GCSE Mathematics – Aiming for an A or Better Grade Criteria and exemplar examination questions to get a Grade A or A* in the following topics: 1. Surds 2. Recurring Decimals 3. Limits of Accuracy 4. Indices 5. Proportionality 6. Rearranging Formulae 7. Algebraic Fractions 8. Using Graphs 9. Quadratic Equations 10. Simultaneous Equations 11. Algebraic Proofs 12. Circle Theorems 13. Trigonometry – for triangles which are not right-angled 14. Vectors
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1.Logic ∃there exist ∀for all p⇒q p implies q / if p‚ then q p⇔q p if and only if q /p is equivalent to q / p and q are equivalent 2.Sets x∈A x belongs to A / x is an element (or a member) of A x∉A x does not belong to A / x is not an element (or a member) of A A⊂B A is contained in B / A is a subset of B A⊃B A contains B / B is a subset of A A∩B A cap B / A meet B / A intersection B A∪B A cup B / A join B / A union B A\B A minus B / the diference between A and B A×B A cross B / the
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CHAPTER 2 WHOLE NUMBERS What have we discussed? 1. The number 1‚ 2‚ 3… which use for counting are known as natural number. 2. If you add 1 to a natural number‚ we get its successor‚ if you subtract 1 from a natural number‚ you get its predecessor. 3. Every natural number has a successor. Every natural number except 1 has a predecessor. 4. If add the number zero to the collection of the natural numbers‚ we get the collection of whole numbers. Thus‚ the number 0‚ 1‚ 2‚ 3… from
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evidence and can never be proved to be true. It is therefore insufficient to argue that a mathematical statement is true simply by experiments and observations. For instance‚ Fermat (1601–1665) conjectured that when n is an integer greater than 2‚ the equation x n + y n = z n admits no solutions in positive integers. Many attempts by mathematicians in finding a counter-example (i.e. a set of positive integer solution) ended up in failure. Despite that‚ we cannot conclude that Fermat’s conjecture
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I. Description Excise tax is a tax applicable to certain specified goods or articles manufactured or produced in the Philippines for domestic sale or consumption or for any other disposition‚ and to things imported into the Philippines. Specific tax - an excise tax imposed on certain goods based on weight or volume capacity or any other physical unit of measurement. It applies to alcohol and alcohol products‚ tobacco and tobacco products‚ and petroleum products. Ad valorem tax - an excise tax
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Property 1.2 Wiener Process 1.3 2. 2.1 2.2 2.3 2.4 2.5 2.6 Taylor Expansion 2.7 3. Stochastic 3.1 3.2 SDE(Stochastic Differential Equation) 4. Stochastic 4.1 Stochastic integration 4.2 Ito Integral 4.3 Ito Integral 4.4 5. Ito’s Lemma 5.1 Stochastic 5.1.1 5.1.2 5.1.3 First Order Term Second Order Term Cross Product Terms “ ” – Ito Integral Riemann (Ordinary Differential Equation) (Chain rule) 5.2 Ito’s Lemma 6. 6.1 6.1.1 6.1.2 Closed-Form Solution Numerical Solution 2 . stochastic process Stochastic
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A P P E N D I X E S Tables and Data Sets A Areas under the Normal Curve B Student’s t Distribution C Data Set 1 — Real Estate D Data Set 2 — Major League Baseball E Data Set 3 — OECD F Data Set 4 — Northwest Ohio School Districts G Critical Values of the F Distribution H Critical Values of Chi-Square I Binomial Probability Distribution J Factors for Control Charts K Poisson Distribution L Table of Random Numbers M Wilcoxon T Values N Banking Data Set — Case 262 Appendixes Appendix A Areas
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controls. Also any changes in price‚ regulations‚ or politics may influence resource recovery since the political and human factors involved are government/political stability; crime control; industrial nationalization; perception and cultural value. 2. As an example of the complex interactions of resource usage‚ please describe the involvement of the varied types of mineral resources in the preparation of a loaf of bread. The involvement of varied mineral resources in the preparation of a loaf
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let f(x) be a quadratic polynomial such that that f(2)= -3 and f(-2)=21‚ then the co-efficient of x in f(x) is a. -3 b. 0 c. -6 d. 2 1. if f(x) =x3 +ax+b is divisible by (x-1) 2 ‚then the remainder obtained when f(x) is divided by (x+2) is ; a. 1 b . 0 c. 3 d. -10 3. the remainder when x1999 is divided
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of C1 and C2. Circle C3 has center A and radius r (therefore circles C1 and C3 are the same size). The point P’ (written P prime) is the intersection of C3 with OP. This is shown in the diagram below. Analytically find OP’ using r=1 and OP=2‚ OP=3‚ and OP=4: First‚ I created a line (see the dashed line in the above figure) between AP’ that creates the ΔAOP’. Because P’ is on the circumference of circle C3 and A is the center of circle C3‚ that means that AP’ is equal to the radius of C3
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