DP Mathematics HL Questionbank
Topic 2 - Core: Functions and equations
Description
The aims of this topic are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of technology in both the development and the application of this topic.
Directly related questions
- 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
- 12M.1.hl.TZ2.11b: Find an expression for the composite function \(f \circ g(x)\) in the form...
- 12N.1.hl.TZ0.3a: Using the information shown in the diagram, find the values of a , b and c .
- 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...
- 08M.2.hl.TZ2.10: Find the set of values of x for which \(\left| {0.1{x^2} - 2x + 3} \right| < {\log _{10}}x\) .
- 11M.1.hl.TZ2.8: A function is defined by...
- SPNone.1.hl.TZ0.13c: Obtain expressions for the inverse function \({f^{ - 1}}\) and state their domains.
- 13M.2.hl.TZ1.12b: Write down the times for which the velocity of the particle is increasing.
- 10M.2.hl.TZ2.11: The function f is defined...
- 10N.1.hl.TZ0.9: Consider the function \(f:x \to \sqrt {\frac{\pi }{4} - \arccos x} \). (a) Find the largest...
- 13M.1.hl.TZ2.12c: State the range of f.
- 13M.1.hl.TZ2.13b: (i) State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them...
- 12M.1.hl.TZ1.4: The graph below shows \(y = f(x)\) , where \(f(x) = x + \ln x\) . (a) On the graph below,...
- 11M.2.hl.TZ1.5a: Write down the quadratic expression \(2{x^2} + x - 3\) as the product of two linear factors.
- 11M.1.hl.TZ1.8b: Find the coordinates of the point where the graph of \(y = f(x)\) and the graph of...
- 14M.1.hl.TZ1.4: The equation \(5{x^3} + 48{x^2} + 100x + 2 = a\) has roots \({r_1}\), \({r_2}\) and...
- 14M.2.hl.TZ1.1: One root of the equation \({x^2} + ax + b = 0\) is \(2 + 3{\text{i}}\) where...
- 14M.1.hl.TZ2.8b: The graph of the function \(g\) is obtained by applying the following transformations to the...
- 15M.1.hl.TZ1.5b: Given that \(g\) is an odd function, find the value of \(r\).
- 15M.1.hl.TZ1.6a: Find an expression for \({f^{ - 1}}(x)\).
- 15M.1.hl.TZ2.10d: Solve the inequality \(\left| {f(x)} \right| < \frac{3}{2}\).
- 15M.1.hl.TZ2.13a: Show that \(\frac{1}{{\sqrt n + \sqrt {n + 1} }} = \sqrt {n + 1} - \sqrt n \) where...
- 15M.2.hl.TZ1.12d: Hence express \(\sin 72^\circ \) in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where...
- 15M.2.hl.TZ1.6: A function \(f\) is defined by \(f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}\)....
- 15M.2.hl.TZ2.12b: Sketch a displacement/time graph for the particle, \(0 \le t \le 5\), showing clearly where the...
- 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.TZ0.7b: The seventh term of the arithmetic sequence is \(3\). The sum of the first \(n\) terms in the...
- 14N.3sp.hl.TZ0.1a: Sketch the graph of \(y = f(x)\).
- 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.TZ1.12a: Write down the sum and the product of the roots of \(P(z) = 0\).
- 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.
- 17N.1.hl.TZ0.3b: Hence or otherwise, factorize \(q(x)\) as a product of linear factors.
- 16N.1.hl.TZ0.5: The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such...
- 16M.1.hl.TZ2.2: The function \(f\) is defined as...
- 18M.2.hl.TZ2.2: The polynomial \({x^4} + p{x^3} + q{x^2} + rx + 6\) is exactly divisible by each...
- 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...
- 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.2.hl.TZ2.6a: Sketch the curve...
- 12N.1.hl.TZ0.12d: (i) State \({F_n}(0){\text{ and }}{F_n}(1)\) . (ii) Show that \({F_n}(x) < x\) ,...
- 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.TZ2.13c: The increasing function f satisfies \(f(0) = 0\) and \(f(a) = b\) , where \(a > 0\) and...
- 09N.1.hl.TZ0.4: Consider the function f , where \(f(x) = \arcsin (\ln x)\). (a) Find the domain of f . (b)...
- 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...
- 13M.1.hl.TZ2.9b: Solve the equation \(f(x) = 4\).
- 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.TZ2.5c: Determine the greatest value of \({u_n} - {v_n}\). Give your answer correct to four significant...
- 11N.1.hl.TZ0.9a: Find the set of values of y for which this equation has real roots.
- 09M.2.hl.TZ1.3: Let \(f(x) = \frac{{1 - x}}{{1 + x}}\) and \(g(x) = \sqrt {x + 1} \), \(x > - 1\). Find the...
- 14M.2.hl.TZ1.6a: Solve the inequality \(f(x) > x\).
- 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.TZ2.14b: Find an expression for the composite function \(h \circ g(x)\) and state its domain.
- 13N.1.hl.TZ0.10e: Sketch the graphs of \(y = f(x)\) and \(y = g(x)\) on the same axes, showing clearly the points...
- 15M.1.hl.TZ1.5a: Given that \(f\) is an even function, show that \(b = 0\).
- 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.TZ1.7a: For the polynomial equation \(p(x) = 0\), state (i) the sum of the roots; (ii) the...
- 15M.1.hl.TZ2.12c: In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form a geometric...
- 15M.2.hl.TZ1.11a: Sketch the graph \(y = f(x)\).
- 15M.2.hl.TZ2.3b: Hence, or otherwise, solve the equation \({(x - 5)^2} - 2\left| {x - 5} \right| - 9 = 0\).
- 15M.2.hl.TZ2.6a: Find the value of \(a\) and the value of \(b\).
- 17M.1.hl.TZ1.11c: Show that \(\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}\).
- 17M.1.hl.TZ1.12c: Find the value of \(b\) and the value of \(c\).
- 17M.1.hl.TZ1.12d: Hence find the complex roots of \(P(z) = 0\).
- 17M.1.hl.TZ2.2a: Write down the range of \(f\).
- 17M.1.hl.TZ2.9a.ii: Showing any \(x\) and \(y\) intercepts, any maximum or minimum points and any asymptotes, sketch...
- 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...
- 16N.1.hl.TZ0.3a: state the value of \(a\) and the value of \(c\);
- 16M.1.hl.TZ1.7b: Find the exact solutions to the equation \(x + 2 = \left| {\frac{7}{{x - 4}}} \right|\).
- 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...
- 18M.2.hl.TZ2.10a.i: Sketch the graph of \(y = f\left( x \right)\)...
- 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.2.hl.TZ2.10d.i: Find \(\alpha \) and β in terms of \(k\).
- 18M.1.hl.TZ2.10b.ii: Sketch the graph of \(y = g\left( x \right)\). State the equations of any asymptotes and...
- 12M.1.hl.TZ2.7a: On the axes below, sketch the graph of \(y = \frac{1}{{f(x)}}\) , clearly showing the coordinates...
- 12N.1.hl.TZ0.3b: If g(x) = 3f(x − 2) , (i) state the coordinates of the points where the graph of g...
- 12N.2.hl.TZ0.2: Show that the quadratic equation \({x^2} - (5 - k)x - (k + 2) = 0\) has two distinct real...
- 08M.1.hl.TZ2.4: Let \(f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1\). If...
- 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...
- 08N.2.hl.TZ0.6: (a) Sketch the curve \(y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1\) ,...
- 11M.1.hl.TZ2.12a: Factorize \({z^3} + 1\) into a linear and quadratic factor.
- 11M.2.hl.TZ2.7b: Calculate the size of the acute angle between the tangents to the two graphs at the point P.
- 13M.2.hl.TZ1.9b: Find the value of k for which the two roots of the equation are closest together.
- 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.TZ1.13d: (i) With f and g as defined in parts (a) and (b), solve \(g \circ f(x) = 2\). (ii) Let...
- 13M.1.hl.TZ2.7b: Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).
- 13M.1.hl.TZ2.12d: (i) Find an expression for \({f^{ - 1}}(x)\). (ii) Sketch the graph of \(y = f(x)\),...
- 13M.2.hl.TZ2.5b: Determine the set of values of n for which \({u_n} > {v_n}\).
- 11N.1.hl.TZ0.4a: sketch the graph of \(f\);
- 13M.2.hl.TZ2.13c: Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).
- 11M.1.hl.TZ1.10b: On the axes below, sketch the graph of \(y = g(x)\) . On the graph, indicate any asymptotes and...
- 11M.2.hl.TZ1.4a: Find the value of \(a\) .
- 09N.2.hl.TZ0.1: Find the values of \(k\) such that the equation \({x^3} + {x^2} - x + 2 = k\) has three distinct...
- 09M.2.hl.TZ2.4b: Solve \(\ln \left( {2x + 1} \right) > 3\cos (x)\), \(x \in [0,10]\).
- 13N.2.hl.TZ0.7b: find the smallest possible positive value of \(\theta \).
- 15M.1.hl.TZ1.9a: Show that \(g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4\).
- 15M.1.hl.TZ2.10e: Solve the inequality \(f\left( {\left| x \right|} \right) < \frac{3}{2}\).
- 15M.1.hl.TZ2.11b: Hence show that \(g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\).
- 15M.1.hl.TZ2.10a: Sketch the graph of \(y = f(x)\), indicating clearly any asymptotes and points of intersection...
- 15M.1.hl.TZ2.10b: Find an expression for \({f^{ - 1}}(x)\).
- 15M.2.hl.TZ1.10b: A function \(g\) is defined by \(g(x) = {x^2} + x - 6,{\text{ }}x \in \mathbb{R}\). Find the...
- 15M.2.hl.TZ2.3a: Sketch the graph of \(y = {(x - 5)^2} - 2\left| {x - 5} \right| - 9,{\text{ for }}0 \le x \le 10\).
- 14N.1.hl.TZ0.2a: Without solving the equation, find the value of (i) \(\alpha + \beta \); (ii) ...
- 15N.1.hl.TZ0.10b: the value of \({a_0}\).
- 15N.1.hl.TZ0.12a: Show that \(f\) is an odd function.
- 15N.2.hl.TZ0.12a: The functions \(u\) and \(v\) are defined as \(u(x) = x - 3,{\text{ }}v(x) = 2x\) where...
- 17M.1.hl.TZ1.11a.i: Express \({x^2} + 3x + 2\) in the form \({(x + h)^2} + k\).
- 17M.2.hl.TZ1.12b: Sketch the graph of \(y = f(x)\) showing clearly the equations of asymptotes and the coordinates...
- 16N.2.hl.TZ0.10e: Find an expression for \({f^{ - 1}}(x)\).
- 16M.2.hl.TZ1.8: When \({x^2} + 4x - b\) is divided by \(x - a\) the remainder is 2. Given that...
- 16M.2.hl.TZ2.5: The function \(f\) is defined as...
- 18M.2.hl.TZ1.2: The equation \({x^2} - 5x - 7 = 0\) has roots \(\alpha \) and \(\beta \). The equation...
- 18M.2.hl.TZ2.10c: Sketch the graph of \(y = g\left( t \right)\) for t ≤ 0. Give the coordinates of any intercepts...
- 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.TZ1.11c: Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.
- 12M.2.hl.TZ2.12b: Sketch the graph of v against t , clearly showing the coordinates of any intercepts, and the...
- 12N.3srg.hl.TZ0.4f: (i) If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) . (ii) Hence show...
- 11M.2.hl.TZ2.5: Sketch the graph of \(f(x) = x + \frac{{8x}}{{{x^2} - 9}}\). Clearly mark the coordinates of the...
- 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.TZ2.12a: Express \(f(x)\) in the form \(A + \frac{B}{{x + 2}}\), where \(A\) and \(B \in \mathbb{Z}\).
- 13M.1.hl.TZ2.12e: (i) On a different diagram, sketch the graph of \(y = f(|x|)\) where \(x \in D\). (ii) ...
- 11N.1.hl.TZ0.9c: Explain why f has no inverse.
- 11N.2.hl.TZ0.1a: Sketch the graph, clearly labelling the x and y intercepts with their values.
- 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...
- 11M.2.hl.TZ1.4b: Show that for this value of \(a\) there is a unique real solution to the equation \(f (x) = 0\) .
- 14M.2.hl.TZ2.7a: (i) Sketch the graph of \(y = f(x)\), clearly indicating any asymptotes and axes...
- 15M.1.hl.TZ1.11d: Find the coordinates of any points of inflexion on the graph of \(y(x)\). Justify whether any...
- 14N.1.hl.TZ0.1b: State the equations of the asymptotes of the graph of \(g\).
- 14N.2.hl.TZ0.13c: Once empty, water is pumped back into the container at a rate of...
- 15N.1.hl.TZ0.12d: Find the range of \(f\).
- 17M.1.hl.TZ1.6a: Sketch the graphs on the same set of axes.
- 17M.2.hl.TZ1.5b: Find the probability that exactly seven rooms will have fewer than three faults in the carpet.
- 17M.2.hl.TZ1.12e: Find the inverse function \({g^{ - 1}}\) and state its domain.
- 16N.2.hl.TZ0.5a: Sketch the graph of \(f\) indicating clearly any intercepts with the axes and the coordinates of...
- 16N.2.hl.TZ0.11c: Deduce that...
- 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.TZ1.11d.ii: Solve \(({f^{ - 1}} \circ g)(x) < 1\).
- 16M.2.hl.TZ1.2a: Express \({x^2} + 4x - 2\) in the form \({(x + a)^2} + b\) where \(a,{\text{ }}b \in \mathbb{Z}\).
- 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
- 11M.1.hl.TZ2.5a: Sketch the graph of \(y = \frac{1}{{f(x)}}\).
- 11M.2.hl.TZ2.7a: Find the coordinates of P.
- 09M.1.hl.TZ2.11: A function is defined as \(f(x) = k\sqrt x \), with \(k > 0\) and \(x \geqslant 0\) . (a) ...
- 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...
- 10M.2.hl.TZ2.8: (a) Simplify the difference of binomial...
- 11N.1.hl.TZ0.9b: Hence determine the range of the function \(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}\).
- 11N.2.hl.TZ0.8b: find the value of x such that \(f(x) = {f^{ - 1}}(x)\).
- 14M.1.hl.TZ2.5a: Sketch the graph of \(y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|\) for...
- 14M.2.hl.TZ2.7b: Find the inverse function \({f^{ - 1}}\), stating its domain.
- 15M.1.hl.TZ1.5c: The function \(h\) is both odd and even, with domain \(\mathbb{R}\). Find \(h(x)\).
- 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.TZ2.10c: Find all values of \(x\) for which \(f(x) = {f^{ - 1}}(x)\).
- 14N.1.hl.TZ0.11a: (i) Find \({f^{ - 1}}(x)\). (ii) State the domain of \({f^{ - 1}}\).
- 15N.2.hl.TZ0.12b: (i) Explain why \(f\) does not have an inverse. (ii) The domain of \(f\) is restricted...
- 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.TZ2.9a.i: 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...
- 17N.1.hl.TZ0.6b: Hence or otherwise, solve the inequality \(\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2\).
- 16N.2.hl.TZ0.10d: Use your answers from parts (b) and (c) to justify that \(f\) has an inverse and state its domain.
- 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.TZ2.12b: (i) By forming a quadratic equation in \({{\text{e}}^x}\), solve the equation \(h(x) = k\),...
- 18M.1.hl.TZ2.2a: Sketch the graphs of \(y = \frac{x}{2} + 1\) and \(y = \left| {x - 2} \right|\) on the following...
- 18M.2.hl.TZ2.10a.ii: With reference to your graph, explain why \(f\) is a function on the given domain.
- 12M.1.hl.TZ1.2a: Draw the graph of y = f (x) on the blank grid below.
- 12M.1.hl.TZ2.11a: State the range of f and of g .
- 12M.2.hl.TZ2.1b: Find the smallest value of n such that the sum of the first n terms is greater than 600.
- 08M.2.hl.TZ2.8: The graph of \(y = f(x){\text{ for }} - 2 \leqslant x \leqslant 8\) is shown. On the set...
- SPNone.1.hl.TZ0.9c: Hence show that \({{\text{e}}^\pi } > {\pi ^{\text{e}}}\) .
- 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.TZ1.12b: The graph of \(y = {x^2}\) is transformed onto the graph of \(y = 4{x^2} - 4x + 5\). Describe a...
- 13M.1.hl.TZ1.12d: Find the range of f.
- 10M.1.hl.TZ1.5: The graph of \(y = \frac{{a + x}}{{b + cx}}\) is drawn below. (a) Find the value of...
- 10M.1.hl.TZ2.10: A function f is defined by \(f(x) = \frac{{2x - 3}}{{x - 1}},{\text{ }}x \ne 1\). (a) Find...
- 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...
- 09M.2.hl.TZ2.4a: The graph of \(y = \ln (x)\) is transformed into the graph of \(y = \ln \left( {2x + 1} \right)\)...
- 13N.1.hl.TZ0.3b: State the range of \({f^{ - 1}}\).
- 13N.1.hl.TZ0.3c: Given that \(f(x) = \ln (ax + b),{\text{ }}x > 1\), find the value of \(a\) and the value of...
- 13N.2.hl.TZ0.3b: Solve the inequality \(\ln x \leqslant {{\text{e}}^{\cos x}},{\text{ }}0 < x \leqslant 10\).
- 15M.1.hl.TZ2.13b: Hence show that \(\sqrt 2 - 1 < \frac{1}{{\sqrt 2 }}\).
- 15M.2.hl.TZ2.8c: Find the value of \(a\).
- 14N.1.hl.TZ0.11b: The function \(g\) is defined as \(g(x) = \ln x,{\text{ }}x \in {\mathbb{R}^ + }\). The graph of...
- 17M.1.hl.TZ1.11e: Sketch the graph of \(y = f\left( {\left| x \right|} \right)\).
- 17M.2.hl.TZ1.12a: Find the largest possible domain \(D\) for \(f\) to be a function.
- 17M.2.hl.TZ1.12d: Explain why the inverse function \({f^{ - 1}}\) does not exist.
- 17N.1.hl.TZ0.3a: Given that \(q(x)\) has a factor \((x - 4)\), find the value of \(k\).
- 17N.2.hl.TZ0.7: In the quadratic equation \(7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})\), one root is three...
- 17N.1.hl.TZ0.11a: Determine whether \({f_n}\) is an odd or even function, justifying your answer.
- 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.TZ0.11b: Find the values of the constants \(a\) and \(b\).
- 16M.2.hl.TZ1.11a: Find the solutions of \(f(x) > 0\).
- 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.
- 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.1.hl.TZ1.5: Solve ...
- 18M.2.hl.TZ2.10a.iii: Explain why \(f\) has no inverse on the given domain.
- 12N.1.hl.TZ0.12c: Show that \({F_{ - n}}(x)\) is an expression for the inverse of \({F_n}\) .
- 08M.2.hl.TZ1.2: (a) Sketch the curve...
- 11M.1.hl.TZ2.5b: Sketch the graph of \(y = x{\text{ }}f(x)\) .
- 09M.1.hl.TZ1.3: Let \(g(x) = {\log _5}\left| {2{{\log }_3}x} \right|\) . Find the product of the zeros of g .
- 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.1.hl.TZ0.2a: Write down the numerical value of the sum and of the product of the roots of this equation.
- SPNone.1.hl.TZ0.5a: Determine whether f is even, odd or neither even nor odd.
- 13M.2.hl.TZ1.12c: Write down the times for which the magnitude of the velocity of the particle is increasing.
- 10M.1.hl.TZ1.2: Shown below are the graphs of \(y = f(x)\) and \(y = g(x)\). If \((f \circ g)(x) = 3\),...
- 10M.2.hl.TZ1.9: Let \(f(x) = \frac{{4 - {x^2}}}{{4 - \sqrt x }}\). (a) State the largest possible domain for...
- 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.
- 11M.1.hl.TZ1.10a: Find the largest possible domain of the function \(g\) .
- 11M.1.hl.TZ1.12d: Now consider the functions \(g(x) = \frac{{\ln \left| x \right|}}{x}\) and...
- 09N.2.hl.TZ0.9: (a) Given that the domain of \(g\) is \(x \geqslant a\) , find the least value of \(a\) such...
- 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.1.hl.TZ2.4: The roots of a quadratic equation \(2{x^2} + 4x - 1 = 0\) are \(\alpha \) and \(\beta...
- 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.TZ2.3a: Find the value of \({x_{\text{A}}}\) and the value of \({x_{\text{B}}}\).
- 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...
- 13N.1.hl.TZ0.9: Solve the following equations: (a) \({\log _2}(x - 2) = {\log _4}({x^2} - 6x + 12)\); (b) ...
- 13N.2.hl.TZ0.3a: Sketch the graph of \(y = f(x)\), stating the coordinates of any maximum and minimum points and...
- 13N.2.hl.TZ0.8b: Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded...
- 13N.1.hl.TZ0.10d: The graph of the function \(g\) is obtained from the graph of \(f\) by stretching it in 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.TZ1.9d: The graph of \(y = g \circ f(x)\) can be obtained by applying four transformations to the graph...
- 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.TZ2.11a: Find an expression for \(g \circ f(x)\), stating its domain.
- 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...
- 14N.1.hl.TZ0.1a: Find an expression for \(g(x)\).
- 14N.2.hl.TZ0.9b: If \(n > 1\) and odd, it can be shown that...
- 15N.1.hl.TZ0.12e: Sketch the graph of \(y = f(x)\) indicating clearly the coordinates of the \(x\)-intercepts and...
- 15N.2.hl.TZ0.12c: Consider the function defined by \(h(x) = \frac{{2x - 5}}{{x + d}}\), \(x \ne - d\) and...
- 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.TZ1.12c: Explain why \(f\) is an even function.
- 17M.2.hl.TZ2.4a: Find the set of values of \(k\) that satisfy the inequality \({k^2} - k - 12 < 0\).
- 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 \(b\).
- 16N.1.hl.TZ0.3b: find the value of \(b\).
- 16M.2.hl.TZ1.11c.i: Write down the largest value of \(a\) for which \(f\) has an inverse. Give your answer correct to...
- 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.2.hl.TZ2.10a.iv: Explain why \(f\) is not a function for...
- 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.TZ2.1: The same remainder is found when \(2{x^3} + k{x^2} + 6x + 32\) and...
- 12M.1.hl.TZ2.10b: Sketch the graph of f .
- 12M.1.hl.TZ2.11c: (i) Find an expression for the inverse function \({f^{ - 1}}(x)\) . (ii) State the...
- 12M.1.hl.TZ2.12B.b: The curve passes through the point \(( - 1,\, - 18)\) . Find \(f(x)\) in the...
- 12M.1.hl.TZ2.12B.c: Find the two complex roots of the equation \(f(x) = 0\) in Cartesian form.
- 12N.1.hl.TZ0.12a: Find an expression for \((f \circ f)(x)\) .
- 08M.1.hl.TZ1.8: The functions f and g are defined as: \[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant...
- 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.TZ2.1b: The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis....
- 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.TZ0.9b: Sketch the graph of f , showing clearly the coordinates of the maximum and minimum.
- 13M.1.hl.TZ1.12c: Sketch the graph of \(y = f(x)\).
- 13M.2.hl.TZ1.9a: Prove that the equation \(3{x^2} + 2kx + k - 1 = 0\) has two distinct real roots for all values...
- 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...
- 10M.2.hl.TZ2.4: (a) Find the solution of the...
- 13M.2.hl.TZ2.11b: A number of distinct points are marked on the circumference of a circle, forming a polygon....
- 11N.2.hl.TZ0.8a: find \({f^{ - 1}}(x)\), stating its domain;
- 11M.1.hl.TZ1.8a: (i) Find \(\left( {g \circ f} \right)\left( x \right)\) and write down the domain of the...
- 14M.2.hl.TZ1.10a: Find the equations of the horizontal and vertical asymptotes of the curve \(y = f(x)\).
- 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.TZ0.3a: Sketch the graph of \(y = {f^{ - 1}}(x)\) on the same axes.
- 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.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\). Justify...
- 15M.1.hl.TZ2.9b: Given that \({\log _x}y = 4{\log _y}x\), find all the possible expressions of \(y\) as a function...
- 15M.1.hl.TZ2.12a: (i) \(p = - (\alpha + \beta + \gamma )\); (ii) ...
- 15N.1.hl.TZ0.10a: the degree of the polynomial;
- 17M.1.hl.TZ1.12b: Show that \((z - 1)\) is a factor of \(P(z)\).
- 17M.2.hl.TZ1.5a: Find the probability that the carpet laid in the first room has fewer than three faults.
- 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.TZ0.6a: 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.
- 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 \(a\).
- 18M.2.hl.TZ2.10d.ii: Show that \(\alpha \) + β < −2.
- 18M.2.hl.TZ2.10b: Show that \(g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}\).
- 18M.1.hl.TZ2.2b: Solve the equation \(\frac{x}{2} + 1 = \left| {x - 2} \right|\).
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 ...
