MYP Mathematics, gradient of a line, perpendicular & parallel lines

MYP Mathematics, gradient of a line, perpendicular & parallel lines

How can we find the slope of the line which is perpendicular to the line y=8x+2 ?

The answer is from www.ibmaths4u.com

If two lines are perpendicular their slopes are negative reciprocals and if two lines are parallel have equal slopes.


Regarding your question about the gradient of the line which is perpendicular to the line y=8x+2,
Let be the required slope then we have the following relation

IB Mathematics HL Option Group Theory

The fundamental theorems and propositions on Group Theory from www.ibmaths4u.com

Re: IB Maths HL Option: Sets, Relations and Groups

1.  is a group under addition modulo n. With identity 0 and the inverse of  is the 

2. The number of elements of a group is its order 

3. The order of an element g, which is denoted by , in a group G is the smallest positive integer n such that 

4. Cyclic group  and g is called a generator of 

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then,  is a subgroup of G.

7. Let  be a cyclic group of order n.
Then  if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any  where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group 

17. A cyclic group of order n is isomorphic to the additive group  of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and . If , then  , where  is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all .

26. Let  be a finite group of order n.
Then 

27. Let  be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element  such that the order of this element equals the order of the group ().

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and 

31.Let G be a group of prime order. Then g is cyclic.

IB Maths HL Option Sets Cayley table

IB Mathematics HL Option: Sets, Relations and groups (Abelian Groups- Cayley table)


How can we show that the set under the addition modulo 3 is an Abelian group? and what is the order of each element?
The answer is here

http://www.ibmaths4u.com/ib-maths-hl-option-sets-cayley-table-t94.html

IB Maths HL Option Sets Isomorphism

IB Mathematics HL Option: Sets, Relations and groups (Groups-Isomorphism)


How can we show that the function defined by is an isomorphism between and

The answer is here

http://www.ibmaths4u.com/ib-maths-hl-option-sets-isomorphism-t95.html

IB Maths HL, Differentiation Quotient Rule

IB Mathematics HL – Derivatives, Differentiation Quotient Rule


How can we find the first derivative function of .  

The answer is here
http://www.ibmaths4u.com/ib-maths-hl-differentiation-quotient-rule-t162.html

Calculus

Calculus
Informal ideas of limit, continuity and convergence. Definition of derivative from first principles.
The derivative interpreted as a gradient function and as a rate of change. Finding equations of tangents and normals.
Identifying increasing and decreasing functions.
The second derivative. Higher derivatives.
Derivatives of xn , sin x , cos x , tan x , and ln x . Differentiation of sums and multiples of functions.
The product and quotient rules. The chain rule for composite functions.
Related rates of change. Implicit differentiation.
Derivatives of sec x , csc x , cot x , , loga x ,arcsin x , arccos x and arctan x .
Local maximum and minimum values. Optimization problems. Points of inflexion with zero and non-zero gradients.
Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ .
Indefinite integration as anti-differentiation. Indefinite integral of x^n , sin x , cos x and .
Anti-differentiation with a boundary condition to determine the constant of integration. Definite integrals.
Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.
Volumes of revolution about the x-axis or y-axis.
Kinematic problems involving displacement s, velocity v and acceleration a. Total distance travelled.
Integration by substitution. Integration by parts.

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Statistics and probability

Statistics and probability


Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
Grouped data: mid-interval values, interval width, upper and lower interval boundaries. Mean, variance, standard deviation.
Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
The probability of an event A. The complementary events A and A′ (not A). Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.
Combined events; the formula for P(A∪ B) . Mutually exclusive events. Conditional probability. Independent events, the definition. Use of Bayes’ theorem for a maximum of three events.
Concept of discrete and continuous random variables and their probability distributions. Definition and use of probability density functions.
Expected value (mean), mode, median, variance and standard deviation.
Binomial distribution, its mean and variance. Poisson distribution, its mean and variance.
Normal distribution, Properties of the normal distribution. Standardization of normal variables.

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Functions and equations

Outline of IB Mathematics Higher Level

Concept of function (domain, range, image),Odd and even functions,Composite functions, Identity function,One-to-one and many-to-one functions
Inverse function, The graph of a function, Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry,and consideration of domain and range.
Transformations of graphs: translations, stretches, reflections in the axes. The graph of the inverse function as a reflection in y = x. Rational function, exponential function and logarithmic function.
Polynomial functions and their graphs. The factor and remainder theorems. The fundamental theorem of algebra.
Solving quadratic equations using the quadratic formula. Use of the discriminant to determine the nature of the roots.
Solving polynomial equations both graphically and algebraically. Sum and product of the roots of polynomial equations.
Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
Graphical or algebraic methods, for simple polynomials up to degree 3. Use of technology for these and other functions.

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Outline of IB Mathematics Higher Level

Outline of IB Mathematics Higher Level

Big overall changes are that matrices have been removed. The two portfolio
pieces have been replaced by a more open-ended mathematical exploration. Applications are mentioned
much more than previously.

Algebra
Arithmetic sequences and series, sum of finite arithmetic series, geometric sequences and series, sum of finite and infinite geometric series,Sigma notation.
Exponents and logarithms,Laws of exponents, laws of logarithms,Change of base.
Counting principles, including permutations and combinations.
The binomial theorem
Proof by mathematical induction.
Complex numbers: the number i
terms real part, imaginary part, conjugate,modulus and argument.
Cartesian form z = a + ib
Sums, products and quotients of complex numbers
Modulus–argument (polar) form, The complex plane.Powers of complex numbers: de Moivre’s theorem.
nth roots of a complex number
Conjugate roots of polynomial equations with real coefficients.
Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

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