This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
This document provides fully worked solutions to exam questions from Form 4 mathematics chapters on standard form, quadratic expressions and equations, sets, mathematical reasoning, the straight line, and statistics. The solutions include:
1) Detailed working to obtain the answers for multiple choice and structured questions.
2) Explanations of mathematical concepts and reasoning such as determining gradients, interpreting graphs, and identifying argument forms.
3) Step-by-step derivations to find equations of lines from given points and gradients.
This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document contains sample problems related to quadratic equations and quadratic functions for Form 4 Additional Mathematics. It is divided into three sections - the first two sections contain sample problems testing concepts related to solving quadratic equations and inequalities. The third section contains sample problems related to identifying properties of quadratic functions such as finding the minimum or maximum value, range of a quadratic function, expressing a quadratic in standard form and sketching its graph.
1. The document is a math test for Additional Mathematics Form 4 consisting of 18 questions. It provides instructions to candidates to answer all questions clearly in the spaces provided and show their working. Diagrams are not drawn to scale unless stated.
2. The questions cover topics on solving simultaneous equations, functions, relations, composite functions, inverse functions and sketching graphs. Candidates are required to find values, images, objects, domains, ranges and relations in function notation.
3. The final two questions involve sketching a graph of a quadratic function given its relation and finding the inverse of a fractional function.
The document provides information about the format and techniques for answering Additional Mathematics SPM Paper 1 exam questions in Malaysia. It discusses the contents and structure of the exam, including the breakdown of question types between knowledge/understanding and application skills. It also offers tips for candidates on how to effectively answer questions, including showing workings, following instructions, and presenting neat and precise solutions.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
The document provides information about quadratic functions including:
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- A quadratic function has a minimum or maximum point which can be used to find the axis of symmetry.
- The relationship between the discriminant (b2 - 4ac) and the position of the graph is explained. If it is greater than 0, the graph cuts the x-axis at two points. If it is equal to 0, the graph touches the x-axis at one point. If it is less than 0, the graph does not cut or touch the x-axis.
- Quadratic inequalities can be solved by sketching
This document provides guidance on solving additional mathematics problems and preparing for the additional mathematics paper 1 and paper 2 exams. It discusses exam formats, common mistakes students make, key strategies for achieving high marks, and examples of solved exam questions. The document emphasizes understanding the problem, planning a strategy, checking answers, and showing working clearly. It also provides tips on time management and the types of questions that may appear.
This document contains information about the format and topics covered in papers 1 and 2 of an exam. Paper 1 has 25 questions to be answered in 2 hours, with 10 questions of low difficulty, 6 of moderate difficulty, and 1 of high difficulty. Paper 2 has 3 sections, with the first section containing 6 questions to answer, the second 5 questions where the test taker must choose 4, and the third 4 questions where they must choose 2. The total time for Paper 2 is 2.5 hours.
The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
1. This document discusses solving quadratic equations by factorizing, using the quadratic formula, and finding the roots given information about the sum and product of the roots.
2. Methods covered include factorizing quadratic expressions, setting each factor equal to zero to find roots, using the quadratic formula, and deriving the quadratic equation when given the sum and product of the roots.
3. Examples are provided to demonstrate each method, such as factorizing (x-1)(x+2)=1 to find the roots x=1 and x=-2, or using the quadratic formula to solve equations like x2-3x-10=0.
Teknik Menjawab Kertas 1 Matematik TambahanZefry Hanif
The document provides information about the format, topics, and analysis of past year mathematics SPM 3472/1 papers from 2008 to 2011. It discusses the number of questions and marks for each paper. It also contains tables analyzing the topics that have appeared each year, including functions, quadratic equations, indices, and other topics. The document advises students to maximize the use of calculators, including the "SOLVE" and "CALC" functions. It provides examples of using these functions to solve equations.
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
This document provides 10 problems related to differentiating functions. The problems cover differentiating various functions with respect to variables like x and y, finding maximum and minimum values, rates of change, and determining equations of tangents and normals. Solutions are not provided. The document is from an Additional Mathematics module for Form 4 students and focuses on differentiation techniques.
The document is a marking scheme for an Additional Mathematics Paper 2 exam from September 2009 in Malaysia. It consists of 13 printed pages detailing the questions, workings, and full marks for each part of the exam. The marking scheme provides the solutions and breakdown of marks to be awarded for students' answers on the Additional Mathematics Paper 2 exam.
Parametric equations define a curve where x and y are defined in terms of a third variable called a parameter. The graph of parametric equations consists of all points (x,y) obtained by allowing the parameter to vary over its domain. Eliminating the parameter between the two equations yields a non-parametric equation of the curve. Examples are provided of eliminating parameters between various parametric equations to obtain the curve and sketching the resulting graphs. Exercises are given to further practice eliminating parameters and sketching curves.
5 marks scheme for add maths paper 2 trial spmzabidah awang
1. The document contains the mark scheme and solutions for an Additional Mathematics exam paper with 10 questions.
2. Question 1 involves solving a pair of simultaneous equations to find the values of x and y. Question 2 involves finding the maximum value and graph of a quadratic function.
3. Question 3 examines areas of shapes in a geometric progression. The final area is found to be 17066 cm^2.
4. Subsequent questions cover topics such as function graphs, probability distributions, logarithmic graphs, trigonometry, and simultaneous equations.
5. The last few questions deal with topics like probability, normal distribution, and kinematic equations. Overall the document provides a comprehensive breakdown of the marking scheme
Lesson 8: Derivatives of Logarithmic and Exponential Functions (worksheet sol...Matthew Leingang
This document provides solutions to derivatives of exponential, logarithmic, and other functions. It includes:
1) The derivatives of functions such as y=e^2x, y=6^x, y=ln(x^3 + 9), and y=log_3(e^x).
2) Using logarithmic differentiation to find the derivatives of functions like y=x^x^2-1 and y=(x-1)(x-2)(x-3).
3) Taking the derivative of functions involving logarithms, exponents, and square roots such as y=sin^2(x)+2sin(x) and y=x(x-1)^3/
5 marks scheme for add maths paper 2 trial spmzabidah awang
1. The document is a mark scheme for an Additional Mathematics exam that provides the solutions and workings for 8 multiple choice questions.
2. It lists the mark allocation for each part of the questions and shows the step-by-step workings to achieve the full marks.
3. The questions cover topics like algebra, calculus, geometry, sequences and series, and logarithms.
The document provides 14 formulae across various topics:
- Algebra formulas for operations, exponents, logarithms
- Calculus formulas for derivatives, integrals, areas under curves
- Statistics formulas for means, standard deviations, probabilities
- Geometry formulas for distances, midpoints, areas of shapes
- Trigonometry formulas for trig functions, angles, triangles
- The symbols used in the formulas are explained.
The document provides 14 formulae across various topics:
- Algebra formulas for operations, exponents, logarithms
- Calculus formulas for derivatives, integrals, areas under curves
- Statistics formulas for means, standard deviations, probabilities
- Geometry formulas for distances, midpoints, areas of shapes
- Trigonometry formulas for trig functions, angles, triangles
- The symbols used in the formulas are explained.
The document contains rules and guidelines for marking the trial SPM Mathematics paper for SBP schools in 2007. It includes:
1) The marking scheme for Section A with 52 marks covering questions 1 to 10, outlining the points and marks awarded.
2) The marking scheme for Section B with 48 marks covering questions 11 to 16, including graphs and diagrams.
3) Examples of student responses with marks awarded for questions involving calculations, graphs, and geometric diagrams.
4) Guidelines specify the level of accuracy for measurements and angles in geometric questions.
In 3 sentences, the document provides the marking scheme and examples to standardize the evaluation of the 2007 trial SPM Mathematics paper for schools under
F4 Final Sbp 2007 Maths Skema P 1 & P2norainisaser
This document contains the marking scheme for the Mathematics Paper 1 exam for Form 4 students in Malaysia in October 2007. It includes the marking schemes for 52 multiple choice questions in Section A worth a total of 52 marks and short answer questions in Section B worth a total of 48 marks. The marking schemes provide the number of marks awarded for each part of each question.
This document provides steps to graph functions of the form y = ax^2 + bx + c. It works through an example problem, graphing y = 2x^2 - 8x + 6. The steps are to: 1) identify the coefficients a, b, and c, 2) find the vertex by calculating x from b/2a and the corresponding y-value, 3) draw the axis of symmetry at x, 4) plot other points and 5) draw the parabola through the points. It then provides guided practice problems for students to practice graphing quadratic functions using the same steps.
Here are the key steps to solve equations in indices that involve logarithms:
1. Isolate the term with the index (e.g. ax) on one side of the equation.
2. Take the logarithm (to an appropriate base) of both sides.
3. Use the property that loga(bx) = loga(b) + xloga(a) to split up logarithms of products/quotients.
4. Simplify the resulting equation so it is in the form of x = value.
5. Solve for x by isolating and evaluating the logarithm.
Some examples:
1) 32x = 8
Take log base
The document is a mathematics test for a pre-university class containing two sections - Section A with 6 multiple choice questions worth a total of 45 marks, and Section B with two extended response questions worth 15 marks each. Question 1 in Section A asks students to express a fraction in partial fractions, question 2 asks them to express a trigonometric expression in an alternative form and solve an equation, and subsequent questions cover logarithms, inequalities, trigonometric substitutions, and factorizing polynomials. Section B offers two word problems, one involving composite functions and another involving factorizing a polynomial based on given information.
The document provides 14 formulae across 4 topics:
1) Algebra - includes formulae for roots of quadratic equations, logarithms, sequences, etc.
2) Calculus - includes formulae for derivatives, integrals, areas under curves, volumes of revolution.
3) Statistics - includes formulae for means, standard deviation, probability, binomial distribution.
4) Geometry - includes formulae for distances, midpoints, areas of triangles, circles, trigonometry ratios.
The document provides 14 formulae across 4 topics:
1) Algebra - includes formulae for roots of quadratic equations, logarithms, sequences, etc.
2) Calculus - includes formulae for derivatives, integrals, areas under curves, volumes of revolution.
3) Statistics - includes formulae for means, standard deviation, probability, binomial distribution.
4) Geometry - includes formulae for distances, midpoints, areas of triangles, circles, trigonometry ratios.
The document contains a midterm exam for an ODE class with 6 problems worth 10 points each. Problem 1 asks to find the general solution of a 7th order linear ODE using the method of undetermined coefficients. Problem 2 asks to solve a 2nd order linear ODE using either variation of parameters or undetermined coefficients. Problem 3 asks to solve a nonlinear 2nd order ODE using a substitution. Problem 4 asks to find the equation of motion for a mass attached to a spring with an external force applied. Problem 5 asks to solve an eigenvalue problem for a CE equation. Problem 6 asks to use variation of parameters to solve a 2nd order nonhomogeneous ODE.
This document contains instructions for 5 assignment questions involving numerical integration and solving differential equations. Question 1 involves using the quad function to evaluate several integrals. Question 2 involves using quad to evaluate Fresnel integrals and plot the results. Question 3 involves using Monte Carlo methods to estimate volumes and double integrals. Question 4 involves using Euler's method to solve an initial value problem and analyze errors. Question 5 involves using lsode to solve a system of differential equations modeling atmospheric circulation and experimenting with initial conditions.
This document contains 3 problems:
1) Showing that a sequence is increasing and less than 3.
2) Calculating the average value of a function over an interval.
3) Finding the solution to a partial differential equation of the form 2xyy' – y^2 + x^2 = 0. The solution is found to be y = √cx - x^2.
1. The document provides examples of graphing systems of inequalities on a coordinate plane. It contains 7 problems where students are asked to shade the region satisfying 3 given inequalities on a graph.
2. The problems involve skills like drawing lines representing linear equations, identifying the region between lines, and determining the intersecting area that satisfies all inequalities simultaneously.
3. Feedback is provided on the answers with notes on common mistakes like drawing lines as solid instead of dashed.
1. The document provides examples of graphing systems of inequalities on a coordinate plane. It contains 7 problems where students are asked to shade the region satisfying 3 given inequalities on a graph.
2. The problems involve skills like drawing lines representing linear equations, identifying the region between lines, and determining which inequalities define the shaded region.
3. Feedback is provided on the answers, noting correct graphing of lines and shading of regions, as well as common errors like drawing lines as solid instead of dashed.
This document contains a midterm exam for an engineering mathematics course. It consists of 4 problems:
1. Finding the general solution to two linear ODEs.
2. Finding the complementary and particular solutions for a given non-homogeneous linear ODE, and using them to solve an IVP.
3. Repeating steps from problem 2 for another given non-homogeneous linear ODE.
4. Modeling and solving an ODE describing the motion of a damped spring-mass system subject to an external force.
El documento proporciona una lista de enlaces repetidos a dos sitios web, http://edu.joshuatly.com/ y http://www.jaarsmtd.blogspot.com/, sin otra información contextual.
This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions. The highest number of marks for a single question is 8 for question 2, which involves calculating statistical measures like the mean, variance, and median of a data set. The document aims to evaluate students' mastery of various mathematical concepts by breaking down the solution steps and assigning partial marks.
This document contains the marking scheme for the Additional Mathematics Paper 1 for the 2010 Kedah Darul Aman National School Principals' Conference Trial SPM Examination. It provides the solutions, marking schemes and allocated marks for each question. The marking scheme has 25 questions and provides the working, method, answer and allocated marks for each. It is meant to guide examiners on how to mark the answers correctly according to the scheme.
This document contains 10 math problems with their solutions. Problem 1 asks to find values of m, n, and h for a given function g(x). The answer is m=-3, n=15, h=-3. Problem 2 asks to find values of x, the common difference d, and the sum of the first 12 terms of an arithmetic progression. The answer is x=11, d=14, sum=148.
The document provides instructions for a quiz competition with 12 multiple choice questions across 3 sections. Participants have 60 seconds to answer each question and can discuss with teammates. Correct answers score 5 marks within 60 seconds or 2 marks within 30 seconds. Incorrect answers score 0 marks but allow a second chance. The sections cover quadratic equations, indices/logarithms, and coordinate geometry/statistics.
The document provides information about answering techniques for the Additional Mathematics SPM Paper 1 exam in Malaysia, including:
1) It outlines the format of Paper 1 which is an objective test consisting of 25 multiple choice questions testing knowledge and application skills.
2) It discusses effective techniques for answering questions such as starting with easier questions, showing working, and presenting neat and precise answers.
3) It provides examples of different types of questions and mistakes to avoid when answering questions involving topics like functions, quadratic equations, graphs and progressions.
The document discusses various topics in coordinate geometry including: distance between two points, division of line segments, midpoints, the ratio theorem, areas of polygons, equations of straight lines, parallel and perpendicular lines, loci involving distance between two points. It also provides notes to candidates on how to approach questions involving diagrams, using formulas correctly, and not accepting solutions by scale drawing alone.
1. MARKING SCHEME
ADDITIONAL MATHEMATICS PAPER 2
SPM TRIAL EXAMINATION 2010
N0. SOLUTION MARKS
1 x = 10 − 2 y P1
K1 Eliminate x
y 2 + (10 − 2 y ) y = 24
y 2 − 10 y + 24 = 0 K1 Solve quadratic
( y − 4) ( y − 6) = 0 equation
y=4 or y=6
N1
x=2 or x = −2
N1
5
2
(a) k=6 P1
(b) Mid point 23 , 28 , 33 , 38 , 43 P1
(i) Mean
=
∑ fx =
1 × 23 + 4 × 28 + 7 × 33 + 5 × 38 + 3 × 43
K1 Use formula and
calculate
∑f 1+ 4 + 7 + 5 + 3
685
= = 34.25 N1
20
(ii) Varian
=
∑ fx2 − x 2
∑f K1 Use formula and
1 × 232 + 4 × 282 + 7 × 332 + 5 × 382 + 3 × 432 calculate
= − 34.252
20
24055
= − 34.252
20 N1
= 29.69
(iii) Median , m
1 1 K1 Use formula and
2N −F 2 (20) − 5 calculate
= L+ C = 30.5 + 5
fm 7
N1
= 34.07
8
2
2. N0. SOLUTION MARKS
3 1
y = x3 − x 2 + 2
3
(a) dy
= x2 − 2 x = 3 K1 Equate and solve
dx
x2 − 2 x − 3 = 0 quadratic
equation
( x + 1) ( x − 3) = 0
x = −1 , 3
2
x = −1 y=
3
x=3 y=2
2
−1, and ( 3, 2 ) N1 N1
3
(b) Equation of normals :
1
mnormal = −
3 K1 Use mnormal to form
2 1 1 equations
y− = − ( x + 1) y−2=− ( x − 3)
3 3 3
1 1 1 N1 N1
y=− x+ or equivalent y = − x+3 or equivalent
3 3 3
6
4
y
(a)
P1 Modulus sine
shape correct.
2 y = 3sin 2 x − 1
P1 Amplitude = 3
[ Maximum = 2
1
and Minimum =
-1]
O π π 3π 2π x P1 Two full cycle in
-1 2 2 0 ≤ x ≤ 2π π
3x
y = 1−
-2
2π P1 Shift down the
graph
3
3. N0. SOLUTION MARKS
4
3x
(b) 3sin 2 x − 1 = 1 −
2π
or N1 For equation
3x
y = 1−
2π
3x
Draw the straight line y = 1− K1 Sketch the
2π straight line
Number of solutions = 5. N1
7
5
(a) Common ratio, r=4 N1
(b) 1 K1
A6 = π ( 32 ) 1
T6 = ar = π ( 4 )
2 5 5
4 OR 4
N1
= 256π = 256π
(c)
S6 − S 2 K1 Use S6 or S2
1
(
π 46 − 1
1
) (
π 42 − 1 ) K1 Use S6 - S2
= 4 −4
4 −1 4 −1
N1
= 341.25π −1.25π
= 340π
6
4
4. N0. SOLUTION MARKS
6
(a) K1 for using vector
(i) uuu uuu uuu
r r r triangle for a(i) or
OD = OC + CD
a(ii)
= 6a + 12b
% % N1
(ii) uuu uuu uuu
r r r
AB = OB − OA
1 uuu uuu
r r
= OD − OA
2
= 3a + 6b − 3a
% % % N1
= 6b
%
OR
uuu 1 uuu
r r
AB = CD = 6b [ K1 N1 ]
2 %
(b)
uuur
AE
uuu uuu
r r
= AB + BE K1 for using vector
uuu
r
1 uuu
r triangle and BE
= 6b + h OD
% 2
= 6b + h ( 3a + 6b )
% % %
a + kb = 3ha + ( 6 + 6h ) b
K1
% % % %
k = 6 + 6h K1 for equating
3h = 1 coefficients
1
1 = 6 + 6 correctly
h= 3
3
=8 N1 N1
8
5
5. N0. SOLUTION MARKS
7
(a)
x 1 2 3 4 5 6
N1 6 correct
log10 y 0.65 0.87 1.08 1.30 1.52 1.74
values of log y
log10 y
(b) K1 Plot log10 y vs x
Correct axes &
uniform scale
N1 6 points plotted
correctly
N1 Line of best-fit
0.43
0 x
(c) log10 y = ( k log10 A ) x + log10 A P1
(i) x = 2.6 N1
(ii) y-intercept = log10 y K1
A = 2.69 N1
gradient = k log10 A K1
gradient
k=
log10 A *
= 0.51 N1
10
6
7. N0. SOLUTION MARKS
9
(a) 4 K1 Use ratio of
cos ∠ POQ =
10 trigonometry or
equivalent
∠ POQ = 1.16 rad.
N1
(b)
( 2π – 1.16 ) rad P1
PQ = 10 ( 2π – 1.16 ) K1 Use s = rθ
= 51.24 cm N1
(c)
P1
10 2 − 4 2
= 9.17 cm
1 K1
Area of trapezium POQR = ( 6 + 10 ) × 9.17 *
2
= 73.36 cm2
1 K1 Use formula
Area of sector POQ = (10) 2 (1.16)
2 1
A = r 2θ
= 58 cm2 2
Area of shaded region
= 73.36 – 58 K1
= 15.36 cm2 N1
10
8
8. N0. SOLUTION MARKS
10.
(a) Equation of str. line PQR :
1 K1
m= −
2
1 N1
y= − x+1
2
(b) 1 K1 solving
2x + 6 = − x+1
2 simultaneous
equation
P( –2, 2) N1
(c) 1( x) + 2(−2) 1( y ) + 2(2) K1 Use the ratio
=0 or =1
1+ 2 1+ 2 rule
R( 4, –1) N1
(d)
(i) 1 K1 Use distance
( x − 4) 2 + ( y + 1) 2 = ( x + 2) 2 + ( y − 2) 2
2 formula
4 [ x2 – 8x + 16 + y2 + 2y +1 ] = x2 + 4x + 4 + y2 – 4y +4
x2 + y2 – 12x + 4y + 15 = 0 N1
(ii) Substitute x = 0, y2 + 4y + 15 = 0 K1 Substitute x = 0
b2 – 4ac = (4)2 – 4(1)(15) 2
and use b – 4ac
= – 44 < 0 to make a
conclusion
⇒ No real root for y,
⇒ The locus does not intercept the y-axis. N1 if b2 – 4ac = -44
9
9. 10
N0. SOLUTION MARKS
11
(a)
µ = 80, σ = 12
65 − 80 X −µ
P ( X ≥ 65 ) = P ( Z ≥ ) K1 Use Z =
12 σ
= P ( Z ≥ − 1.25 )
= 1 – 0.1056 K1 Use 1 – Q(Z)
= 0.8944 N1
(b) 0.1056 × 4000 K1
= 422 or 423 N1
(c) 200 P1
= 0.05
4000
Q( Z ) = 0.05
Z = 1.645 K1 Find value of Z
m − 80 m−µ
= − 1.645 K1 Use
12 σ
K1 Use negative
value
m = 60.26 g N1
10
10
12. 10
N0. SOLUTION MARKS
14
(a) y ≥ 200 N1
x+y ≤ 800 N1
4x + y ≤ 1400 N1
(b) y
1000
4x + y = 1400
900
800
700
600 (200,600)
500
400
R
300
y = 200
200
100
x + y = 800
x
100 200 300 400 500 600 700 800 900
• At least one straight line is drawn correctly from inequalities K1
involving x and y.
N1
• All the three straight lines are drawn correctly
• Region is correctly shaded N1
(c)
(i)
650 N1
(ii)
Maximum point (200, 600) N1
Maximum profit = 20(200) + 6(600) K1
= RM 7600 N1
13
13. 10
N0. SOLUTION MARKS
15
(a) TQ2 = 92 + 62 – 2(9)(6)cos56o K1
TQ = 7.524 cm N1
(b) sin ∠QTR sin 56 0 K1
=
6 7.524
∠QTR = 41o 23’ N1
(c) 1 K1
42.28 = ( RS )(6 )sin 56 o
2
RS = 17
ST = 17 − 9 (or ST + 9 in formula of area) K1
= 8 cm N1
(d) 1
Area ∆ QTR = (9)(6) sin 56 0
2 K1
2
= 22.38 cm
Area of quadrilateral PQTS = 2(42.28) – 22.38 K1
= 62.18 cm2 N1
10
END OF MARKING SCHEME
14