10 and 200, which are divisible by 7.    ...
• 13N.1.hl.TZ0.7: The sum of the first two terms of a geometric series is 10 and the sum of the first four terms is...
• 13N.2.hl.TZ0.2: The fourth term in an arithmetic sequence is 34 and the tenth term is 76. (a)     Find the first...
• 14N.2.hl.TZ0.7a: Show that \(d = \frac{a}{2}\).
• 14N.2.hl.TZ0.7b: The seventh term of the arithmetic sequence is \(3\). The sum of the first \(n\) terms in the...
• 14N.2.hl.TZ0.12d: Find the probability that Ava eventually wins.
• 14N.3ca.hl.TZ0.4a: Consider the infinite geometric...
• 15M.1.hl.TZ1.12a: (i)     Show that \(\frac{{{v_{n + 1}}}}{{{v_n}}}\) is a constant. (ii)     Write down the first...
• 15M.1.hl.TZ1.12b: Let \({S_n}\) be the sum of the first \(n\) terms of the sequence \(\{ {v_n}\} \). (i)     Find...
• 15M.1.hl.TZ1.12c: Let \(\{ {w_n}\} ,{\text{ }}n \in {\mathbb{Z}^ + }\), be a geometric sequence with first term...
• 15M.1.hl.TZ2.12b: It is now given that \(p =  - 6\) and \(q = 18\) for parts (b) and (c) below. (i)     In the...
• 15M.1.hl.TZ2.12c: In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form a geometric...
• 15N.1.hl.TZ0.6b: The game is now changed so that the ball chosen is replaced after each turn. Darren still plays...
• 15N.1.hl.TZ0.10a: the degree of the polynomial;
• 16M.1.hl.TZ1.1: The fifth term of an arithmetic sequence is equal to 6 and the sum of the first 12 terms is...
• 16M.2.hl.TZ2.4: The sum of the second and third terms of a geometric sequence is 96. The sum to infinity of this...
• 16N.1.hl.TZ0.6a: Write down the value of \({u_1}\).
• 16N.1.hl.TZ0.6b: Find the value of \({u_6}\).
• 16N.1.hl.TZ0.6c: Prove that \(\{ {u_n}\} \) is an arithmetic sequence, stating clearly its common difference.
• 16N.2.hl.TZ0.12a: Find an expression for \({A_1}\) and show that \({A_2} = {1.004^2}x + 1.004x\).
• 16N.2.hl.TZ0.12b: (i)     Write down a similar expression for \({A_3}\) and \({A_4}\). (ii)     Hence show that...
• 16N.2.hl.TZ0.12c: Write down an expression for \({A_n}\) in terms of \(x\) on the day before Mary turned 18 years...
• 16N.2.hl.TZ0.12d: Mary’s grandparents wished for the amount in her account to be at least \(\$ 20\,000\) the day...
• 16N.2.hl.TZ0.12e: As soon as Mary was 18 she decided to invest \(\$ 15\,000\) of this money in an account of the...
• 17M.1.hl.TZ1.7a: find the value of \(d\).
• 17M.1.hl.TZ1.7b: determine the value of \(\sum\limits_{r = 1}^N {{u_r}} \).
• 17M.1.hl.TZ2.3a: the value of \(d\);
• 17M.1.hl.TZ2.3b: the value of \(r\);
• 17N.2.hl.TZ0.12a: Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.
• 17N.2.hl.TZ0.12b: Show that the total value of Phil’s savings after 20 years is...
• 17N.2.hl.TZ0.12c: Given that Phil’s aim is to own the house after 20 years, find the value for \(P\) to the nearest...
• 17N.2.hl.TZ0.12d.i: David wishes to withdraw $5000 at the end of each year for a period of \(n\) years. Show that an...
• 17N.2.hl.TZ0.12d.ii: Hence or otherwise, find the minimum value of \(Q\) that would permit David to withdraw annual...
• 18M.2.hl.TZ1.1a: Find the first term and the common difference of the sequence.
• 18M.2.hl.TZ1.1b: Calculate the number of positive terms in the sequence.
• 18M.2.hl.TZ1.7a: Show that there will be approximately 2645 fish in the lake at the start of 2020.
• 18M.2.hl.TZ1.7b: Find the approximate number of fish in the lake at the start of 2042.
• 18M.1.hl.TZ2.5a: Show that A is an arithmetic sequence, stating its common difference d in terms of r.
• 18M.1.hl.TZ2.5b: A particular geometric sequence has u1 = 3 and a sum to infinity of 4. Find the value of d.

1.2

• 12M.1.hl.TZ1.8: Solve the equation \(2 - {\log _3}(x + 7) = {\log _{\tfrac{1}{3}}}2x\) .
• 12M.2.hl.TZ1.6: Let \(f(x) = \ln x\) . The graph of f is transformed into the graph of the function g by a...
• 13M.1.hl.TZ1.8: The first terms of an arithmetic sequence are...
• 10M.1.hl.TZ1.4: Solve the equation \({4^{x - 1}} = {2^x} + 8\).
• 11M.2.hl.TZ1.9: Solve the following system of equations. \[{\log _{x + 1}}y = 2\]\[{\log _{y + 1}}x = \frac{1}{4}\]
• 14M.1.hl.TZ1.3: Consider...
• 14M.1.hl.TZ2.2: Solve the equation \({8^{x - 1}} = {6^{3x}}\). Express your answer in terms of \(\ln 2\) and...
• 15M.1.hl.TZ1.12a: (i)     Show that \(\frac{{{v_{n + 1}}}}{{{v_n}}}\) is a constant. (ii)     Write down the first...
• 15M.1.hl.TZ1.12b: Let \({S_n}\) be the sum of the first \(n\) terms of the sequence \(\{ {v_n}\} \). (i)     Find...
• 15M.1.hl.TZ1.12c: Let \(\{ {w_n}\} ,{\text{ }}n \in {\mathbb{Z}^ + }\), be a geometric sequence with first term...
• 16M.2.hl.TZ1.7a: Use these results to find estimates for the value of \(a\) and the value of \(b\). Give your...
• 16M.2.hl.TZ1.7b: Use this model to estimate the mean time for the finalists in an Olympic race for boats with 8...
• 16M.2.hl.TZ1.7c: Calculate the error in your estimate as a percentage of the actual value.
• 16M.2.hl.TZ1.7d: Comment on the likely validity of the model as \(N\) increases beyond 8.
• 16M.1.hl.TZ1.6: Find integer values of \(m\) and \(n\) for which \[m - n{\log _3}2 = 10{\log _9}6\]
• 16M.2.hl.TZ2.3: Solve the simultaneous equations \[\ln \frac{y}{x} = 2\] \[\ln {x^2} + \ln {y^3} = 7.\]
• 16N.1.hl.TZ0.7: Solve the equation \({4^x} + {2^{x + 2}} = 3\).
• 17M.1.hl.TZ1.1: Find the solution of \({\log _2}x - {\log _2}5 = 2 + {\log _2}3\).
• 17M.1.hl.TZ2.7a: The random variable \(X\) has the Poisson distribution \({\text{Po}}(m)\). Given that...
• 17M.2.hl.TZ2.6: Given that...
• 17N.1.hl.TZ0.1: Solve the equation \({\log _2}(x + 3) + {\log _2}(x - 3) = 4\).
• 18M.1.hl.TZ1.5: Solve ...
• 18M.1.hl.TZ2.11a: Show that \({\text{lo}}{{\text{g}}_{{r^2}}}x =...
• 18M.1.hl.TZ2.11b: Express \(y\) in terms of \(x\). Give your answer in the form \(y = p{x^q}\), where p , q...

1.3

• 12M.2.hl.TZ1.9: Find the constant term in the expansion of...
• 12M.1.hl.TZ2.4a: Expand and simplify \({\left( {x - \frac{2}{x}} \right)^4}\).
• 12M.1.hl.TZ2.4b: Hence determine the constant term in the expansion...
• 12M.2.hl.TZ2.4a: In how many ways can they be seated in a single line so that the boys and girls are in two...
• 12M.2.hl.TZ2.4b: Two boys and three girls are selected to go the theatre. In how many ways can this selection be...
• 12N.1.hl.TZ0.2: Expand and simplify \({\left( {\frac{x}{y} - \frac{y}{x}} \right)^4}\).
• 08M.2.hl.TZ1.1: Determine the first three terms in the expansion of \({(1 - 2x)^5}{(1 + x)^7}\) in ascending...
• 13M.1.hl.TZ1.13b: (i)     Explain why the total number of possible outcomes for the results of the 6 games is...
• 13M.2.hl.TZ1.8a: Find the number of ways this can be done if the three girls must sit together.
• 13M.2.hl.TZ1.8b: Find the number of ways this can be done if the three girls must all sit apart.
• 10M.2.hl.TZ1.7: Three Mathematics books, five English books, four Science books and a dictionary are to be placed...
• 10N.1.hl.TZ0.3: Expand and simplify \({\left( {{x^2} - \frac{2}{x}} \right)^4}\).
• 13M.1.hl.TZ2.3: Expand \({(2 - 3x)^5}\) in ascending powers of x, simplifying coefficients.
• 13M.2.hl.TZ2.11b: A number of distinct points are marked on the circumference of a circle, forming a polygon....
• 11N.3ca.hl.TZ0.2a: Show that \(n! \geqslant {2^{n - 1}}\), for \(n \geqslant 1\).
• 11M.2.hl.TZ1.5b: Hence, or otherwise, find the coefficient of \(x\) in the expansion of...
• 09N.2.hl.TZ0.4: (a)     Find the value of \(n\) . (b)     Hence, find the coefficient of \({x^2}\) .
• 09M.2.hl.TZ2.8: Six people are to sit at a circular table. Two of the people are not to sit immediately beside...
• 14M.2.hl.TZ1.3: Find the number of ways in which seven different toys can be given to three children, if the...
• 14M.2.hl.TZ1.8a: Find the term in \({x^5}\) in the expansion of \((3x + A){(2x + B)^6}\).
• 14M.2.hl.TZ2.5: Find the coefficient of \({x^{ - 2}}\) in the expansion of...
• 13N.1.hl.TZ0.12g: (i)     Write down an expression for the constant term in the expansion of...
• 13N.1.hl.TZ0.12b: Expand \({\left( {z + {z^{ - 1}}} \right)^4}\).
• 14N.1.hl.TZ0.10a: Find the number of selections Grace could make if the largest integer drawn among the four cards...
• 14N.1.hl.TZ0.10b: Find the number of selections Grace could make if at least two of the four integers drawn are even.
• 15M.1.hl.TZ1.4a: Expand \({(x + h)^3}\).
• 15M.1.hl.TZ2.2: Expand \({(3 - x)^4}\) in ascending powers of \(x\) and simplify your answer.
• 15M.2.hl.TZ1.12a: (i)     Use the binomial theorem to expand \({(\cos \theta  + {\text{i}}\sin \theta )^5}\). (ii)...
• 15M.2.hl.TZ2.2a: Determine how many possible groups can be chosen.
• 15M.2.hl.TZ2.2b: Determine how many groups can be formed consisting of two males and two females.
• 15M.2.hl.TZ2.2c: Determine how many groups can be formed consisting of at least one female.
• 15N.1.hl.TZ0.3a: Write down and simplify the expansion of \({(2 + x)^4}\) in ascending powers of \(x\).
• 15N.1.hl.TZ0.3b: Hence find the exact value of \({(2.1)^4}\).
• 16M.1.hl.TZ2.6a: Write down the first four terms of the expansion.
• 16M.1.hl.TZ2.6b: (i)     Show that \({n^3} - 9{n^2} + 14n = 0\). (ii)     Hence find the value of \(n\).
• 16N.2.hl.TZ0.4: Find the constant term in the expansion of \({\left( {4{x^2} - \frac{3}{{2x}}} \right)^{12}}\).
• 17M.1.hl.TZ1.4: Three girls and four boys are seated randomly on a straight bench. Find the probability that the...
• 17M.1.hl.TZ2.1: Find the term independent of \(x\) in the binomial expansion of...
• 17M.2.hl.TZ1.3: The coefficient of \({x^2}\) in the expansion of \({\left( {\frac{1}{x} + 5x} \right)^8}\) is...
• 17M.2.hl.TZ2.8: In a trial examination session a candidate at a school has to take 18 examination papers...
• 17N.1.hl.TZ0.4: Find the coefficient of \({x^8}\) in the expansion of \({\left( {{x^2} - \frac{2}{x}} \right)^7}\).
• 17N.1.hl.TZ0.10a: Show that the probability that Chloe wins the game is \(\frac{3}{8}\).
• 17N.2.hl.TZ0.9a: Find the number of ways the twelve students may be arranged in the exam hall.
• 17N.2.hl.TZ0.9b: Find the number of ways the students may be arranged if Helen and Nicky must sit so that one is...
• 17N.2.hl.TZ0.9c: Find the number of ways the students may be arranged if Helen and Nicky must not sit next to each...
• 18M.2.hl.TZ2.5a: Express the binomial...
• 18M.2.hl.TZ2.5b: Hence find the least value of \(n\) for which...

1.4

• 12M.1.hl.TZ2.13b: Prove by induction that the \({n^{{\text{th}}}}\) derivative of \({(2x + 1)^{ - 1}}\) is...
• 12N.1.hl.TZ0.12b: Let \({F_n}(x) = \frac{x}{{{2^n} - ({2^n} - 1)x}}\), where \(0 \leqslant x \leqslant 1\). Use...
• 08M.1.hl.TZ2.12b: Use mathematical induction to prove that for \(n \in {\mathbb{Z}^ + }\)...
• 11M.2.hl.TZ2.13A: Prove by mathematical induction that, for \(n \in {\mathbb{Z}^ +...
• 09M.1.hl.TZ2.8: Prove by mathematical induction \(\sum\limits_{r = 1}^n {r(r!) = (n + 1)! - 1} \),...
• 09N.1.hl.TZ0.11d: Using mathematical induction, prove...
• SPNone.1.hl.TZ0.12c: Suggest an expression for \({f^{(2n)}}(x)\), \(n \in {\mathbb{Z}^ + }\), and prove your...
• 10M.1.hl.TZ1.13: (a)     Show that...
• 10M.1.hl.TZ2.11: (a)     Consider the following sequence of equations.    ...
• 13M.2.hl.TZ2.8: Use the method of mathematical induction to prove that \({5^{2n}} - 24n - 1\) is divisible by 576...
• 11N.1.hl.TZ0.6: Given that \(y = \frac{1}{{1 - x}}\), use mathematical induction to prove that...
• 14M.2.hl.TZ1.7: Prove, by mathematical induction, that \({7^{8n + 3}} + 2,{\text{ }}n \in \mathbb{N}\), is...
• 14M.2.hl.TZ2.13a: Consider \(z = r(\cos \theta  + {\text{i}}\sin \theta ),{\text{ }}z \in \mathbb{C}\). Use...
• 13N.1.hl.TZ0.6: Prove by mathematical induction that \({n^3} + 11n\) is divisible by 3 for all...
• 14N.1.hl.TZ0.8: Use mathematical induction to prove that...
• 15M.1.hl.TZ1.11b: Prove by induction that...
• 15M.1.hl.TZ2.13c: Prove, by mathematical induction, that...
• 15N.3ca.hl.TZ0.3a: Prove by induction that \(n! > {3^n}\), for \(n \ge 7,{\text{ }}n \in \mathbb{Z}\).
• 15N.1.hl.TZ0.8b: Consider \(f(x) = \sin (ax)\) where \(a\) is a constant. Prove by mathematical induction that...
• 16M.1.hl.TZ2.8: Use mathematical induction to prove that \(n({n^2} + 5)\) is divisible by 6 for...
• 16M.1.hl.TZ1.12b: Use mathematical induction to prove...
• 16N.1.hl.TZ0.13c: Use the principle of mathematical induction to prove...
• 17M.1.hl.TZ1.8: Use the method of mathematical induction to prove that \({4^n} + 15n - 1\) is divisible by 9 for...
• 17M.1.hl.TZ2.8: Prove by mathematical induction that...
• 17N.1.hl.TZ0.11b: By using mathematical induction, prove...
• 18M.1.hl.TZ1.6: Use the principle of mathematical induction to prove...
• 18M.2.hl.TZ2.6: Use mathematical induction to prove that \({\left( {1 - a} \right)^n} > 1 - na\)...

1.5

• 12M.1.hl.TZ1.3: If \({z_1} = a + a\sqrt 3 i\) and \({z_2} = 1 - i\), where a is a real constant, express...
• 12M.1.hl.TZ1.7: Given that z is the complex number \(x + {\text{i}}y\) and that...
• 12M.1.hl.TZ2.6a: m and n are real numbers;
• 12M.1.hl.TZ2.6b: m and n are conjugate complex numbers.
• 12M.1.hl.TZ2.12A.a: Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in...
• 12M.1.hl.TZ2.12A.c: For any complex number z , show that \({(z^*)^2} = ({z^2})^*\) .
• 12N.1.hl.TZ0.10a: (i)     Write down \({z_1}\) in Cartesian form. (ii)     Hence determine...
• 12N.2.hl.TZ0.10: Let \(\omega = \cos \theta  + {\text{i}}\sin \theta \) . Find, in terms of \(\theta \) , the...
• 08M.1.hl.TZ1.1: Express...
• 08M.2.hl.TZ1.10: Find, in its simplest form, the argument of...
• 08M.2.hl.TZ1.14: \({z_1} = {(1 + {\text{i}}\sqrt 3 )^m}{\text{ and }}{z_2} = {(1 - {\text{i}})^n}\) . (a)    ...
• 08M.2.hl.TZ2.9: Consider...
• 08N.1.hl.TZ0.13Part B: (a)     Expand and simplify \((x - 1)({x^4} + {x^3} + {x^2} + x + 1)\) . (b)     Given that b is...
• 11M.1.hl.TZ2.4a: Find AB, giving your answer in the form \(a\sqrt {b - \sqrt 3 } \) , where a ,...
• 11M.1.hl.TZ2.4b: Calculate \({\rm{A\hat OB}}\) in terms of \(\pi \).
• 11M.1.hl.TZ2.12b: Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\). (i)     Show that \(\gamma \) is one of the...
• 09M.1.hl.TZ1.1: Consider the complex numbers \(z = 1 + 2{\text{i}}\) and \(w = 2 + a{\text{i}}\) , where...
• 09M.1.hl.TZ1.13Part A: If z is a non-zero complex number, we define \(L(z)\) by the...
• 09N.1.hl.TZ0.2: Find the values of n such that \({\left( {1 + \sqrt 3 {\text{i}}} \right)^n}\) is a real number.
• 09N.1.hl.TZ0.13a: Let \(z = x + {\text{i}}y\) be any non-zero complex number. (i)     Express \(\frac{1}{z}\) in...
• SPNone.2.hl.TZ0.4a: Find the modulus and argument of z , giving the argument in degrees.
• 13M.1.hl.TZ1.1a: If w = 2 + 2i , find the modulus and argument of w.
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 10M.1.hl.TZ2.13: Consider...
• 10M.2.hl.TZ2.9: Given that \(z = \cos \theta + {\text{i}}\sin \theta \) show that (a)    ...
• 10N.1.hl.TZ0.11: Consider the complex number \(\omega = \frac{{z + {\text{i}}}}{{z + 2}}\), where...
• 13M.1.hl.TZ2.7a: Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).
• 11N.2.hl.TZ0.6: The complex numbers \({z_1}\) and \({z_2}\) have arguments between 0 and \(\pi \) radians. Given...
• 11N.2.hl.TZ0.10: Given that...
• 11N.2.hl.TZ0.14d: Hence, show that...
• 11M.1.hl.TZ1.2: Given that \(\frac{z}{{z + 2}} = 2 - {\text{i}}\) , \(z \in \mathbb{C}\) , find z in the form...
• 11M.1.hl.TZ1.13a: Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the...
• 14M.1.hl.TZ1.13: A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by...
• 14M.1.hl.TZ2.7: Consider the complex numbers \(u = 2 + 3{\text{i}}\) and \(v = 3 + 2{\text{i}}\). (a)     Given...
• 13N.2.hl.TZ0.6: A complex number z is given by...
• 15M.2.hl.TZ1.12a: (i)     Use the binomial theorem to expand \({(\cos \theta  + {\text{i}}\sin \theta )^5}\). (ii)...
• 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...
• 16N.1.hl.TZ0.12b: Show that \((\omega  - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13\).
• 16N.1.hl.TZ0.12c: Find the values of \(x\) that satisfy the equation \(\left| p \right| = \left| q \right|\).
• 16N.1.hl.TZ0.12d: Solve the inequality...
• 18M.1.hl.TZ2.7a: Find the real part of \(\frac{{z + w}}{{z - w}}\).
• 18M.1.hl.TZ2.7b: Find the value of the real part of \(\frac{{z + w}}{{z - w}}\)...

1.6

• 12M.1.hl.TZ1.3: If \({z_1} = a + a\sqrt 3 i\) and \({z_2} = 1 - i\), where a is a real constant, express...
• 12M.1.hl.TZ2.12B.d: Draw the four roots on the complex plane (the Argand diagram).
• 12M.1.hl.TZ2.12B.e: Express each of the four roots of the equation in the form \(r{{\text{e}}^{{\text{i}}\theta }}\) .
• 12N.1.hl.TZ0.10b: (i)     Write \({z_2}\) in modulus-argument form. (ii)     Hence solve the equation...
• 12N.1.hl.TZ0.10c: Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\)  and...
• 12N.1.hl.TZ0.10d: Find the smallest positive value of n for which...
• 11M.1.hl.TZ2.4a: Find AB, giving your answer in the form \(a\sqrt {b - \sqrt 3 } \) , where a ,...
• 09M.1.hl.TZ2.7: Given that \({z_1} = 2\) and \({z_2} = 1 + \sqrt 3 {\text{i}}\) are roots of the cubic equation...
• 09N.1.hl.TZ0.2: Find the values of n such that \({\left( {1 + \sqrt 3 {\text{i}}} \right)^n}\) is a real number.
• SPNone.1.hl.TZ0.10a: Calculate \(\frac{{{z_1}}}{{{z_2}}}\) giving your answer both in modulus-argument form and...
• SPNone.1.hl.TZ0.10b: Using your results, find the exact value of tan 75° , giving your answer in the form...
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 10M.1.hl.TZ2.13: Consider...
• 13M.1.hl.TZ2.13a: (i)     Express each of the complex numbers...
• 13M.1.hl.TZ2.13b: (i)     State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them...
• 11N.2.hl.TZ0.14a: Show that \(\left| {{{\text{e}}^{{\text{i}}\theta }}} \right| = 1\).
• 14M.1.hl.TZ1.13: A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by...
• 14M.1.hl.TZ2.7: Consider the complex numbers \(u = 2 + 3{\text{i}}\) and \(v = 3 + 2{\text{i}}\). (a)     Given...
• 14M.2.hl.TZ2.13a: Consider \(z = r(\cos \theta  + {\text{i}}\sin \theta ),{\text{ }}z \in \mathbb{C}\). Use...
• 14M.2.hl.TZ2.13b: Given \(u = 1 + \sqrt 3 {\text{i}}\) and \(v = 1 - {\text{i}}\), (i)     express \(u\) and \(v\)...
• 14M.2.hl.TZ2.13d: Find the area of triangle O\({\text{A}}'\)\({\text{B}}'\).
• 13N.1.hl.TZ0.12d: Show that...
• 14M.2.hl.TZ2.13c: Plot point A and point B on the Argand diagram.
• 13N.1.hl.TZ0.12c: Hence show that \({\cos ^4}\theta  = p\cos 4\theta  + q\cos 2\theta  + r\), where...
• 15M.1.hl.TZ2.7b: The roots are represented by the vertices of a triangle in an Argand diagram. Show that the area...
• 15N.1.hl.TZ0.11a: Solve the equation \({z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}\) giving your answers in the...
• 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...
• 17M.1.hl.TZ1.2a.i: By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the modulus of \(w\);
• 17M.1.hl.TZ1.2a.ii: By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the argument of \(w\).
• 17M.1.hl.TZ2.5: In the following Argand diagram the point A represents the complex number \( - 1 + 4{\text{i}}\)...
• 17M.1.hl.TZ2.11c.i: Find the modulus and argument of \(z\) in terms of \(\theta \). Express each answer in its...
• 18M.1.hl.TZ1.11a.i: Express w2 and w3 in modulus-argument form.
• 18M.1.hl.TZ1.11a.ii: Sketch on an Argand diagram the points represented by w0 , w1 , w2 and w3.
• 18M.2.hl.TZ2.1a: Express \(z\) in the form \(a + {\text{i}}b\), where \(a,\,b \in \mathbb{Q}\).
• 18M.2.hl.TZ2.1b: Find the exact value of the modulus of \(z\).
• 18M.2.hl.TZ2.1c: Find the argument of \(z\), giving your answer to 4 decimal places.

1.7

• 12M.1.hl.TZ2.12A.b: Hence find the two square roots of \( - 5 + 12{\text{i}}\) .
• 12M.1.hl.TZ2.12A.d: Hence write down the two square roots of \( - 5 - 12{\text{i}}\) .
• 12N.1.hl.TZ0.10b: (i)     Write \({z_2}\) in modulus-argument form. (ii)     Hence solve the equation...
• 12N.1.hl.TZ0.10c: Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\)  and...
• 12N.1.hl.TZ0.10d: Find the smallest positive value of n for which...
• 08M.2.hl.TZ1.14: \({z_1} = {(1 + {\text{i}}\sqrt 3 )^m}{\text{ and }}{z_2} = {(1 - {\text{i}})^n}\) . (a)    ...
• 08M.1.hl.TZ2.14: Let \(w = \cos \frac{{2\pi }}{5} + {\text{i}}\sin \frac{{2\pi }}{5}\). (a)     Show that w is a...
• 08N.1.hl.TZ0.13Part A: (a)     Use de Moivre’s theorem to find the roots of the equation \({z^4} = 1 - {\text{i}}\)...
• 09M.1.hl.TZ2.12: The complex number z is defined as \(z = \cos \theta + {\text{i}}\sin \theta \) . (a)     State...
• 09N.1.hl.TZ0.13b: Let \(w = \cos \theta + {\text{i}}\sin \theta \) . (i)     Show that...
• SPNone.2.hl.TZ0.4b: Find the cube root of z which lies in the first quadrant of the Argand diagram, giving your...
• SPNone.2.hl.TZ0.4c: Find the smallest positive integer n for which \({z^n}\) is a positive real number.
• 13M.1.hl.TZ1.1b: Given...
• 10M.2.hl.TZ1.4: (a)     Solve the equation \({z^3} = - 2 + 2{\text{i}}\), giving your answers in modulus-argument...
• 13M.1.hl.TZ2.13a: (i)     Express each of the complex numbers...
• 13M.1.hl.TZ2.13b: (i)     State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them...
• 11N.1.hl.TZ0.2: Find the cube roots of i in the form \(a + b{\text{i}}\), where \(a,{\text{ }}b \in \mathbb{R}\).
• 11M.1.hl.TZ1.13b: Hence show that \(\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \) .
• 11M.1.hl.TZ1.13c: Similarly show that \(\cos 5\theta = 16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta \) .
• 13N.1.hl.TZ0.12a: Use De Moivre’s theorem to show that...
• 14N.1.hl.TZ0.13a: (i)     Show that...
• 15M.1.hl.TZ2.7a: Find three distinct roots of the equation \(8{z^3} + 27 = 0,{\text{ }}z \in \mathbb{C}\) giving...
• 15M.2.hl.TZ1.12a: (i)     Use the binomial theorem to expand \({(\cos \theta  + {\text{i}}\sin \theta )^5}\). (ii)...
• 15M.2.hl.TZ1.12b: Find the value of \(r\) and the value of \(\alpha \).
• 15N.1.hl.TZ0.11a: Solve the equation \({z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}\) giving your answers in the...
• 15N.1.hl.TZ0.11b: Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and...
• 16M.1.hl.TZ2.12a: Verify that \(w\) is a root of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\).
• 16M.1.hl.TZ2.12b: (i)     Expand \((w - 1)(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6})\). (ii)     Hence deduce...
• 16M.1.hl.TZ2.12c: Write down the roots of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\) in terms of...
• 16M.1.hl.TZ1.12a: Use de Moivre’s theorem to find the value of...
• 16N.1.hl.TZ0.12a: Determine the value of (i)     \(1 + \omega  + {\omega ^2}\); (ii)    ...
• 17M.1.hl.TZ1.2b: Find the smallest positive integer value of \(n\), such that \({w^n}\) is a real number.
• 17M.1.hl.TZ2.11c.ii: Hence find the cube roots of \(z\) in modulus-argument form.
• 17N.1.hl.TZ0.8: Determine the roots of the equation \({(z + 2{\text{i}})^3} = 216{\text{i}}\),...
• 18M.1.hl.TZ1.11a.i: Express w2 and w3 in modulus-argument form.
• 18M.1.hl.TZ1.11a.ii: Sketch on an Argand diagram the points represented by w0 , w1 , w2 and w3.
• 18M.1.hl.TZ1.11c: Let...

1.8

• 09M.1.hl.TZ2.7: Given that \({z_1} = 2\) and \({z_2} = 1 + \sqrt 3 {\text{i}}\) are roots of the cubic equation...
• 10N.2.hl.TZ0.6: Consider the polynomial \(p(x) = {x^4} + a{x^3} + b{x^2} + cx + d\), where a, b, c, d...
• 09N.2.hl.TZ0.7: (a) the other two roots; (b) \(a\) , \(b\) and \(c\) .
• 09M.2.hl.TZ1.2: (a)     Show that the complex number i is a root of the...
• 14M.2.hl.TZ1.1: One root of the equation \({x^2} + ax + b = 0\) is \(2 + 3{\text{i}}\) where...
• 14M.2.hl.TZ2.13e: Given that \(u\) and \(v\) are roots of the equation \({z^4} + b{z^3} + c{z^2} + dz + e = 0\),...
• 16M.2.hl.TZ1.9a: Write down the other two roots in terms of \(c\) and \(d\).
• 16M.2.hl.TZ1.9b: Find the value of \(c\) and the value of \(d\).
• 16M.1.hl.TZ2.12d: (i)     Given that \(\alpha  = w + {w^2} + {w^4}\), show that...
• 16M.1.hl.TZ2.12e: Using the values for \(b\) and \(c\) obtained in part (d)(ii), find the imaginary part of...
• 16M.1.hl.TZ1.12c: Find an expression in terms of \(\theta \) for...
• 16M.1.hl.TZ1.12d: (i)     Show that \(zz{\text{*}} = 1\). (ii)     Write down the binomial expansion of...
• 16M.1.hl.TZ1.12e: Hence solve \(4{\cos ^3}\theta  - 2{\cos ^2}\theta  - 3\cos \theta  + 1 = 0\) for...

1.9

• 12M.2.hl.TZ2.11a: Find the values of k for which the following system of equations has no solutions and the value...
• 12N.1.hl.TZ0.6a: If each of these equations defines a plane, show that, for any value of a , the planes do not...
• 12N.1.hl.TZ0.6b: Find the value of b for which the intersection of the planes is a straight line.
• 08N.2.hl.TZ0.5: (a)     Find the set of values of k for which the following system of equations has no...
• SPNone.1.hl.TZ0.7a: Show that this system does not have a unique solution for any value of \(\lambda \) .
• SPNone.1.hl.TZ0.7b: (i)     Determine the value of \(\lambda \) for which the system is consistent. (ii)     For...
• 13M.2.hl.TZ1.2: Find the value of k such that the following system of equations does not have a unique...
• 10M.2.hl.TZ1.2: The system of equations \[2x - y + 3z = 2\] \[3x + y + 2z = - 2\] \[ - x + 2y + az = b\] is...
• 10M.2.hl.TZ2.7a: Find the value of each of a, b and c.
• 10M.2.hl.TZ2.7b: The equations of three planes are      \(2x - 4y - 3z = 4\)      \( - x + 3y + 5z = - 2\)     ...
• 13M.2.hl.TZ2.2a: Express the system of equations in matrix form.
• 13M.2.hl.TZ2.2b: Find the solution to the system of equations.
• 11M.2.hl.TZ1.11a: Given that \(a = 0\) , show that the three planes intersect at a point.
• 11M.2.hl.TZ1.11b: Find the value of a such that the three planes do not meet at a point.
• 11M.2.hl.TZ1.11c: Given a such that the three planes do not meet at a point, find the value of \(k\) such that the...
• 14M.2.hl.TZ1.4: A system of equations is given below.      \(x + 2y - z = 2\)      \(2x + y + z = 1\)     ...
• 14M.1.hl.TZ2.3: (a)     Show that the following system of equations has an infinite number of solutions.     ...
• 15M.2.hl.TZ2.7a: Find conditions on \(\alpha \) and \(\beta \) for which (i)     the system has no...
• 15M.2.hl.TZ2.7b: In the case where the number of solutions is infinite, find the general solution of the system of...
• 15N.2.hl.TZ0.2: The three planes having Cartesian equations \(2x + 3y - z = 11,{\text{ }}x + 2y + z = 3\) and...
• 16M.1.hl.TZ2.1: The following system of equations represents three planes in space. \[x + 3y + z =  -...
• 17N.2.hl.TZ0.1: Boxes of mixed fruit are on sale at a local supermarket. Box A contains 2 bananas, 3 kiwifruit...

1.0

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