DP Mathematics SL Questionbank
Topic 1 - Algebra
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Description
The aim of this topic is to introduce students to some basic algebraic concepts and applications.
Directly related questions
- 08N.1.sl.TZ0.1b: Consider the infinite geometric sequence...
- 08N.2.sl.TZ0.2b: Find the term in \({x^3}\) in \((3x + 4){(x - 2)^4}\) .
- 12M.2.sl.TZ1.6b: Find k.
- 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
- 09N.1.sl.TZ0.7b: Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .
- 09M.1.sl.TZ2.4a: Find \({\log _2}32\) .
- SPNone.2.sl.TZ0.1a: Find the common difference.
- 11N.2.sl.TZ0.5a: Write down the number of terms in the expansion.
- 11N.2.sl.TZ0.8b(i) and (ii): Consider an arithmetic sequence with n terms, with first term (\( - 36\)) and eighth term...
- 11N.1.sl.TZ0.10a: Find the equation of L .
- 11M.2.sl.TZ1.3a: Find the value of the common difference.
- 14M.1.sl.TZ2.2a: \({\log _6}36\)
- 13N.1.sl.TZ0.9a(ii): Hence, show that \(m\) satisfies the equation \({m^2} + 3m - 40 = 0\).
- 13N.1.sl.TZ0.9c(ii): The sequence has a finite sum. Calculate the sum of the sequence.
- 15N.1.sl.TZ0.7: An arithmetic sequence has the first term \(\ln a\) and a common difference \(\ln 3\). The 13th...
- 15M.1.sl.TZ1.10c: Find the probability that Ann wins the game.
- 16N.1.sl.TZ0.9b: Show that the sum of the infinite sequence is \(4{\log _2}x\).
- 16N.1.sl.TZ0.9e: Given that \({S_{12}}\) is equal to half the sum of the infinite geometric sequence, find \(x\),...
- 16N.2.sl.TZ0.8b: During week 2, the students worked on a major project and they each spent an additional five...
- 17M.1.sl.TZ2.7: Solve \({\log _2}(2\sin x) + {\log _2}(\cos x) = - 1\), for...
- 17N.1.sl.TZ0.2b: Find the tenth term.
- 17N.1.sl.TZ0.2c: Find the sum of the first ten terms.
- 17N.1.sl.TZ0.10a: The following diagram shows [AB], with length 2 cm. The line is divided into an infinite number...
- 12M.2.sl.TZ1.6a: Find b.
- 10N.1.sl.TZ0.1b: Find \({u_6}\) .
- 10M.1.sl.TZ1.3b: Hence, find the term in \({x^2}\) in \({(2 + x)^4}\left( {1 + \frac{1}{{{x^2}}}} \right)\) .
- 09M.1.sl.TZ2.4b: Given that \({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right)\) can be written as \(px + qy\)...
- 13M.2.sl.TZ2.6: The constant term in the expansion of \({\left( {\frac{x}{a} + \frac{{{a^2}}}{x}} \right)^6}\)...
- 14M.1.sl.TZ1.2a: Find the common difference.
- 14M.1.sl.TZ1.2b: Find the first term.
- 14M.1.sl.TZ1.4a(iii): (iii) \({\log _{16}}4\).
- 14M.1.sl.TZ2.2c: \({\log _6}2 - {\log _6}12\)
- 14M.2.sl.TZ2.7: Consider the expansion of \({x^2}{\left( {3{x^2} + \frac{k}{x}} \right)^8}\). The constant term...
- 13N.1.sl.TZ0.9a(i): Write down an expression for the common ratio, \(r\).
- 13M.2.sl.TZ1.1c: Given that \({u_n} = 1502\) , find the value of \(n\) .
- 14N.1.sl.TZ0.2a: Find the common difference.
- 14N.1.sl.TZ0.2c: Find the sum of the first eight terms of the sequence.
- 14N.1.sl.TZ0.2b: Find the eighth term.
- 15N.2.sl.TZ0.4b: Find the value of \({S_6}\).
- 15M.2.sl.TZ1.3a: Write down the value of the common difference.
- 15M.2.sl.TZ2.6: Ramiro walks to work each morning. During the first minute he walks \(80\) metres. In each...
- 16M.2.sl.TZ1.5b: The equation of the line of best fit is \(\ln M = - 0.12t + 4.67\). Given that...
- 16M.2.sl.TZ1.7a: (i) Find the value of \(k\). (ii) Interpret the meaning of the value of \(k\).
- 16M.2.sl.TZ1.7b: Find the least number of whole years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).
- 16M.2.sl.TZ2.5b: Find the coefficient of \({x^8}\).
- 17M.1.sl.TZ2.1a: Find the common difference.
- 17N.2.sl.TZ0.6: In the expansion of \(a{x^3}{(2 + ax)^{11}}\), the coefficient of the term in \({x^5}\) is 11880....
- 08N.1.sl.TZ0.1a: Consider the infinite geometric sequence...
- 08M.2.sl.TZ2.1c: Find the exact sum of the infinite sequence.
- 12M.2.sl.TZ1.1b(i) and (ii): (i) Show that \({S_n} = 2{n^2} + 34n\) . (ii) Hence, write down the value of \({S_{14}}\) .
- 10M.2.sl.TZ2.4: Find the term in \({x^4}\) in the expansion of \({\left( {3{x^2} - \frac{2}{x}} \right)^5}\) .
- 11M.2.sl.TZ1.10c(i), (ii) and (iii): The function f can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\) . (i) ...
- 11M.2.sl.TZ1.10d: Write down the value of \({f^{ - 1}}(0)\) .
- 14M.1.sl.TZ1.4a(ii): (ii) \({\log _8}\frac{1}{8}\);
- 14M.1.sl.TZ1.10b: The process described above is repeated. Find \({A_6}\).
- 13M.1.sl.TZ2.3b: Find \({\log _3}\left( {\frac{p}{q}} \right)\) .
- 13M.2.sl.TZ1.1b: Find (i) \({u_{100}}\) ; (ii) \({S_{100}}\) .
- 14N.2.sl.TZ0.6: Consider the expansion of \({\left( {\frac{{{x^3}}}{2} + \frac{p}{x}} \right)^8}\). The constant...
- 14N.2.sl.TZ0.9a: (i) Find the common ratio. (ii) Hence or otherwise, find \({u_5}\).
- 15M.2.sl.TZ1.2a: Write down the number of terms in this expansion.
- 15M.2.sl.TZ1.2b: Find the term in \({x^3}\).
- 16M.1.sl.TZ1.4b: Find the total number of line segments in the first 200 figures.
- 16M.2.sl.TZ1.4b: Hence, find the term in \({x^7}\) in the expansion of \(5x{(x + 2)^9}\).
- 16N.2.sl.TZ0.8a: Find the mean number of hours spent browsing the Internet.
- 17M.2.sl.TZ1.6: Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative,...
- 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
- 18M.2.sl.TZ1.7b: Hence find the value of n such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).
- 18M.2.sl.TZ1.8b: Use the regression equation to estimate the value of y when x = 3.57.
- 18M.2.sl.TZ2.4a: Find the common ratio.
- 18M.2.sl.TZ2.4c: Find the least value of n for which Sn > 163.
- 12M.2.sl.TZ2.3b: The first term of a geometric sequence is 200 and the sum of the first four terms is 324.8. Find...
- 08M.2.sl.TZ2.2: Find the term \({x^3}\) in the expansion of \({\left( {\frac{2}{3}x - 3} \right)^8}\) .
- 09M.1.sl.TZ2.3a: Write down the value of \(n\).
- 09M.2.sl.TZ2.5b: (i) Find the value of \(\sum\limits_{r = 4}^{30} {{2^r}} \) . (ii) Explain why...
- 10N.2.sl.TZ0.3a: Write down the common difference.
- 10N.2.sl.TZ0.3b(i) and (ii): (i) Given that the nth term of this sequence is 115, find the value of n . (ii) For this...
- 11N.2.sl.TZ0.5b: Find the term in \({x^4}\) .
- 11M.2.sl.TZ1.3b: Find the value of n .
- 11M.2.sl.TZ1.10a: Show that \(f(x) = {\log _3}2x\) .
- 11M.1.sl.TZ2.5b: The graph of g is a transformation of the graph of f . Give a full geometric description of this...
- 11M.1.sl.TZ2.1c: Find \({S_{20}}\) .
- 14M.1.sl.TZ1.4a(i): (i) \({\log _3}27\);
- 14M.1.sl.TZ1.10c: Consider an initial square of side length \(k {\text{ cm}}\). The process described above is...
- 14M.2.sl.TZ1.2a: Write down the number of terms in this expansion.
- 13M.2.sl.TZ1.1a: Write down the value of \(d\) .
- 13M.2.sl.TZ1.3a: Write down the value of \(p\) , of \(q\) and of \(r\) .
- 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
- 15M.1.sl.TZ1.3a: Given that \({2^m} = 8\) and \({2^n} = 16\), write down the value of \(m\) and of \(n\).
- 16M.2.sl.TZ1.5a: State two words that describe the linear correlation between \(\ln M\) and \(t\).
- 16M.2.sl.TZ2.1a: Find the common difference.
- 16M.2.sl.TZ2.5a: Write down the number of terms of this expansion.
- 16N.1.sl.TZ0.3b: Hence or otherwise, find the term in \({x^3}\) in the expansion of \({(2x + 3)^5}\).
- 18M.2.sl.TZ1.8a: Find the value of a and of b.
- 12N.2.sl.TZ0.1c: The first three terms of an arithmetic sequence are \(5\) , \(6.7\) , \(8.4\) . Find the sum of...
- 12N.2.sl.TZ0.4: The third term in the expansion of \({(2x + p)^6}\) is \(60{x^4}\) . Find the possible values of p .
- 12M.2.sl.TZ1.1a(i) and (ii): (i) Write down the value of d . (ii) Find \({u_8}\) .
- 10N.1.sl.TZ0.1a: Write down the value of r .
- 10N.1.sl.TZ0.1c: Find the sum to infinity of this sequence.
- 10M.1.sl.TZ1.3a: Expand \({(2 + x)^4}\) and simplify your result.
- 10M.1.sl.TZ2.6: Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .
- 09M.2.sl.TZ1.1a: Find the common difference.
- 09M.2.sl.TZ1.6a: Find the value of \(r\) .
- 09M.1.sl.TZ2.3c: Write down an expression for the sixth term in the expansion.
- 09M.2.sl.TZ2.5a: Expand \(\sum\limits_{r = 4}^7 {{2^r}} \) as the sum of four terms.
- 10M.2.sl.TZ1.2a: Write down the common difference.
- 10M.2.sl.TZ1.2c: Find the sum of the sequence.
- SPNone.2.sl.TZ0.1b: Find the value of the 78th term.
- 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
- 11N.2.sl.TZ0.8a(i) and (ii): Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) . (i) ...
- 11M.2.sl.TZ1.10b: Find the value of \(f(0.5)\) and of \(f(4.5)\) .
- 13M.1.sl.TZ1.7a: Find the value of \({\log _2}40 - {\log _2}5\) .
- 13M.1.sl.TZ1.7b: Find the value of \({8^{{{\log }_2}5}}\) .
- 09N.1.sl.TZ0.7a: Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .
- 14M.1.sl.TZ1.2c: Find the sum of the first 20 terms of the sequence.
- 14M.1.sl.TZ2.7b: Find a general expression for \({u_n}\).
- 13N.1.sl.TZ0.9c(i): The sequence has a finite sum. State which value of \(r\) leads to this sum and justify your...
- 14N.1.sl.TZ0.4a: Write the expression \(3\ln 2 - \ln 4\) in the form \(\ln k\), where \(k \in \mathbb{Z}\).
- 15N.1.sl.TZ0.6: In the expansion of \({(3x + 1)^n}\), the coefficient of the term in \({x^2}\) is \(135n\), where...
- 16N.1.sl.TZ0.9d: Show that \({S_{12}} = 12{\log _2}x - 66\).
- 16N.2.sl.TZ0.8c: During week 3 each student spent 5% less time browsing the Internet than during week 1. For week...
- 17M.1.sl.TZ1.7a: Find the common ratio.
- 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
- 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
- 18M.1.sl.TZ2.7a: Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).
- 08N.2.sl.TZ0.2a: Expand \({(x - 2)^4}\) and simplify your result.
- 08M.1.sl.TZ1.3a: Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) ....
- 08M.1.sl.TZ1.3b: Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) ....
- 08M.2.sl.TZ2.1a: Find the common ratio.
- 08M.2.sl.TZ2.1b: Find the 10th term.
- 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
- 09N.2.sl.TZ0.1: In an arithmetic sequence, \({S_{40}} = 1900\) and \({u_{40}} = 106\) . Find the value of...
- 09M.2.sl.TZ1.6b: Find the smallest value of n for which \({S_n} > 40\) .
- 09M.1.sl.TZ2.3b: Write down a and b, in terms of p and/or q.
- 10M.2.sl.TZ2.2a: Find \({u_1}\).
- 13M.2.sl.TZ2.5: The sum of the first three terms of a geometric sequence is \(62.755\), and the sum of the...
- 14M.1.sl.TZ2.7a: Given that \({u_1} = 1 + k\), find \({u_2},{\text{ }}{u_3}\) and \({u_4}\).
- 13N.1.sl.TZ0.9b(ii): Find the possible values of \(r\).
- 13M.1.sl.TZ2.3c: Find \({\log _3}(9p)\) .
- 13M.1.sl.TZ2.3a: Find \({\log _3}{p^2}\) .
- 09M.2.sl.TZ1.10a: Expand \({(x + h)^3}\) .
- 15M.2.sl.TZ1.3b: Find the first term.
- 16M.1.sl.TZ1.4a: Given that Figure \(n\) contains 801 line segments, show that \(n = 200\).
- 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
- 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
- 16M.1.sl.TZ2.3a: \(\ln \left( {\frac{5}{3}} \right)\).
- 16M.1.sl.TZ2.4: Three consecutive terms of a geometric sequence are \(x - 3\), 6 and \(x + 2\). Find the...
- 16M.2.sl.TZ2.1b: Find the 30th term of the sequence.
- 16N.1.sl.TZ0.9c: Find \(d\), giving your answer as an integer.
- 17M.1.sl.TZ1.7b: Solve \(\sum\limits_{k = 1}^\infty {{2^{5 - k}}\ln x = 64} \).
- 17M.1.sl.TZ2.1b: Find the tenth term.
- 17M.2.sl.TZ2.5: Consider a geometric sequence where the first term is 768 and the second term is 576. Find the...
- 18M.2.sl.TZ1.7a: Given that xk + 1 = xk + a, find a.
- 18M.1.sl.TZ2.7b: Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).
- 18M.2.sl.TZ2.4b: Find the sum of the first 8 terms.
- 12M.1.sl.TZ2.7: Given that \({\left( {1 + \frac{2}{3}x} \right)^n}{(3 + nx)^2} = 9 + 84x + \ldots \) , find the...
- 12M.2.sl.TZ2.3a: The first term of a geometric sequence is 200 and the sum of the first four terms is 324.8. Find...
- 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
- 09M.2.sl.TZ1.1b: Find the value of the 78th term.
- 10M.2.sl.TZ2.2b(i) and (ii): (i) Given that \({u_n} = 516\) , find the value of n . (ii) For this value of n , find...
- 11N.2.sl.TZ0.8c: The sum of the infinite geometric sequence is equal to twice the sum of the arithmetic sequence....
- 11M.2.sl.TZ1.10e: The point A lies on the graph of f . At A, \(x = 4.5\) . On your diagram, sketch the graph of...
- 11M.1.sl.TZ2.1a: Find d .
- 15N.2.sl.TZ0.4c: Find the least value of \(n\) such that \({S_n} > 75\,000\).
- 15M.2.sl.TZ1.3c: Find the sum of the first 50 terms of the sequence.
- 15M.2.sl.TZ2.4: The third term in the expansion of \({(x + k)^8}\) is \(63{x^6}\). Find the possible values of...
- 16N.1.sl.TZ0.3a: Write down the values in the fifth row of Pascal’s triangle.
- 16M.2.sl.TZ1.4a: Find the term in \({x^6}\) in the expansion of \({(x + 2)^9}\).
- 16N.2.sl.TZ0.8d: (i) Find the number of students who spent between 25 and 30 hours browsing the...
- 17N.1.sl.TZ0.10b: The following diagram shows [CD], with length \(b{\text{ cm}}\), where \(b > 1\). Squares with...
- 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.
- 18M.2.sl.TZ1.8c: The relationship between x and y can be modelled using the formula y = kxn, where k ≠ 0 , n ≠ 0 ,...
- 18M.2.sl.TZ2.5: Consider the expansion of \({\left( {2x + \frac{k}{x}} \right)^9}\), where k > 0 . The...
- 12N.2.sl.TZ0.1a: The first three terms of an arithmetic sequence are 5 , 6.7 , 8.4 . Find the common difference.
- 12N.2.sl.TZ0.1b: The first three terms of an arithmetic sequence are 5 , 6.7 , 8.4 . Find the 28th term of the...
- 08M.2.sl.TZ2.10a(i) and (ii): (i) Find the number of taxis in the city at the end of 2005. (ii) Find the year in...
- 09M.1.sl.TZ1.6b: Solve the equation \({f^{ - 1}}(x) = \ln \frac{1}{x}\) .
- 10M.2.sl.TZ1.2b: Find the number of terms in the sequence.
- 11M.1.sl.TZ2.1b: Find \({u_{20}}\) .
- 11M.1.sl.TZ2.5a: Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .
- 14M.1.sl.TZ1.10a: The following table gives the values of \({x_n}\) and \({A_n}\), for...
- 14M.2.sl.TZ1.2b: Find the term containing \({x^3}\).
- 14M.1.sl.TZ2.2b: \({\log _6}4 + {\log _6}9\)
- 13N.1.sl.TZ0.9b(i): Find the two possible values of \(m\).
- 13M.2.sl.TZ1.3b: Find the coefficient of the term in \({x^5}\) .
- 14N.1.sl.TZ0.4b: Hence or otherwise, solve \(3\ln 2 - \ln 4 = - \ln x\).
- 15N.2.sl.TZ0.4a: Find the value of \(r\).
- 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
- 16M.2.sl.TZ1.6: In a geometric sequence, the fourth term is 8 times the first term. The sum of the first 10 terms...
- 16M.1.sl.TZ2.3b: \(\ln 45\).
- 16M.2.sl.TZ2.1c: Find the sum of the first 30 terms.
- 16N.1.sl.TZ0.9a: Find \(r\).
- 17M.1.sl.TZ2.1c: Find the sum of the first ten terms of the sequence.
- 17N.1.sl.TZ0.2a: Find the common difference.
Sub sections and their related questions
1.1
- 12N.2.sl.TZ0.1a: The first three terms of an arithmetic sequence are 5 , 6.7 , 8.4 . Find the common difference.
- 12N.2.sl.TZ0.1b: The first three terms of an arithmetic sequence are 5 , 6.7 , 8.4 . Find the 28th term of the...
- 12N.2.sl.TZ0.1c: The first three terms of an arithmetic sequence are \(5\) , \(6.7\) , \(8.4\) . Find the sum of...
- 12M.2.sl.TZ2.3a: The first term of a geometric sequence is 200 and the sum of the first four terms is 324.8. Find...
- 12M.2.sl.TZ2.3b: The first term of a geometric sequence is 200 and the sum of the first four terms is 324.8. Find...
- 08N.1.sl.TZ0.1a: Consider the infinite geometric sequence...
- 08N.1.sl.TZ0.1b: Consider the infinite geometric sequence...
- 08M.1.sl.TZ1.3a: Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) ....
- 08M.1.sl.TZ1.3b: Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) ....
- 08M.2.sl.TZ2.1a: Find the common ratio.
- 08M.2.sl.TZ2.1b: Find the 10th term.
- 08M.2.sl.TZ2.1c: Find the exact sum of the infinite sequence.
- 08M.2.sl.TZ2.10a(i) and (ii): (i) Find the number of taxis in the city at the end of 2005. (ii) Find the year in...
- 12M.2.sl.TZ1.1a(i) and (ii): (i) Write down the value of d . (ii) Find \({u_8}\) .
- 12M.2.sl.TZ1.1b(i) and (ii): (i) Show that \({S_n} = 2{n^2} + 34n\) . (ii) Hence, write down the value of \({S_{14}}\) .
- 10N.1.sl.TZ0.1a: Write down the value of r .
- 10N.1.sl.TZ0.1b: Find \({u_6}\) .
- 10N.1.sl.TZ0.1c: Find the sum to infinity of this sequence.
- 09N.2.sl.TZ0.1: In an arithmetic sequence, \({S_{40}} = 1900\) and \({u_{40}} = 106\) . Find the value of...
- 09M.2.sl.TZ1.1a: Find the common difference.
- 09M.2.sl.TZ1.1b: Find the value of the 78th term.
- 09M.2.sl.TZ1.6a: Find the value of \(r\) .
- 09M.2.sl.TZ1.6b: Find the smallest value of n for which \({S_n} > 40\) .
- 09M.2.sl.TZ2.5a: Expand \(\sum\limits_{r = 4}^7 {{2^r}} \) as the sum of four terms.
- 09M.2.sl.TZ2.5b: (i) Find the value of \(\sum\limits_{r = 4}^{30} {{2^r}} \) . (ii) Explain why...
- 10N.2.sl.TZ0.3a: Write down the common difference.
- 10N.2.sl.TZ0.3b(i) and (ii): (i) Given that the nth term of this sequence is 115, find the value of n . (ii) For this...
- 10M.2.sl.TZ1.2a: Write down the common difference.
- 10M.2.sl.TZ1.2b: Find the number of terms in the sequence.
- 10M.2.sl.TZ1.2c: Find the sum of the sequence.
- 10M.2.sl.TZ2.2a: Find \({u_1}\).
- 10M.2.sl.TZ2.2b(i) and (ii): (i) Given that \({u_n} = 516\) , find the value of n . (ii) For this value of n , find...
- SPNone.2.sl.TZ0.1a: Find the common difference.
- SPNone.2.sl.TZ0.1b: Find the value of the 78th term.
- 11N.2.sl.TZ0.8a(i) and (ii): Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) . (i) ...
- 11N.2.sl.TZ0.8b(i) and (ii): Consider an arithmetic sequence with n terms, with first term (\( - 36\)) and eighth term...
- 11N.2.sl.TZ0.8c: The sum of the infinite geometric sequence is equal to twice the sum of the arithmetic sequence....
- 11M.2.sl.TZ1.3a: Find the value of the common difference.
- 11M.2.sl.TZ1.3b: Find the value of n .
- 11M.1.sl.TZ2.1a: Find d .
- 11M.1.sl.TZ2.1b: Find \({u_{20}}\) .
- 11M.1.sl.TZ2.1c: Find \({S_{20}}\) .
- 13M.2.sl.TZ2.5: The sum of the first three terms of a geometric sequence is \(62.755\), and the sum of the...
- 14M.1.sl.TZ1.2a: Find the common difference.
- 14M.1.sl.TZ1.2b: Find the first term.
- 14M.1.sl.TZ1.2c: Find the sum of the first 20 terms of the sequence.
- 14M.1.sl.TZ1.10a: The following table gives the values of \({x_n}\) and \({A_n}\), for...
- 14M.1.sl.TZ1.10b: The process described above is repeated. Find \({A_6}\).
- 14M.1.sl.TZ1.10c: Consider an initial square of side length \(k {\text{ cm}}\). The process described above is...
- 14M.1.sl.TZ2.7a: Given that \({u_1} = 1 + k\), find \({u_2},{\text{ }}{u_3}\) and \({u_4}\).
- 14M.1.sl.TZ2.7b: Find a general expression for \({u_n}\).
- 13N.1.sl.TZ0.9a(i): Write down an expression for the common ratio, \(r\).
- 13N.1.sl.TZ0.9a(ii): Hence, show that \(m\) satisfies the equation \({m^2} + 3m - 40 = 0\).
- 13N.1.sl.TZ0.9b(i): Find the two possible values of \(m\).
- 13N.1.sl.TZ0.9b(ii): Find the possible values of \(r\).
- 13N.1.sl.TZ0.9c(i): The sequence has a finite sum. State which value of \(r\) leads to this sum and justify your...
- 13N.1.sl.TZ0.9c(ii): The sequence has a finite sum. Calculate the sum of the sequence.
- 13M.2.sl.TZ1.1a: Write down the value of \(d\) .
- 13M.2.sl.TZ1.1b: Find (i) \({u_{100}}\) ; (ii) \({S_{100}}\) .
- 13M.2.sl.TZ1.1c: Given that \({u_n} = 1502\) , find the value of \(n\) .
- 14N.1.sl.TZ0.2a: Find the common difference.
- 14N.1.sl.TZ0.2b: Find the eighth term.
- 14N.1.sl.TZ0.2c: Find the sum of the first eight terms of the sequence.
- 14N.2.sl.TZ0.9a: (i) Find the common ratio. (ii) Hence or otherwise, find \({u_5}\).
- 15M.1.sl.TZ1.10c: Find the probability that Ann wins the game.
- 15M.2.sl.TZ1.3a: Write down the value of the common difference.
- 15M.2.sl.TZ1.3b: Find the first term.
- 15M.2.sl.TZ1.3c: Find the sum of the first 50 terms of the sequence.
- 15M.2.sl.TZ2.6: Ramiro walks to work each morning. During the first minute he walks \(80\) metres. In each...
- 15N.1.sl.TZ0.7: An arithmetic sequence has the first term \(\ln a\) and a common difference \(\ln 3\). The 13th...
- 15N.2.sl.TZ0.4a: Find the value of \(r\).
- 15N.2.sl.TZ0.4b: Find the value of \({S_6}\).
- 15N.2.sl.TZ0.4c: Find the least value of \(n\) such that \({S_n} > 75\,000\).
- 16M.1.sl.TZ1.4a: Given that Figure \(n\) contains 801 line segments, show that \(n = 200\).
- 16M.1.sl.TZ1.4b: Find the total number of line segments in the first 200 figures.
- 16M.2.sl.TZ1.6: In a geometric sequence, the fourth term is 8 times the first term. The sum of the first 10 terms...
- 16M.2.sl.TZ1.7a: (i) Find the value of \(k\). (ii) Interpret the meaning of the value of \(k\).
- 16M.2.sl.TZ1.7b: Find the least number of whole years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).
- 16M.1.sl.TZ2.4: Three consecutive terms of a geometric sequence are \(x - 3\), 6 and \(x + 2\). Find the...
- 16M.2.sl.TZ2.1a: Find the common difference.
- 16M.2.sl.TZ2.1b: Find the 30th term of the sequence.
- 16M.2.sl.TZ2.1c: Find the sum of the first 30 terms.
- 16N.1.sl.TZ0.9a: Find \(r\).
- 16N.1.sl.TZ0.9b: Show that the sum of the infinite sequence is \(4{\log _2}x\).
- 16N.1.sl.TZ0.9c: Find \(d\), giving your answer as an integer.
- 16N.1.sl.TZ0.9d: Show that \({S_{12}} = 12{\log _2}x - 66\).
- 16N.2.sl.TZ0.8a: Find the mean number of hours spent browsing the Internet.
- 16N.2.sl.TZ0.8b: During week 2, the students worked on a major project and they each spent an additional five...
- 16N.2.sl.TZ0.8c: During week 3 each student spent 5% less time browsing the Internet than during week 1. For week...
- 16N.2.sl.TZ0.8d: (i) Find the number of students who spent between 25 and 30 hours browsing the...
- 17M.1.sl.TZ1.7a: Find the common ratio.
- 17M.1.sl.TZ1.7b: Solve \(\sum\limits_{k = 1}^\infty {{2^{5 - k}}\ln x = 64} \).
- 17M.1.sl.TZ2.1a: Find the common difference.
- 17M.1.sl.TZ2.1b: Find the tenth term.
- 17M.1.sl.TZ2.1c: Find the sum of the first ten terms of the sequence.
- 17M.2.sl.TZ2.5: Consider a geometric sequence where the first term is 768 and the second term is 576. Find the...
- 17N.1.sl.TZ0.2a: Find the common difference.
- 17N.1.sl.TZ0.2b: Find the tenth term.
- 17N.1.sl.TZ0.2c: Find the sum of the first ten terms.
- 17N.1.sl.TZ0.10a: The following diagram shows [AB], with length 2 cm. The line is divided into an infinite number...
- 17N.1.sl.TZ0.10b: The following diagram shows [CD], with length \(b{\text{ cm}}\), where \(b > 1\). Squares with...
- 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
- 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
- 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
- 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.
- 18M.2.sl.TZ1.7a: Given that xk + 1 = xk + a, find a.
- 18M.2.sl.TZ1.7b: Hence find the value of n such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).
- 18M.1.sl.TZ2.7a: Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).
- 18M.1.sl.TZ2.7b: Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).
- 18M.2.sl.TZ2.4a: Find the common ratio.
- 18M.2.sl.TZ2.4b: Find the sum of the first 8 terms.
- 18M.2.sl.TZ2.4c: Find the least value of n for which Sn > 163.
1.2
- 08N.2.sl.TZ0.2b: Find the term in \({x^3}\) in \((3x + 4){(x - 2)^4}\) .
- 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
- 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
- 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
- 10M.1.sl.TZ2.6: Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .
- 09N.1.sl.TZ0.7a: Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .
- 09N.1.sl.TZ0.7b: Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .
- 09M.1.sl.TZ1.6b: Solve the equation \({f^{ - 1}}(x) = \ln \frac{1}{x}\) .
- 09M.1.sl.TZ2.4a: Find \({\log _2}32\) .
- 09M.1.sl.TZ2.4b: Given that \({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right)\) can be written as \(px + qy\)...
- 11N.1.sl.TZ0.10a: Find the equation of L .
- 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
- 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
- 11M.2.sl.TZ1.10a: Show that \(f(x) = {\log _3}2x\) .
- 11M.2.sl.TZ1.10b: Find the value of \(f(0.5)\) and of \(f(4.5)\) .
- 11M.2.sl.TZ1.10c(i), (ii) and (iii): The function f can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\) . (i) ...
- 11M.2.sl.TZ1.10d: Write down the value of \({f^{ - 1}}(0)\) .
- 11M.2.sl.TZ1.10e: The point A lies on the graph of f . At A, \(x = 4.5\) . On your diagram, sketch the graph of...
- 11M.1.sl.TZ2.5a: Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .
- 11M.1.sl.TZ2.5b: The graph of g is a transformation of the graph of f . Give a full geometric description of this...
- 13M.1.sl.TZ1.7a: Find the value of \({\log _2}40 - {\log _2}5\) .
- 13M.1.sl.TZ1.7b: Find the value of \({8^{{{\log }_2}5}}\) .
- 14M.1.sl.TZ1.4a(i): (i) \({\log _3}27\);
- 14M.1.sl.TZ1.4a(ii): (ii) \({\log _8}\frac{1}{8}\);
- 14M.1.sl.TZ1.4a(iii): (iii) \({\log _{16}}4\).
- 14M.1.sl.TZ2.2a: \({\log _6}36\)
- 14M.1.sl.TZ2.2b: \({\log _6}4 + {\log _6}9\)
- 14M.1.sl.TZ2.2c: \({\log _6}2 - {\log _6}12\)
- 13M.1.sl.TZ2.3a: Find \({\log _3}{p^2}\) .
- 13M.1.sl.TZ2.3b: Find \({\log _3}\left( {\frac{p}{q}} \right)\) .
- 13M.1.sl.TZ2.3c: Find \({\log _3}(9p)\) .
- 14N.1.sl.TZ0.4a: Write the expression \(3\ln 2 - \ln 4\) in the form \(\ln k\), where \(k \in \mathbb{Z}\).
- 14N.1.sl.TZ0.4b: Hence or otherwise, solve \(3\ln 2 - \ln 4 = - \ln x\).
- 15M.1.sl.TZ1.3a: Given that \({2^m} = 8\) and \({2^n} = 16\), write down the value of \(m\) and of \(n\).
- 15N.1.sl.TZ0.7: An arithmetic sequence has the first term \(\ln a\) and a common difference \(\ln 3\). The 13th...
- 16M.1.sl.TZ1.9a: Find the \(x\)-coordinate of P.
- 16M.1.sl.TZ1.9b: Find \(f(x)\), expressing your answer as a single logarithm.
- 16M.1.sl.TZ1.9c: The graph of \(f\) is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\)....
- 16M.2.sl.TZ1.5a: State two words that describe the linear correlation between \(\ln M\) and \(t\).
- 16M.2.sl.TZ1.5b: The equation of the line of best fit is \(\ln M = - 0.12t + 4.67\). Given that...
- 16M.1.sl.TZ2.3a: \(\ln \left( {\frac{5}{3}} \right)\).
- 16M.1.sl.TZ2.3b: \(\ln 45\).
- 16N.1.sl.TZ0.9a: Find \(r\).
- 16N.1.sl.TZ0.9c: Find \(d\), giving your answer as an integer.
- 16N.1.sl.TZ0.9e: Given that \({S_{12}}\) is equal to half the sum of the infinite geometric sequence, find \(x\),...
- 17M.1.sl.TZ1.7a: Find the common ratio.
- 17M.1.sl.TZ1.7b: Solve \(\sum\limits_{k = 1}^\infty {{2^{5 - k}}\ln x = 64} \).
- 17M.1.sl.TZ2.7: Solve \({\log _2}(2\sin x) + {\log _2}(\cos x) = - 1\), for...
- 18M.2.sl.TZ1.8a: Find the value of a and of b.
- 18M.2.sl.TZ1.8b: Use the regression equation to estimate the value of y when x = 3.57.
- 18M.2.sl.TZ1.8c: The relationship between x and y can be modelled using the formula y = kxn, where k ≠ 0 , n ≠ 0 ,...
- 18M.1.sl.TZ2.7a: Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).
- 18M.1.sl.TZ2.7b: Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).
1.3
- 12N.2.sl.TZ0.4: The third term in the expansion of \({(2x + p)^6}\) is \(60{x^4}\) . Find the possible values of p .
- 12M.1.sl.TZ2.7: Given that \({\left( {1 + \frac{2}{3}x} \right)^n}{(3 + nx)^2} = 9 + 84x + \ldots \) , find the...
- 08N.2.sl.TZ0.2a: Expand \({(x - 2)^4}\) and simplify your result.
- 08M.2.sl.TZ2.2: Find the term \({x^3}\) in the expansion of \({\left( {\frac{2}{3}x - 3} \right)^8}\) .
- 12M.2.sl.TZ1.6a: Find b.
- 12M.2.sl.TZ1.6b: Find k.
- 10M.1.sl.TZ1.3a: Expand \({(2 + x)^4}\) and simplify your result.
- 10M.1.sl.TZ1.3b: Hence, find the term in \({x^2}\) in \({(2 + x)^4}\left( {1 + \frac{1}{{{x^2}}}} \right)\) .
- 09M.2.sl.TZ1.10a: Expand \({(x + h)^3}\) .
- 09M.1.sl.TZ2.3a: Write down the value of \(n\).
- 09M.1.sl.TZ2.3b: Write down a and b, in terms of p and/or q.
- 09M.1.sl.TZ2.3c: Write down an expression for the sixth term in the expansion.
- 10M.2.sl.TZ2.4: Find the term in \({x^4}\) in the expansion of \({\left( {3{x^2} - \frac{2}{x}} \right)^5}\) .
- 11N.2.sl.TZ0.5a: Write down the number of terms in the expansion.
- 11N.2.sl.TZ0.5b: Find the term in \({x^4}\) .
- 13M.2.sl.TZ2.6: The constant term in the expansion of \({\left( {\frac{x}{a} + \frac{{{a^2}}}{x}} \right)^6}\)...
- 14M.2.sl.TZ1.2a: Write down the number of terms in this expansion.
- 14M.2.sl.TZ1.2b: Find the term containing \({x^3}\).
- 14M.2.sl.TZ2.7: Consider the expansion of \({x^2}{\left( {3{x^2} + \frac{k}{x}} \right)^8}\). The constant term...
- 13M.2.sl.TZ1.3a: Write down the value of \(p\) , of \(q\) and of \(r\) .
- 13M.2.sl.TZ1.3b: Find the coefficient of the term in \({x^5}\) .
- 14N.2.sl.TZ0.6: Consider the expansion of \({\left( {\frac{{{x^3}}}{2} + \frac{p}{x}} \right)^8}\). The constant...
- 15M.2.sl.TZ1.2a: Write down the number of terms in this expansion.
- 15M.2.sl.TZ1.2b: Find the term in \({x^3}\).
- 15M.2.sl.TZ2.4: The third term in the expansion of \({(x + k)^8}\) is \(63{x^6}\). Find the possible values of...
- 15N.1.sl.TZ0.6: In the expansion of \({(3x + 1)^n}\), the coefficient of the term in \({x^2}\) is \(135n\), where...
- 16M.2.sl.TZ1.4a: Find the term in \({x^6}\) in the expansion of \({(x + 2)^9}\).
- 16M.2.sl.TZ1.4b: Hence, find the term in \({x^7}\) in the expansion of \(5x{(x + 2)^9}\).
- 16M.2.sl.TZ2.5a: Write down the number of terms of this expansion.
- 16M.2.sl.TZ2.5b: Find the coefficient of \({x^8}\).
- 16N.1.sl.TZ0.3a: Write down the values in the fifth row of Pascal’s triangle.
- 16N.1.sl.TZ0.3b: Hence or otherwise, find the term in \({x^3}\) in the expansion of \({(2x + 3)^5}\).
- 17M.2.sl.TZ1.6: Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative,...
- 17N.2.sl.TZ0.6: In the expansion of \(a{x^3}{(2 + ax)^{11}}\), the coefficient of the term in \({x^5}\) is 11880....
- 18M.2.sl.TZ2.5: Consider the expansion of \({\left( {2x + \frac{k}{x}} \right)^9}\), where k > 0 . The...
