DP Mathematics HL Questionbank

• Syllabus

1.1

Sub sections and their related questions

Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.

• 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...
• 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.

Sigma notation.

• 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.TZ2.8a: 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....
• 11M.3ca.hl.TZ0.4c: Hence determine the exact value of \(\int_0^\infty  {{{\text{e}}^{ - x}}|\sin x|{\text{d}}x} \) .
• 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...

Applications.

• 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....
• 11M.3ca.hl.TZ0.4c: Hence determine the exact value of \(\int_0^\infty  {{{\text{e}}^{ - x}}|\sin x|{\text{d}}x} \) .