DP Mathematics SL Questionbank

• Syllabus

Topic 1 - Algebra

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...