The function $$$\mathrm{f}$$$ is defined by $$$\mathrm{f}(x)=\frac{2 x}{3 x-1}$$$ for $$$x>\frac{1}{3}$$$. Show that $$$\frac{2}{3}+\frac{2}{3(3 x-1)}$$$ can be expressed as $$$\frac{2 x}{3 x-1}$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_13 Year：2020 Question No：6(b)

Answer:

$\left[\frac{2(3 x-1)+2}{3(3 x-1)}\right]=\left[\frac{6 x}{3(3 x-1)}=\frac{2 x}{3 x-1}\right]$

Knowledge points:

1.1.1 carry out the process of completing the square for a quadratic polynomial and use a completed square form

1.1.2 find the discriminant of a quadratic polynomial and use the discriminant

1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)

1.1.4 solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic

1.1.5 recognise and solve equations in which are quadratic in some function of

1.2.1 understand the terms function, domain, range, one-one function, inverse function and composition of functions

1.2.2 identify the range of a given function in simple cases, and find the composition of two given functions

1.2.3 determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases

1.2.4 illustrate in graphical terms the relation between a one-one function and its inverse (Sketches should include an indication of the mirror line .)

1.2.5 understand and use the transformations of the graph of and simple combinations of these. (Including use of the terms ‘translation’, ‘reflection’ and ‘stretch’ in describing transformations. Questions may involve algebraic or trigonometric functions, or other graphs with given features.)

1.3.1 find the equation of a straight line given sufficient information

1.3.2 interpret and use any of the forms in solving problems (Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines.)

1.3.3 understand that the equation represents the circle with centre and radius (Including use of the expanded form.)

1.3.4 use algebraic methods to solve problems involving lines and circles (Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry.) (Implicit differentiation is not included.)

1.3.5 understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. (e.g. to determine the set of values of for which the line intersects, touches or does not meet a quadratic curve.)

1.4.1 understand the definition of a radian, and use the relationship between radians and degrees

1.4.2 use the formulae in solving problems concerning the arc length and sector area of a circle (Including calculation of lengths and angles in triangles and areas of triangles.)

1.5.1 sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians)

1.5.2 use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles

1.5.3 use the notations to denote the principal values of the inverse trigonometric relations (No specialised knowledge of these functions is required, but understanding of them as examples of inverse functions is expected.)

1.5.4 use the identities

1.5.5 find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).

1.6.1 use the expansion of , where is a positive integer (Including the notations and n!) (Knowledge of the greatest term and properties of the coefficients are not required.)

1.6.2 recognise arithmetic and geometric progressions

1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)

1.6.4 use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

1.7.1 understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations for first and second derivatives (Only an informal understanding of the idea of a limit is expected.)

1.7.2 use the derivative of (for any rational ), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule

1.7.3 apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (Including connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of the area for a specific value of one of the variables.)

1.7.4 locate stationary points and determine their nature, and use information about stationary points in sketching graphs. (Including use of the second derivative for identifying maxima and minima; alternatives may be used in questions where no method is specified.) (Knowledge of points of inflexion is not included.)

1.8.1 understand integration as the reverse process of differentiation, and integrate (for any rational n except-1 , together with constant multiples, sums and differences

1.8.2 solve problems involving the evaluation of a constant of integration

1.8.3 evaluate definite integrals (Including simple cases of ‘improper’ integrals, such as)

1.8.4.1 the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves

1.8.4.2 a volume of revolution about one of the axes. (A volume of revolution may involve a region not bounded by the axis of rotation, e.g. the region between and y = 5 rotated about the x-axis.)

Answer:

Knowledge points:

Solution:

Related Question