Mathematics Basics of the 2024 Class 199 Management Joint Entrance Examination (Part 1)

Introduction to management exams

• Management Comprehensive 200 points, time 3 hours
  • Mathematics: 75 points/25 questions, which is the core module that widens the gap
    • Problem solving questions: 15, choose one from 5
    • Condition adequacy judgment: 10, choose the answer based on two conditions
      • Condition 1 is sufficient, condition 2 is insufficient: A
      • Condition 1 is insufficient, Condition 2 is sufficient: B
      • Condition one is sufficient, condition two is sufficient: D
      • Condition 1 is insufficient, condition 2 is insufficient, and the combination is sufficient: C
      • Condition 1 is insufficient, condition 2 is insufficient, and the combination is insufficient: E
  • Logic: 60 points/30 questions, form, argumentation, analytical reasoning [mostly 40-50 points]
  • Writing: 65 points/2 essays, 600 words of argument effectiveness analysis, 700 words of argumentative essay [Mostly 35-45 points 】
• English II 100 points, 3 hours

Mathematics Overview

Arithmetic(Average 1-2 problems)

Rational and irrational numbers

Real numbers are divided into rational numbers and irrational numbers. The difference between the two lies in whether they can be written as the ratio of two integers.

• Rational numbers include integers and fractions
  • Integer: positive integer, 0, negative integer
  • Natural numbers: 0 and positive integers are collectively called natural numbers, that is, non-negative integers
• Irrational numbers: infinite non-cyclic decimals, such as e, log, etc.
• Often check that a formula contains both rational and irrational parts, and the whole is equal to 0
  • Problem-solving ideas: Combine the rational number parts and the irrational number parts, respectively, to 0

Prime and composite numbers

• Prime number/prime number: an integer greater than 1, a number that is not divisible by other positive integers except 1 and itself (that is, the only divisors are 1 and the prime number itself )
• Composite number: an integer greater than 1, a number that can be divisible by other positive integers in addition to 1 and itself (that is, the divisors include 1, itself and other divisors< /span>)
• Notice
  • 1 is neither a prime number nor a composite number, 2 is the only even prime number;A prime factor represents a number that is both a divisor of a number and a prime number
  • 8 major numbers within 20 : 2,3,5,7,11,13,17,19

odd even number

• Odd number: an integer that is not divisible by 2, expressed as 2k+1
• Even number: an integer divisible by 2, includes 0, is expressed as 2k

Divisors/divisors/multiples

• Characteristics of divisibility of common numbers
  • Numbers divisible by 2: single digits are 0, 2, 4, 6, 8
  • Numbers divisible by 3:The sum of each digit must be divisible by 3
  • Numbers divisible by 4: The last two digits must be divisible by 4
  • A number divisible by 5: the single digit is 0 or 5
  • Numbers divisible by 6: Divisible by 2 and 3 at the same time
  • Numbers divisible by 8: The last three digits must be divisible by 8
  • Numbers divisible by 9:The sum of each digit is divisible by 9
  • Numbers divisible by 10: the ones digit is 0
• Common divisor: The common divisor of several natural numbers is called the common divisor of these natural numbers. The greatest common divisor among the common divisors is called the greatest common divisor of these natural numbers.
• Least common multiple: a multiple common to several natural numbers (excluding 0)

  

Absolute value non-negativity

• The absolute value of a positive number is itself, a negative number is its opposite, and the absolute value of zero is still 0
• Geometric meaning: represents the distance value from the point corresponding to a real number a on the number axis to the origin 0. For example, |x-b| =a means that the distance from point b is a.
• Non-negative signs: absolute value, even power, open even root
• Common test scenarios: Containing absolute values ​​and even roots or even powers
• Problem-solving ideas: Just keep each item at 0

Absolute value of two models

• and model (the function curve is pan-shaped): there is a minimum value
  • |x-a|+|x-b|The geometric meaning represents the sum of the distances from x to points a and b on the number axis,When x has a minimum value between point a and point b , that is, the minimum distance value is |a-b|, if x is not between the two, it tends to positive infinity.
  • If f(x) = |x-a|+|x-b|+|x-c|, there is no maximum value,When x is between a and c and x=b There is a minimum value |c-a|
  • That is, the odd number of points takes the middle point to have the minimum value, and the even number of points takes the middle point between the two numbers to have the minimum value.
• Difference model (function curve is Z-shaped): there is both a minimum value and a maximum value
  • |x-a|-|x-b|The geometric meaning represents the difference between the distance from x to points a and b on the number axis.When x is outside point a and point b, there is a maximum The value |a-b| and the minimum distance value is -|a-b|

Triangle Inequality (Find the Maximum Value)

• |a + b| <= |a|+|b|; The equal sign holds condition ab>=0;
• |a - b| <= |a|+|b|; The equal sign holds condition ab<=0;
• Triangle inequality mainly examines the conditions for obtaining the equal sign, and elimination of parameters is the core

ratio and proportion

• Basic properties of proportion
  • The first and last terms of the ratio are multiplied or divided by the same number without 0 at the same time, and the ratio remains unchanged.
  • a:b=c:d <==> b:a=d:c<=> a:c=b:d<=>c:a=d:b
• proportional theorem
  • Combination ratio theorem: a/b = c/d <=> (a+b)/b=(c+d)/d
  • Ratio theorem: a/b = c/d <=> (a-b)/b=(c-d)/d
  • The sum ratio theorem: a/b = c/d <=> (a+b)/a-b=(c+d)/c-d
  • Theorem of equal proportions: a/b = c/d = e/f =a+c+e/b+d+f(b+d+f !=0)
• Problem-solving ideas
  • Fractional ratios are converted into whole number ratios
  • It is necessary to introduce the proportion coefficient k to turn the abstract proportion into a concrete numerical calculation solution.

    

Mean value theorem (mean inequality)

• Arithmetic mean: x1+x2+...+xn/n
• Geometric mean: Let npositive numbersx1...xn be called x=n times √ radical x1..xn
• Fundamental theorem
  • When x1...xn is a positive number, their arithmetic mean is not less than the geometric mean, that is, x1...xn/n>=n times√if and only if x1=...=xn, the equal sign holdsx1...xn under the radical sign,
  • If a>0,b>0, then a+b/2>=√The root sign is ab(a+b> =2√​​​​​​The root sign is ab), the equal sign is true if and only if a=b ( One is positive, two is definite, and three are equal)
    • The product is a constant value, then the sum has a minimum value
    • The sum is a constant value, then the product has a maximum value
  • a+1/a>=2(a>0) takes the minimum value 2 if and only if a=1, that is, for positive numbers, the sum of two numbers that are reciprocals of each other is not less than 2
  • Expand
    • a+b+c>=3*3 times √abc(a,b,c>0) under the root sign obtains the equal sign if and only if a=b=c
    • a+b+c+d>=4*4 times √abcd(a,b,c,d>0) under the root sign obtains the equal sign if and only if a=b=c=d

Proper equation (Average 1-2 scale)

factoring

• Concept: To convert a polynomial into the product of several integers, the essence is to convert the sum into a product, for example, x^2 + 3x +2 = (x+2)(x+1)
  • Note: Factorization must be decomposed within the specified range until it can no longer be decomposed.
• Common methods
  • Group decomposition method: for example am+bm+an+bn=(a+b)(m+n)
  • Square difference formula: a^2-b^2=(a-b)(a+b)
  • Perfect square formula: (a+_b)^2=a^2+-2ab+b^2
  • Formulas for sum of cubes and difference of cubes: a^3+-b^3=(a+-b)(a^2-+ab+b^2)
  • Three-term perfect sum of squares formula: (a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc
  • Complete cube sum formula: (a+b)^3=a^3+3a^2b+3ab^2+b^3
  • Extended formula: (a-b)^2+(b-c)^2+(a-c)^2=2[a^2+b^2+c^2-ab-ac-bc]
  • Cross multiplication method: such as x^2+5x+6=(x+2)(x+3)

factor theorem

• question mode
  • An algebraic expression is divisible by a certain expression
  • A certain expression is a factor of an algebraic expression
  • The algebraic expression contains a certain factor
• Problem solving mode
  • Let the factor be zero and find the value of x
  • The factor is a root, the root is brought into the original formula, and the algebraic formula is zero

algebraic simplification

Split-term cancellation method (summation of a sequence or simplification of fractions)

• Concept: The essence is a form of transformation of factorization, adding several fractions, the numerator of each fraction is 1, and then recombining it to eliminate some terms, and finally achieve the purpose of summation
• 1/n(n+k)=1/k(1/n-1/(n+k)), that is, 1/(big)(small)=1/big-small(1/small-1/big)

Set given function(average 1-2 scale)

gather

• nature
  • Deterministic: The element is in a set or not in a set, it cannot be ambiguous
  • Dissimilarity: elements in the set cannot be repeated
  • Disordered:There is no order requirement for the elements in the set
• Most of the operations in the set, including union, intersection, and complement, are represented by Venn diagrams.

linear function of one variable

• Definition: Generally, there are two variables x and y in a certain change process. If an x ​​is given and a y value is determined accordingly, then y is said to be a function of x, x is the independent variable, and y is the dependent variable.
• If the relationship between two variables x and y can be expressed as y=kx+b (k and b are constants, k!=0), then y is said to be a linear function of x. When b=0, then y =kx(k!=0) says that y is a proportional function of x. Note: linear functions are all straight lines

Quadratic function of one variable

• basic definition
  • y=ax^2+bx+c (a!=0, a, b, c are constants), symmetry axis x=-b/2a, vertex coordinates (-b/2a, 4ac-b^2/4a), The maximum value is obtained at the vertex (a>0 is the minimum value, a<0 means the function has the maximum value)
• expression
  • General formula: y=ax^2+bx+c
  • Vertex formula: y=a(x+b/2a)^2+4ac-b2/4a
  • Two radical formula: y=a(x-x1)(x-x2)
    • x1, x2 represent the abscissa of the intersection of the function and the x-axis/the zero point of the function/corresponding to the two roots of the quadratic equation of one variable
• Properties of quadratic functions of one variable
  • a determines the opening direction of the parabola, a>0 is upward, a<0 is downward
  • Symmetry axis-b/2a>0 means on the right side of the y-axis, <0 means on the left side of the y-axis, =0 means that the symmetry axis is the y-axis
  • c>0 means that the intersection point of the parabola and the y-axis is above the origin, =0 means that it passes the origin, <0 means that it is below the origin.
  • The discriminant b^2-4ac determines the number of intersections between the parabola and the x-axis, >0 means there are two, =0 means one and the vertex is on the x-axis, <0 means no intersection
  • If a+b+c=0, the parabola passes through the point (1,0); if a-b+c=0, the parabola passes through the point (-1,0)

Exponential and logarithmic functions

• exponential function
  • Definition: y=a^x (a>0 and a!=1) is called an exponential function, x is the independent variable< /span>
  • algorithm
    • a^m * a^n = a^m+n
    • a^m / a^n = a^m-n
    • (a^m)^n = a^mn
    • (ab)^m = a^m*b^m
    • a^0 = 1;a^-p = 1/a^p(a!=0)

      

      

• Logarithmic function
  • Definition: y=logaX(a>0 and a!=1,x>0), a is the base, x is a real number, y is the logarithm of x with a as the base
    • It is an inverse function of the exponential: a^m = b "="m=logab
  • algorithm
    • logaMN=logaM+logaN
    • logaM/N=logaM-logaN
    • logaM^n=nlogaM
    • log^Nb^M=M/Nlogab
  • Bottom changing formula
    • logN=logbN/logba
    • loga1=0;logaa=1;lg2+lg5=1;

Equation inequality (average 2-4 dimensions)

Discriminant of Equations and Roots

• A linear equation of one variable: it contains only one unknown, and the highest degree of the unknown is 1. The equation ax+b=0 (x is the unknown number, a!=0) is a standard linear equation of one variable. The solution of the equation is x=-b/ a;
• Quadratic equation of one variable: ax^2+bx+c=0(a!=0), a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term
  • The discriminant of the root b^2-4ac
    • >0, the equation has two unequal real roots, the root x=[-b±√(b²-4ac)]/2a
    • =0, there are two equal real roots, x=-b/2a
    • When <0, the equation has no real roots.
  • b^2-4ac natural language expression in the joint entrance examination
    • b^2-4ac=0
      • The equation has two equal real roots/multiple real roots
      • The function parabola has one and only one intersection/zero point with the x-axis
      • The parabola is tangent to the x-axis
      • The function is a perfect square formula
      • The maximum/minimum value of the function parabola is 0
      • There is only one x such that ax^2+bx+c=0 holds
    • b^2-4ac>0
      • The equation has two unequal real roots
      • ​The parabola intersects the x-axis/has two intersection points
      • A function or equation has two zeros
      • A straight line and a parabola have two intersection points
    • b^2-4ac<0
      • Equation has no real roots
      • The function parabola has no intersection/zero point with the x-axis
      • The parabola is separated from the x-axis
      • A straight line has no intersection with a parabola
      • The graph of the quadratic function is always above/below the x-axis
    • b^2-4ac>=0
      • The equation has two real roots
      • The equation has two positive roots
      • The equation has two negative roots
      • The equation has roots
• Whether to discuss the quadratic term coefficient a=0
  • If the question clearly states quadratic functions, quadratic equations, quadratic inequalities, parabolas, etc., the default is a!=0
  • If the knowledge representation of the topic is a function, equation, or inequality, then it is necessary to conduct a classified discussion on whether a is 0.

Vedic theorem

• Definition: The two roots of ax^2+bx+c=0(a!=0) are x1 and x2, then
  • x1+x2=-b/a
  • x1*x2=c/a
  • extended formula
    • x1^2+x2^2=(x1+x2)^2-2x1x2
    • x1^3+x2^3=(x1+x2)[(x1+x2)^2-3x1x2]
    • |x1-x2|=(√radical lower b^2-4ac)/|a|
      • |x1-x2|Different natural language expressions
        • The absolute value of the difference between the two roots of the equation
        • The distance between the two roots of the equation
        • The length of the function parabola intercepts the x-axis
        • The length of the base of the triangle enclosed by the function parabola and the two coordinate axes

distribution of roots of equations

• The distribution of roots is a comprehensive problem that requires the use of discriminants and Vedic theorem at the same time
  • The one-dimensional equation ax^2+bx+c=0(a!=0) has two positive roots
    • b^2-4ac>=0
    • x1+x2=-b/a>0
    • x1x2=c/a>0
  • The one-dimensional equation ax^2+bx+c=0(a!=0) has two negative roots
    • b^2-4ac>=0
    • x1+x2=-b/a<0
    • x1x2=c/a>0
  • The linear equation ax^2+bx+c=0(a!=0) has one negative root and one positive root
    • b^2-4ac>0
    • x1x2=c/a<0
  • The one-dimensional equation ax^2+bx+c=0 (a!=0) has two roots x1, x2 and satisfies m<x1<n, p<x2<q

    

     
    • f(m)f(n)<0
    • f(p)f(q)<0
  • The one-dimensional equation ax^2+bx+c=0(a!=0) has two roots x1, x2, one is greater than k, and the other is less than k
    • Then whether a>0 or a<0, af(k)<0

Fractional equation

• The idea is to convert fractional equations into integral equations
• Special solution: substitution method (holistic thinking), you also need to consider whether there is root augmentation, etc.

Solutions to systems of equations

Properties of Inequalities

Solutions to quadratic inequalities of one variable

• Basic concept: The inequality contains an unknown number, the degree of the unknown is 2, and both sides of the inequality are integers.
• Problem-solving ideas: Treat inequalities as equations, and then combine them with the parabola method (the idea of ​​combining numbers and shapes)
• Solution to Quadratic Inequality of One Variable

  

   
  • The boundary of the solution set of the inequality is the root/zero point/intersection point of the equation and the x-axis

Conclusion inequality(Difficulty)

• Problem-solving ideas: The key to solving inequalities containing absolute values ​​is to remove the absolute value sign in the formula. Commonly used methods are as follows
  • Square method:You need to ensure that both sides are non-negative to square
  • Definition Classification Discussion
    • |f(x)|>f(x),如|x/x-1|>x/x-1
      • When f(x)>=0|f(x)|=f(x)
      • f(x)<0时|f(x)|=-f(x)
    • When |f(x)|<g(x), please note that g(x) has its own domain
  • formula method
    • |f(x)|<a,a>0, then -a<f(x)<a
    • |f(x)|>a, a>0, then f(x)>a or f(x)<-a

Sequence(Average 2-3 steps)

Definitions related to sequence

• Definition: A sequence is a sequence of numbers arranged in a certain order.
  • The concepts of an and {an} are different. an represents the nth item of the sequence, and {an} represents the sequence.
  • The difference between sequence and set
    • The set elements are definite, unordered, and different from each other
    • Sequence elements are definite, ordered, and repeatable
  • anをSn(前n项和)关system
    • an=S1(n=1)
    • an=Sn-S(n-1)(n>=2)

arithmetic progression

• Definition: If a sequence starts from the second item, the difference between each item and its previous item is equal to the same constant
  • This constant is the tolerance d, which can be positive or negative, or 0. When it is 0, it represents a constant term.
  • Tongyu official
    • an = a1 +(n-1)d
    • an = am +(n-m)d 《=》d=an-am / n-m(n!=m)
• Expand
  • Constant sequence idea: There is only a single condition in the question. At this time, each sequence item is regarded as a constant item.
  • Arithmetic median/mean: If a, b, c are arithmetic sequences, then 2b=a+c
  • The sum formula of the first n terms of the arithmetic sequence
    • sn=n(a1+an)/2 "="sn=d/2n^2+(a1-d/2)n, that is, sn is about n Quadratic function (constant term is 0)
      • It can be concluded that d>0, Sn has a minimum value, and d<0 has a maximum value
    • sn=An^2+Bn+C, if C=0, it is an arithmetic sequence, if C!=0, it is an arithmetic sequence starting from the second term
  • The new sequence formed by the sum of consecutive items in the arithmetic sequence is still an arithmetic sequence, that is, Sn, S2n-Sn...
• Arithmetic Sequence Determination Method
  • Definition method: an-an-1=d, then it is an arithmetic sequence
  • General term formula method: an = Pn+q, then it is an arithmetic sequence
  • Middle term formula method: 2an+1=an + an+2, then it is an arithmetic sequence
  • Sum formula method of the first n terms: sn=An^2 + Bn, which is an arithmetic sequence

geometric sequence

• Definition: If a sequence starts from the second item, the ratio of each item to its previous item is equal to the same constant
  • This constant is the common ratio q, q!=0. When q=1, it represents a non-zero constant sequence.
  • Tongyu official
    • an = a1 * q^n-1
    • an = am * q^n-m
  • The sum formula of the first n terms of the geometric sequence
    • q!=1时, Sn=a1(1-q^n)/1-q=(a1-a1q^n)/1-q
    • When q=1, Sn=na1
    • When q!=0 and q!=1, then Sn=A-Aq^n《=》Sn=A+Bq^n(A+B=0)
  • Properties of geometric sequence:If m+n=p+k, then am*an=ap*ak, especially if m+n=2p, then am*an =ap^2
• Geometric Sequence Judgment Method
  • Definition method: an+1/an=q, then it is a geometric sequence
  • General term formula method: an = cq^n, then it is a geometric sequence
  • Middle term formula method: an+1^2=an*an+2, which is a geometric sequence
  • Sum formula method of first n terms: Sn=A-Aq^n, then it is a geometric sequence

sequence synthesis