DP Mathematics HL Questionbank

• Syllabus

Topic 2 - Core: Functions and equations

Sub sections and their related questions

2.1

• 12M.1.hl.TZ2.11b: Find an expression for the composite function $f \circ g(x)$ in the form...
• 12M.1.hl.TZ2.11c: (i)     Find an expression for the inverse function ${f^{ - 1}}(x)$ . (ii)     State the...
• 12N.1.hl.TZ0.12a: Find an expression for $(f \circ f)(x)$ .  
• 12N.1.hl.TZ0.12c: Show that ${F_{ - n}}(x)$ is an expression for the inverse of ${F_n}$ .
• 12N.1.hl.TZ0.12d: (i)     State ${F_n}(0){\text{ and }}{F_n}(1)$ . (ii)     Show that ${F_n}(x) < x$ ,...
• 08M.1.hl.TZ1.8: The functions f and g are defined as: \[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant...
• 08M.1.hl.TZ2.4: Let $f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1$. If...
• 11M.1.hl.TZ2.8: A function is defined by...
• 09M.1.hl.TZ2.11: A function is defined as $f(x) = k\sqrt x$, with $k > 0$ and $x \geqslant 0$ . (a)    ...
• 09N.1.hl.TZ0.4: Consider the function f , where $f(x) = \arcsin (\ln x)$. (a)     Find the domain of f . (b)...
• SPNone.1.hl.TZ0.5a: Determine whether f is even, odd or neither even nor odd.
• SPNone.1.hl.TZ0.13c: Obtain expressions for the inverse function ${f^{ - 1}}$ and state their domains.
• 13M.1.hl.TZ1.12d: Find the range of f.
• 13M.2.hl.TZ1.13d: (i)     With f and g as defined in parts (a) and (b), solve $g \circ f(x) = 2$. (ii)     Let...
• 10M.1.hl.TZ1.2: Shown below are the graphs of $y = f(x)$ and $y = g(x)$.     If $(f \circ g)(x) = 3$,...
• 10M.1.hl.TZ2.10: A function f is defined by $f(x) = \frac{{2x - 3}}{{x - 1}},{\text{ }}x \ne 1$. (a)     Find...
• 10N.1.hl.TZ0.9: Consider the function $f:x \to \sqrt {\frac{\pi }{4} - \arccos x}$. (a)     Find the largest...
• 13M.1.hl.TZ2.12d: (i)     Find an expression for ${f^{ - 1}}(x)$. (ii)     Sketch the graph of $y = f(x)$,...
• 11N.1.hl.TZ0.9b: Hence determine the range of the function $f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}$.
• 11N.1.hl.TZ0.9c: Explain why f has no inverse.
• 11N.2.hl.TZ0.8a: find ${f^{ - 1}}(x)$, stating its domain;
• 11M.1.hl.TZ1.8b: Find the coordinates of the point where the graph of $y = f(x)$ and the graph of...
• 11M.1.hl.TZ1.8a: (i)     Find $\left( {g \circ f} \right)\left( x \right)$ and write down the domain of the...
• 11M.1.hl.TZ1.10a: Find the largest possible domain of the function $g$ .
• 09N.2.hl.TZ0.9: (a)     Given that the domain of $g$ is $x \geqslant a$ , find the least value of $a$ such...
• 14M.2.hl.TZ1.12: Let $f(x) = \left| x \right| - 1$. (a)     The graph of $y = g(x)$ is drawn below.      ...
• 14M.1.hl.TZ2.14b: Find an expression for the composite function $h \circ g(x)$ and state its domain.
• 14M.1.hl.TZ2.14d: Nigel states that $f$ is an odd function and Tom argues that $f$ is an even function. (i)  ...
• 14M.2.hl.TZ2.7b: Find the inverse function ${f^{ - 1}}$, stating its domain.
• 14N.1.hl.TZ0.11a: (i)     Find ${f^{ - 1}}(x)$. (ii)     State the domain of ${f^{ - 1}}$.
• 15M.1.hl.TZ1.5a: Given that $f$ is an even function, show that $b = 0$.
• 15M.1.hl.TZ1.5b: Given that $g$ is an odd function, find the value of $r$.
• 15M.1.hl.TZ1.5c: The function $h$ is both odd and even, with domain $\mathbb{R}$. Find $h(x)$.
• 15M.1.hl.TZ1.6a: Find an expression for ${f^{ - 1}}(x)$.
• 15M.1.hl.TZ1.6b: Given that $f(x)$ can be written in the form $f(x) = A + \frac{B}{{2x - 1}}$, find the values...
• 15M.1.hl.TZ1.9a: Show that $g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4$.
• 15M.1.hl.TZ1.9b: Find the range of $g \circ f$.
• 15M.1.hl.TZ1.9c: Given that $g \circ f\left( {\frac{{3\pi }}{{20}}} \right) = 7$, find the next value of $x$,...
• 15M.1.hl.TZ2.10b: Find an expression for ${f^{ - 1}}(x)$.
• 15M.1.hl.TZ2.10c: Find all values of $x$ for which $f(x) = {f^{ - 1}}(x)$.
• 15M.1.hl.TZ2.11a: Find an expression for $g \circ f(x)$, stating its domain.
• 15M.1.hl.TZ2.11b: Hence show that $g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}$.
• 15M.1.hl.TZ2.13a: Show that $\frac{1}{{\sqrt n  + \sqrt {n + 1} }} = \sqrt {n + 1}  - \sqrt n$ where...
• 15M.1.hl.TZ2.13b: Hence show that $\sqrt 2  - 1 < \frac{1}{{\sqrt 2 }}$.
• 15M.2.hl.TZ1.6: A function $f$ is defined by $f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}$....
• 15N.1.hl.TZ0.12a: Show that $f$ is an odd function.
• 15N.1.hl.TZ0.12d: Find the range of $f$.
• 15N.2.hl.TZ0.12a: The functions $u$ and $v$ are defined as $u(x) = x - 3,{\text{ }}v(x) = 2x$ where...
• 15N.2.hl.TZ0.12b: (i)     Explain why $f$ does not have an inverse. (ii)     The domain of $f$ is restricted...
• 15N.2.hl.TZ0.12c: Consider the function defined by $h(x) = \frac{{2x - 5}}{{x + d}}$, $x \ne  - d$ and...
• 16M.2.hl.TZ1.2a: Express ${x^2} + 4x - 2$ in the form ${(x + a)^2} + b$ where $a,{\text{ }}b \in \mathbb{Z}$.
• 16M.2.hl.TZ1.2b: If $f(x) = x + 2$ and $(g \circ f)(x) = {x^2} + 4x - 2$ write down $g(x)$.
• 16M.2.hl.TZ1.5a: Prove that $f$ is an even function.
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.2.hl.TZ1.11c.i: Write down the largest value of $a$ for which $f$ has an inverse. Give your answer correct to...
• 16M.2.hl.TZ1.11c.ii: For this value of a sketch the graphs of $y = f(x)$ and $y = {f^{ - 1}}(x)$ on the same set...
• 16M.2.hl.TZ1.11c.iii: Solve ${f^{ - 1}}(x) = 1$.
• 16M.2.hl.TZ1.11d.i: Find an expression for ${g^{ - 1}}(x)$, stating the domain.
• 16M.2.hl.TZ1.11d.ii: Solve $({f^{ - 1}} \circ g)(x) < 1$.
• 16M.2.hl.TZ2.5: The function $f$ is defined as...
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.
• 16N.2.hl.TZ0.10d: Use your answers from parts (b) and (c) to justify that $f$ has an inverse and state its domain.
• 16N.2.hl.TZ0.10e: Find an expression for ${f^{ - 1}}(x)$.
• 17M.1.hl.TZ2.2a: Write down the range of $f$.
• 17M.1.hl.TZ2.2b: Find an expression for ${f^{ - 1}}(x)$.
• 17M.1.hl.TZ2.2c: Write down the domain and range of ${f^{ - 1}}$.
• 17M.2.hl.TZ1.12a: Find the largest possible domain $D$ for $f$ to be a function.
• 17M.2.hl.TZ1.12c: Explain why $f$ is an even function.
• 17M.2.hl.TZ1.12d: Explain why the inverse function ${f^{ - 1}}$ does not exist.
• 17M.2.hl.TZ1.12e: Find the inverse function ${g^{ - 1}}$ and state its domain.
• 17N.1.hl.TZ0.11a: Determine whether ${f_n}$ is an odd or even function, justifying your answer.
• 18M.1.hl.TZ2.10a: Find the inverse function ${f^{ - 1}}$, stating its domain.
• 18M.1.hl.TZ2.10c: The function $h$ is defined by $h\left( x \right) = \sqrt x$, for $x$ ≥ 0. State the...
• 18M.2.hl.TZ2.10a.i: Sketch the graph of $y = f\left( x \right)$...
• 18M.2.hl.TZ2.10a.ii: With reference to your graph, explain why $f$ is a function on the given domain.
• 18M.2.hl.TZ2.10a.iii: Explain why $f$ has no inverse on the given domain.
• 18M.2.hl.TZ2.10a.iv: Explain why $f$ is not a function for...

2.2

• 12M.1.hl.TZ1.2a: Draw the graph of y = f (x) on the blank grid below.
• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.2.hl.TZ1.11c: Find the coordinates of the point, on $y = f(x)$ , where the gradient of the graph is 3.
• 12M.1.hl.TZ2.7a: On the axes below, sketch the graph of $y = \frac{1}{{f(x)}}$ , clearly showing the coordinates...
• 12M.1.hl.TZ2.11a: State the range of f and of g .
• 12M.2.hl.TZ2.6a: Sketch the curve...
• 12M.2.hl.TZ2.12b: Sketch the graph of v against t , clearly showing the coordinates of any intercepts, and the...
• 12N.1.hl.TZ0.3a: Using the information shown in the diagram, find the values of a , b and c .
• 08M.2.hl.TZ1.2: (a)     Sketch the curve...
• 08M.2.hl.TZ2.8: The graph of $y = f(x){\text{ for }} - 2 \leqslant x \leqslant 8$ is shown.     On the set...
• 08N.2.hl.TZ0.6: (a)     Sketch the curve $y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1$ ,...
• 11M.1.hl.TZ2.3a: Sketch the graph of the function. You are not required to find the coordinates of the maximum.
• 11M.1.hl.TZ2.5a: Sketch the graph of $y = \frac{1}{{f(x)}}$.
• 11M.1.hl.TZ2.5b: Sketch the graph of  $y = x{\text{ }}f(x)$ .
• 11M.2.hl.TZ2.5: Sketch the graph of $f(x) = x + \frac{{8x}}{{{x^2} - 9}}$. Clearly mark the coordinates of the...
• 09M.1.hl.TZ2.11: A function is defined as $f(x) = k\sqrt x$, with $k > 0$ and $x \geqslant 0$ . (a)    ...
• SPNone.1.hl.TZ0.9b: Sketch the graph of f , showing clearly the coordinates of the maximum and minimum.
• SPNone.1.hl.TZ0.9c: Hence show that ${{\text{e}}^\pi } > {\pi ^{\text{e}}}$ .
• SPNone.2.hl.TZ0.6: The function f is of the form $f(x) = \frac{{x + a}}{{bx + c}}$, $x \ne - \frac{c}{b}$. Given...
• 13M.1.hl.TZ1.12c: Sketch the graph of $y = f(x)$.
• 13M.2.hl.TZ1.12a: Sketch the graph of ${v_A} = {t^3} - 5{t^2} + 6t$ for $t \geqslant 0$, with ${v_A}$ on the...
• 13M.2.hl.TZ1.12b: Write down the times for which the velocity of the particle is increasing.
• 13M.2.hl.TZ1.12c: Write down the times for which the magnitude of the velocity of the particle is increasing.
• 10M.1.hl.TZ1.5: The graph of $y = \frac{{a + x}}{{b + cx}}$ is drawn below.     (a)     Find the value of...
• 10N.2.hl.TZ0.8: The diagram shows the graphs of a linear function f and a quadratic function g.     On the...
• 13M.1.hl.TZ2.12e: (i)     On a different diagram, sketch the graph of $y = f(|x|)$ where $x \in D$. (ii)    ...
• 13M.2.hl.TZ2.13c: Sketch the graph of $\theta$, for $0 \leqslant x \leqslant 20$.
• 11N.1.hl.TZ0.4a: sketch the graph of $f$;  
• 11N.2.hl.TZ0.1a: Sketch the graph, clearly labelling the x and y intercepts with their values.
• 11M.1.hl.TZ1.10b: On the axes below, sketch the graph of $y = g(x)$ . On the graph, indicate any asymptotes and...
• 11M.1.hl.TZ1.12d: Now consider the functions $g(x) = \frac{{\ln \left| x \right|}}{x}$ and...
• 14M.2.hl.TZ1.10a: Find the equations of the horizontal and vertical asymptotes of the curve $y = f(x)$.
• 14M.2.hl.TZ1.12: Let $f(x) = \left| x \right| - 1$. (a)     The graph of $y = g(x)$ is drawn below.      ...
• 14M.1.hl.TZ2.5a: Sketch the graph of $y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|$ for...
• 14M.2.hl.TZ2.7a: (i)     Sketch the graph of $y = f(x)$, clearly indicating any asymptotes and axes...
• 14M.2.hl.TZ2.14a: Sketch the graph of $y = v(t)$. Indicate clearly the local maximum and write down its coordinates.
• 13N.1.hl.TZ0.3b: State the range of ${f^{ - 1}}$.
• 13N.2.hl.TZ0.3a: Sketch the graph of $y = f(x)$, stating the coordinates of any maximum and minimum points and...
• 14N.1.hl.TZ0.1b: State the equations of the asymptotes of the graph of $g$.
• 14N.3sp.hl.TZ0.1a: Sketch the graph of $y = f(x)$.
• 15M.1.hl.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of $y(x)$. Justify...
• 15M.1.hl.TZ1.11d: Find the coordinates of any points of inflexion on the graph of $y(x)$. Justify whether any...
• 15M.1.hl.TZ2.10a: Sketch the graph of $y = f(x)$, indicating clearly any asymptotes and points of intersection...
• 15M.2.hl.TZ1.11a: Sketch the graph $y = f(x)$.
• 15M.2.hl.TZ2.3a: Sketch the graph of $y = {(x - 5)^2} - 2\left| {x - 5} \right| - 9,{\text{ for }}0 \le x \le 10$.
• 15M.2.hl.TZ2.12b: Sketch a displacement/time graph for the particle, $0 \le t \le 5$, showing clearly where the...
• 15N.1.hl.TZ0.12e: Sketch the graph of $y = f(x)$ indicating clearly the coordinates of the $x$-intercepts and...
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.1.hl.TZ2.2: The function $f$  is defined as...
• 16M.1.hl.TZ1.7a: Sketch on the same axes the curve $y = \left| {\frac{7}{{x - 4}}} \right|$ and the line...
• 16M.1.hl.TZ1.13a: Given that the curve passes through the point $(a,{\text{ }}0)$, state the value of $a$.
• 16M.1.hl.TZ1.7b: Find the exact solutions to the equation $x + 2 = \left| {\frac{7}{{x - 4}}} \right|$.
• 16N.1.hl.TZ0.3a: state the value of $a$ and the value of $c$;
• 16N.1.hl.TZ0.3b: find the value of $b$.
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.
• 16N.2.hl.TZ0.5a: Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of...
• 17M.1.hl.TZ1.6a: Sketch the graphs on the same set of axes.
• 17M.1.hl.TZ1.6b: Given that the graphs enclose a region of area 18 square units, find the value of b.
• 17M.1.hl.TZ1.11a.i: Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$.
• 17M.1.hl.TZ1.11a.ii: Factorize ${x^2} + 3x + 2$.
• 17M.1.hl.TZ1.11b: Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates...
• 17M.1.hl.TZ1.11c: Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$.
• 17M.1.hl.TZ1.11e: Sketch the graph of $y = f\left( {\left| x \right|} \right)$.
• 17M.1.hl.TZ2.9a.i: Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch...
• 17M.1.hl.TZ2.9a.ii: Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch...
• 17M.1.hl.TZ2.9a.iii: Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch...
• 17M.2.hl.TZ1.5a: Find the probability that the carpet laid in the first room has fewer than three faults.
• 17M.2.hl.TZ1.12b: Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates...
• 17N.1.hl.TZ0.6a: Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$, showing clearly any asymptotes and stating...
• 17N.2.hl.TZ0.10b: Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
• 18M.1.hl.TZ1.9c: Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.
• 18M.1.hl.TZ2.2a: Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following...
• 18M.1.hl.TZ2.10b.i: Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
• 18M.1.hl.TZ2.10b.ii: Sketch the graph of $y = g\left( x \right)$. State the equations of any asymptotes and...
• 18M.2.hl.TZ2.10b: Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$.
• 18M.2.hl.TZ2.10c: Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts...

2.3

• 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...
• 12M.1.hl.TZ2.10b: Sketch the graph of f .
• 12N.1.hl.TZ0.3b: If g(x) = 3f(x − 2) , (i)     state the coordinates of the points where the graph of g...
• 11M.1.hl.TZ2.1b: The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis....
• 11M.1.hl.TZ2.13c: The increasing function f satisfies $f(0) = 0$ and $f(a) = b$ , where $a > 0$ and...
• 13M.1.hl.TZ1.12b: The graph of $y = {x^2}$ is transformed onto the graph of $y = 4{x^2} - 4x + 5$. Describe a...
• 10M.1.hl.TZ2.2: (a)     Express the quadratic $3{x^2} - 6x + 5$ in the form $a{(x + b)^2} + c$, where a, b, c...
• 13M.1.hl.TZ2.12d: (i)     Find an expression for ${f^{ - 1}}(x)$. (ii)     Sketch the graph of $y = f(x)$,...
• 12M.1.hl.TZ1.4: The graph below shows $y = f(x)$ , where $f(x) = x + \ln x$ .   (a) On the graph below,...
• 09M.2.hl.TZ2.4a: The graph of $y = \ln (x)$ is transformed into the graph of $y = \ln \left( {2x + 1} \right)$...
• 14M.1.hl.TZ2.8b: The graph of the function $g$ is obtained by applying the following transformations to the...
• 13N.1.hl.TZ0.3a: Sketch the graph of $y = {f^{ - 1}}(x)$ on the same axes.
• 13N.1.hl.TZ0.10e: Sketch the graphs of $y = f(x)$ and $y = g(x)$ on the same axes, showing clearly the points...
• 13N.1.hl.TZ0.10d: The graph of the function $g$ is obtained from the graph of $f$ by stretching it in the...
• 14N.1.hl.TZ0.1a: Find an expression for $g(x)$.
• 15M.1.hl.TZ1.9d: The graph of $y = g \circ f(x)$ can be obtained by applying four transformations to the graph...
• 15M.2.hl.TZ2.6a: Find the value of $a$ and the value of $b$.
• 15N.2.hl.TZ0.12b: (i)     Explain why $f$ does not have an inverse. (ii)     The domain of $f$ is restricted...
• 17M.2.hl.TZ1.5b: Find the probability that exactly seven rooms will have fewer than three faults in the carpet.

2.4

• 08N.1.hl.TZ0.2: Write $\ln ({x^2} - 1) - 2\ln (x + 1) + \ln ({x^2} + x)$ as a single logarithm, in its simplest...
• 09M.1.hl.TZ1.3: Let $g(x) = {\log _5}\left| {2{{\log }_3}x} \right|$ . Find the product of the zeros of g .
• 10M.2.hl.TZ2.4: (a)     Find the solution of the...
• 13M.1.hl.TZ2.12a: Express $f(x)$ in the form $A + \frac{B}{{x + 2}}$, where $A$ and $B \in \mathbb{Z}$.
• 13M.1.hl.TZ2.12c: State the range of f.
• 13M.1.hl.TZ2.12d: (i)     Find an expression for ${f^{ - 1}}(x)$. (ii)     Sketch the graph of $y = f(x)$,...
• 13N.1.hl.TZ0.3c: Given that $f(x) = \ln (ax + b),{\text{ }}x > 1$, find the value of $a$ and the value of...
• 14N.1.hl.TZ0.11b: The function $g$ is defined as $g(x) = \ln x,{\text{ }}x \in {\mathbb{R}^ + }$. The graph of...
• 15M.1.hl.TZ2.9a: State the set of values of $a$ for which the function $x \mapsto {\log _a}x$ exists, for all...
• 15M.1.hl.TZ2.9b: Given that ${\log _x}y = 4{\log _y}x$, find all the possible expressions of $y$ as a function...

2.5

• 10M.1.hl.TZ1.1: Given that $A{x^3} + B{x^2} + x + 6$ is exactly divisible by $(x +1)(x − 2)$, find the value...
• 12M.1.hl.TZ2.1: The same remainder is found when $2{x^3} + k{x^2} + 6x + 32$ and...
• 12M.1.hl.TZ2.12B.a: Explain why, of the four roots of the equation $f(x) = 0$ , two are real and two are complex.
• 12M.1.hl.TZ2.12B.b: The curve passes through the point $( - 1,\, - 18)$ . Find $f(x)$ in the...
• 08M.1.hl.TZ2.2: The polynomial $P(x) = {x^3} + a{x^2} + bx + 2$ is divisible by (x +1) and by (x − 2) . Find...
• 08N.1.hl.TZ0.1: When $f(x) = {x^4} + 3{x^3} + p{x^2} - 2x + q$ is divided by (x − 2) the remainder is 15, and...
• 11M.1.hl.TZ2.12a: Factorize ${z^3} + 1$ into a linear and quadratic factor.
• 09M.1.hl.TZ2.1: When the function $q(x) = {x^3} + k{x^2} - 7x + 3$ is divided by (x + 1) the remainder is seven...
• 09N.1.hl.TZ0.1: When $3{x^5} - ax + b$ is divided by x −1 and x +1 the remainders are equal. Given that a ,...
• SPNone.2.hl.TZ0.1: Given that (x − 2) is a factor of $f(x) = {x^3} + a{x^2} + bx - 4$ and that division $f(x)$...
• 13M.1.hl.TZ1.12a: Express $4{x^2} - 4x + 5$ in the form $a{(x - h)^2} + k$ where a, h, $k \in \mathbb{Q}$.
• 13M.2.hl.TZ1.6: A polynomial $p(x)$ with real coefficients is of degree five. The equation $p(x) = 0$ has a...
• 13M.1.hl.TZ2.13b: (i)     State the solutions of the equation ${z^7} = 1$ for $z \in \mathbb{C}$, giving them...
• 11M.2.hl.TZ1.4a: Find the value of $a$ .
• 14M.1.hl.TZ1.1: When the polynomial $3{x^3} + ax + b$ is divided by $(x - 2)$, the remainder is 2, and when...
• 14M.2.hl.TZ1.1: One root of the equation ${x^2} + ax + b = 0$ is $2 + 3{\text{i}}$ where...
• 13N.1.hl.TZ0.1: The cubic polynomial $3{x^3} + p{x^2} + qx - 2$ has a factor $(x + 2)$ and leaves a remainder...
• 14N.2.hl.TZ0.6: Consider $p(x) = 3{x^3} + ax + 5a,\;\;\;a \in \mathbb{R}$. The polynomial $p(x)$ leaves a...
• 16M.2.hl.TZ1.8: When ${x^2} + 4x - b$ is divided by $x - a$ the remainder is 2. Given that...
• 17M.1.hl.TZ1.12b: Show that $(z - 1)$ is a factor of $P(z)$.
• 17M.1.hl.TZ1.12e.i: Show that the graph of $y = q(x)$ is concave up for $x > 1$.
• 17M.1.hl.TZ1.12e.ii: Sketch the graph of $y = q(x)$ showing clearly any intercepts with the axes.
• 17M.2.hl.TZ2.11a: Given that ${x^2} - 1$ is a factor of $f(x)$ find the value of $a$ and the value of $b$.
• 17M.2.hl.TZ2.11b: Factorize $f(x)$ into a product of linear factors.
• 17M.2.hl.TZ2.11c: Sketch the graph of $y = f(x)$, labelling the maximum and minimum points and the $x$ and...
• 17M.2.hl.TZ2.11d: Using your graph state the range of values of $c$ for which $f(x) = c$ has exactly two...
• 17N.1.hl.TZ0.3a: Given that $q(x)$ has a factor $(x - 4)$, find the value of $k$.
• 17N.1.hl.TZ0.3b: Hence or otherwise, factorize $q(x)$ as a product of linear factors.
• 18M.1.hl.TZ1.1: Let f(x) = x4 + px3 + qx + 5 where p, q are constants. The remainder when f(x) is divided by (x...
• 18M.2.hl.TZ2.2: The polynomial ${x^4} + p{x^3} + q{x^2} + rx + 6$ is exactly divisible by each...

2.6

• 12M.2.hl.TZ1.1: Given that the graph of $y = {x^3} - 6{x^2} + kx - 4$ has exactly one point at which...
• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.2.hl.TZ1.11c: Find the coordinates of the point, on $y = f(x)$ , where the gradient of the graph is 3.
• 12M.1.hl.TZ2.12B.c: Find the two complex roots of the equation $f(x) = 0$ in Cartesian form.
• 12N.2.hl.TZ0.2: Show that the quadratic equation ${x^2} - (5 - k)x - (k + 2) = 0$ has two distinct real...
• 08N.2.hl.TZ0.6: (a)     Sketch the curve $y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1$ ,...
• 11M.2.hl.TZ2.7a: Find the coordinates of P.
• 11M.2.hl.TZ2.7b: Calculate the size of the acute angle between the tangents to the two graphs at the point P.
• 09M.1.hl.TZ1.10: The diagram below shows a solid with volume V , obtained from a cube with edge $a > 1$ when...
• SPNone.1.hl.TZ0.2a: Write down the numerical value of the sum and of the product of the roots of this equation.
• 13M.2.hl.TZ1.9a: Prove that the equation $3{x^2} + 2kx + k - 1 = 0$ has two distinct real roots for all values...
• 13M.2.hl.TZ1.9b: Find the value of k for which the two roots of the equation are closest together.
• 10M.2.hl.TZ2.11: The function f is defined...
• 13M.1.hl.TZ2.7b: Find the minimum value of $\left| {{z_1} + \alpha{z_2}} \right|$, where $\alpha \in \mathbb{R}$.
• 13M.2.hl.TZ2.4b: Determine the value of m if $\int_0^m {x{{\sec }^2}x{\text{d}}x = 0.5}$, where m > 0.
• 13M.2.hl.TZ2.11b: A number of distinct points are marked on the circumference of a circle, forming a polygon....
• 11N.1.hl.TZ0.9a: Find the set of values of y for which this equation has real roots.
• 11N.2.hl.TZ0.8b: find the value of x such that $f(x) = {f^{ - 1}}(x)$.
• 11M.2.hl.TZ1.4b: Show that for this value of $a$ there is a unique real solution to the equation $f (x) = 0$ .
• 11M.2.hl.TZ1.5a: Write down the quadratic expression $2{x^2} + x - 3$ as the product of two linear factors.
• 09N.2.hl.TZ0.1: Find the values of $k$ such that the equation ${x^3} + {x^2} - x + 2 = k$ has three distinct...
• 14M.1.hl.TZ1.4: The equation $5{x^3} + 48{x^2} + 100x + 2 = a$ has roots ${r_1}$, ${r_2}$ and...
• 14M.1.hl.TZ2.4: The roots of a quadratic equation $2{x^2} + 4x - 1 = 0$ are $\alpha$ and \(\beta...
• 14M.2.hl.TZ2.3a: Find the value of ${x_{\text{A}}}$ and the value of ${x_{\text{B}}}$.
• 13N.1.hl.TZ0.9: Solve the following equations: (a)     ${\log _2}(x - 2) = {\log _4}({x^2} - 6x + 12)$; (b)  ...
• 13N.2.hl.TZ0.7b: find the smallest possible positive value of $\theta$.
• 13N.2.hl.TZ0.8b: Find the value of $\theta$ for which the shaded area is equal to half that of the unshaded...
• 14N.1.hl.TZ0.2a: Without solving the equation, find the value of (i)     $\alpha  + \beta$; (ii)    ...
• 14N.1.hl.TZ0.2b: Another quadratic equation ${x^2} + px + q = 0,{\text{ }}p,{\text{ }}q \in \mathbb{Z}$ has...
• 14N.2.hl.TZ0.6: Consider $p(x) = 3{x^3} + ax + 5a,\;\;\;a \in \mathbb{R}$. The polynomial $p(x)$ leaves a...
• 14N.2.hl.TZ0.13c: Once empty, water is pumped back into the container at a rate of...
• 15M.1.hl.TZ1.7a: For the polynomial equation $p(x) = 0$, state (i)     the sum of the roots; (ii)     the...
• 15M.1.hl.TZ1.7b: A new polynomial is defined by $q(x) = p(x + 4)$. Find the sum of the roots of the equation...
• 15M.1.hl.TZ2.12a: (i)     $p =  - (\alpha  + \beta  + \gamma )$; (ii)    ...
• 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...
• 15M.2.hl.TZ1.12d: Hence express $\sin 72^\circ$ in the form $\frac{{\sqrt {a + b\sqrt c } }}{d}$ where...
• 15M.2.hl.TZ2.3b: Hence, or otherwise, solve the equation ${(x - 5)^2} - 2\left| {x - 5} \right| - 9 = 0$.
• 15M.2.hl.TZ2.8c: Find the value of $a$.
• 15N.1.hl.TZ0.10a: the degree of the polynomial;
• 15N.1.hl.TZ0.10b: the value of ${a_0}$.
• 16M.2.hl.TZ1.11a: Find the solutions of $f(x) > 0$.
• 16M.2.hl.TZ2.12b: (i)     By forming a quadratic equation in ${{\text{e}}^x}$, solve the equation $h(x) = k$,...
• 16N.1.hl.TZ0.5: The quadratic equation ${x^2} - 2kx + (k - 1) = 0$ has roots $\alpha$ and $\beta$ such...
• 16N.2.hl.TZ0.11b: Find the values of the constants $a$ and $b$.
• 16N.2.hl.TZ0.11c: Deduce that...
• 17M.1.hl.TZ1.12a: Write down the sum and the product of the roots of $P(z) = 0$.
• 17M.1.hl.TZ1.12c: Find the value of $b$ and the value of $c$.
• 17M.1.hl.TZ1.12d: Hence find the complex roots of $P(z) = 0$.
• 17N.2.hl.TZ0.7: In the quadratic equation $7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})$, one root is three...
• 18M.2.hl.TZ1.2: The equation ${x^2} - 5x - 7 = 0$ has roots $\alpha$ and $\beta$. The equation...
• 18M.1.hl.TZ2.2b: Solve the equation $\frac{x}{2} + 1 = \left| {x - 2} \right|$.
• 18M.2.hl.TZ2.10d.i: Find $\alpha$ and β in terms of $k$.
• 18M.2.hl.TZ2.10d.ii: Show that $\alpha$ + β < −2.

2.7

• 12M.2.hl.TZ2.1b: Find the smallest value of n such that the sum of the first n terms is greater than 600.
• 12N.3srg.hl.TZ0.4f: (i)     If $x,{\text{ }}y \in G$ explain why $(c - x)(c - y) > 0$ . (ii)     Hence show...
• 08M.2.hl.TZ2.10: Find the set of values of x for which $\left| {0.1{x^2} - 2x + 3} \right| < {\log _{10}}x$ .
• 08N.2.hl.TZ0.6: (a)     Sketch the curve $y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1$ ,...
• 10M.2.hl.TZ1.9: Let $f(x) = \frac{{4 - {x^2}}}{{4 - \sqrt x }}$. (a)     State the largest possible domain for...
• 10M.2.hl.TZ2.8: (a)     Simplify the difference of binomial...
• 10N.1.hl.TZ0.1: Find the set of values of x for which $\left| {x - 1} \right| > \left| {2x - 1} \right|$.
• 13M.1.hl.TZ2.9a: Show that $f(x) > 1$ for all x > 0.
• 13M.1.hl.TZ2.9b: Solve the equation $f(x) = 4$.
• 13M.2.hl.TZ2.5b: Determine the set of values of n for which ${u_n} > {v_n}$.
• 13M.2.hl.TZ2.5c: Determine the greatest value of ${u_n} - {v_n}$. Give your answer correct to four significant...
• 11M.1.hl.TZ1.12d: Now consider the functions $g(x) = \frac{{\ln \left| x \right|}}{x}$ and...
• 09M.2.hl.TZ1.3: Let $f(x) = \frac{{1 - x}}{{1 + x}}$ and $g(x) = \sqrt {x + 1}$, $x > - 1$. Find the...
• 09M.2.hl.TZ2.4b: Solve $\ln \left( {2x + 1} \right) > 3\cos (x)$, $x \in [0,10]$.
• 14M.2.hl.TZ1.6a: Solve the inequality $f(x) > x$.
• 13N.2.hl.TZ0.3b: Solve the inequality $\ln x \leqslant {{\text{e}}^{\cos x}},{\text{ }}0 < x \leqslant 10$.
• 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.TZ0.9b: If $n > 1$ and odd, it can be shown that...
• 15M.1.hl.TZ2.10d: Solve the inequality $\left| {f(x)} \right| < \frac{3}{2}$.
• 15M.1.hl.TZ2.10e: Solve the inequality $f\left( {\left| x \right|} \right) < \frac{3}{2}$.
• 15M.2.hl.TZ1.10a: A function $f$ is defined by $f(x) = (x + 1)(x-1)(x-5),{\text{ }}x \in \mathbb{R}$. Find the...
• 15M.2.hl.TZ1.10b: A function $g$ is defined by $g(x) = {x^2} + x - 6,{\text{ }}x \in \mathbb{R}$. Find the...
• 17M.2.hl.TZ2.4a: Find the set of values of $k$ that satisfy the inequality ${k^2} - k - 12 < 0$.
• 17N.1.hl.TZ0.6b: Hence or otherwise, solve the inequality $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$.
• 18M.1.hl.TZ1.5: Solve ...