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Numbers and Algebra

Fundamental Algebra - Algebraic Equations (Linear, Quadratic)

Q.01

'Solve the following equations:'

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Q.02

'Determining coefficients from the imaginary solutions of an equation'

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Q.03

'Let x be the amount to be repaid at the end of each year, find the value of x such that the balance at the end of each year is zero.'

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Q.04

'Find the cubic equation with roots 1, 2, and 3.'

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Q.05

'Find the equations of the following lines:\n(1) A line passing through the point (6,-4) and parallel to the line 3x + y - 7 = 0\n(2) A line passing through the point (-1,3) and perpendicular to the line x - 5y + 2 = 0'

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Q.06

'Therefore, the value of the given expression is'

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Q.07

'12 \\times 3 \\cdot 2^{k-1}=3 \\cdot 2^{k}'

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Q.08

'Determine the types of solutions for the following quadratic equations. Where a is a constant. (1) 3x^2-5x+3=0 (2) 2x^2-(a+2)x+a-1=0 (3) x^2-(a-2)x+(9-2a)=0'

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Q.09

'For the equation x22(k3)x+4k=0 x^{2}-2(k-3) x+4 k=0 , determine the range of constant k k so that the equation has the following roots:'

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Q.10

'Solve the following system of simultaneous equations.'

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Q.11

'When 0 ≤ α < π/2, sin α is the y-coordinate of point P in figure [1], and 2β (0 ≤ 2β ≤ 2π) represents the angles of radii OQ and OR.\n∠ AOQ=∠ BOP= π/2 - α, hence\n2β₁ = π/2 - α, 2β₂ = 2π - (π/2 - α)\nTherefore, β₁ = π/4 - α/2, β₂ = 3/4π + α/2\n\nWhen π/2 ≤ α ≤ π, sin α is the y-coordinate of point P in figure [2], and 2β (0 ≤ 2β ≤ 2π) represents the angles of radii OQ and OR. ∠AOQ=∠BOP=α - π/2, thus 2β₁ = α - π/2, 2β₂ = 2π - (α - π/2)\n\nTherefore, β₁ = -π/4 + α/2, β₂ = 5/4π - α/2\nFor 0 ≤ α < π/2\nα + β₁/2 + β₂/3 = α + 1/2 ( π/4 - α/2 ) + 1/3 ( 3/4π + α/2 ) = 11/12α + 3/8π\n\nHence, 3/8π ≤ α + β₁/2 + β₂/3 < 5/6π\nFor π/2 ≤ α ≤ π\nα + β₁/2 + β₂/3 = α + 1/2 ( -π/4 + α/2 ) + 1/3 ( 5/4π - α/2 ) = 13/12α + 7/24π\nTherefore, 5/6π ≤ α + β₁/2 + β₂/3 ≤ 11/8π\nFrom (1) and (2), for 0 ≤ α ≤ π, 3/8π ≤ α + β₁/2 + β₂/3 ≤ 11/8π\ny = sin(α + β₁/2 + β₂/3) is maximized\nα + β₁/2 + β₂/3 = π/2, i.e., 11/12α + 3/8π = π/2, thus α = 3/22π and the value of y at this point is 1.'

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Q.12

'Determine the types of solutions for the following quadratic equations.'

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Q.13

'Practice solving the following equations and inequalities.'

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Q.14

'Find the values of the constant mm for which the quadratic equation x2mx+3m=0x^2 - mx + 3m = 0 has only integer solutions, and determine all such integer solutions.'

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Q.15

'Find the sum and product of the two solutions of the following quadratic equations.'

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Q.16

'Solve the following practice problem: Find the solutions to the quadratic equation.'

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Q.17

'Important Example 23 | Solutions of Quadratic Equations and the Value of Equations Let the two solutions of the quadratic equation x2+nx+p=0 x^{2}+n x+p=0 be a,b a, b , and let the two solutions of x2+nx+q=0 x^{2}+n x+q=0 be c,d c, d . Here, p,q p, q are integers, and n n is a real number. (1) Express (ca)(cb) (c-a)(c-b) in terms of p,q p, q . (2) Prove that (ac)(bd)(ad)(bc) (a-c)(b-d)(a-d)(b-c) is a perfect square (can be expressed as the square of an integer).'

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Q.18

'Calculate the probability p_{n+2} after (n+2) seconds using p_n and p_{n+1}.'

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Q.19

'(1) Let D be the discriminant of the quadratic equation x^2-k x+3 k-4=0 (1), then D=(-k)^2-4(3 k-4)=k^2-12 k+16. For the quadratic equation (1) to have complex solutions, the condition is D<0, so k^2-12 k+16<0.'

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Q.20

'Given the three lines, where a and b are constants: x-y+1=0, x-3y+5=0, ax+by=1. Prove that when these three lines pass through the same point, the three points (-1,1), (3,-1), (a, b) are collinear.'

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Q.21

'Determine the values of the constants a, b such that the equation (a-2b+4)x + (a-3b+7) = 0 becomes an identity with respect to x.'

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Q.22

'Solve trigonometric equations using sum and product formulas.'

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Q.23

'(1) Solve the equation: 6x261x+153=0 6x^{2}-61x+153=0 .'

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Q.24

'In sequence, 4x + 3y - 17 = 0, 3x - 4y + 6 = 0'

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Q.25

'Based on the following conditions, solve the problem.'

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Q.26

'Find the equation of the line passing through the point \\( (x_{1}, y_{1}) \\) and perpendicular to the \ x \ axis.'

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Q.27

'When a cubic equation with real coefficients ax^{3}+bx^{2}+cx+d=0 has an imaginary solution α, explain about the conjugate complex numbers and demonstrate their properties.'

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Q.28

'(1) Let alpha\\alpha and \eta\eta be the two solutions of the quadratic equation x2+3x6=0x^{2}+3x-6=0. Determine a new quadratic equation with solutions 2alpha+\eta2\\alpha+\eta and alpha+2\eta\\alpha+2\eta. (2) If alpha\\alpha and \eta\eta are the solutions of the quadratic equation x2+px+q=0x^{2}+px+q=0, and one of the quadratic equations with solutions alpha2\\alpha^{2} and \eta2\eta^{2} is x24x+36=0x^{2}-4x+36=0, find the values of the real constants pp and qq.'

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Q.29

'Find the values of α, β, γ that satisfy the following equations: \ \egin{\overlineray}{l} \\alpha^{3}=2 \\alpha^{2}+4, \eta^{3}=2 \eta^{2}+4, \\gamma^{3}=2 \\gamma^{2}+4 \\end{\overlineray} \'

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Q.30

'Find all integer values of mm for which the quadratic equation x2+(2m+5)x+m+3=0x^{2} + (2m + 5)x + m + 3 = 0 has integer solutions.'

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Q.31

'Practice: Starting from the origin O on the number line, throw a coin, moving 2 units in the positive direction if it lands heads, and 481 units in the positive direction if it lands tails. Let the probability of reaching point n be denoted as pn. Here, n is a natural number.\n(1) Determine the relationship between pn, pn-1, and pn-2 for n greater than or equal to 3.\n(2) Find the value of pn.'

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Q.32

'Translate the given text into multiple languages.'

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Q.33

'(1) (6, 4) (2) In order (4x + 3y -17 = 0, 3x - 4y + 6 = 0)'

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Q.34

'Mathematics II'

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Q.35

'Taking the reciprocal of both sides of the recurrence relation gives \\ \\frac{1}{a_{n+1}}=4+\\frac{3}{a_{n}} \\ Letting \\ \\frac{1}{a_{n}}=b_{n} \\ we have \\ b_{n+1}=4 + 3 b_{n} \\ Rearranging this gives \\ b_{n+1} + 2=3 (b_{n}+2) \\ Also, \\ b_{1}+2 = \\frac{1}{a_{1}} + 2 = \\frac{1}{\\frac{2}{3}} + 2 = 3 Therefore, the sequence \\ \\{b_{n}+2\\} forms a geometric sequence with first term 3 and common ratio 3, where \\ b_{n}+2 = 3 \\cdot 3^{n-1} \\ which implies \\ b_{n} = 3^{n} - 2 \\ Hence, \\ a_{n} = \\frac{1}{b_{n}} = \\frac{1}{3^{n} - 2}'

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Q.36

'(2) Let the x-coordinates of two intersection points A and B be α and β, respectively. By eliminating y from y=x^{2} and y=m(x+2), we get x^{2}-mx-2m=0. α and β are two different real solutions to this quadratic equation. Let D be the discriminant, then D=(-m)^{2}-4\\cdot 1\\cdot(-2m)=m(m+8). Since D>0, we have m(m+8)>0, which implies m<-8 and 0<m. Also, based on the relationship between the solutions and coefficients, α+β=m. Therefore, if we let the coordinates of the midpoint of line segment AB be (x, y), then x=(α+β)/2=m/2. Additionally, y=m(x+2). By eliminating m from (2) and (3), we get y=2x(x+2), which is y=2x^{2}+4x. Furthermore, from (1) and (2), we know that x<-4 and 0<x. Therefore, the sought trajectory is the part of the parabola y=2x^{2}+4x where x<-4 and 0<x.'

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Q.37

'When the equation of a circle is transformed, \\((x - 2)^2 + (y - 1)^2 = 2\\), the center of the circle C is at the point (2,1), and the radius is \\\sqrt{2}\.'

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Q.38

'Let 14k be a real number, consider the quadratic equation in x, x^{2}-kx+3k-4=0.'

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Q.39

'Consider the following conditions for integers a, b, c (*). ∫(x²+bx)dx = ∫(x²+ax)dx when integrated from a to c and b to c. (1) Express c² in terms of a, b when integers a, b, c satisfy (*) and a≠b. (2) Find all pairs of integers (a, b) that satisfy (*) and a<b when c=3. (3) Show that when integers a, b, c satisfy (*) and a≠b, c is a multiple of 3.'

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Q.40

'Find the general term of the sequence {an} defined by the following conditions.'

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Q.41

'Practice 2 curves y = 2x^{3} + 2x^{2} + a, y = x^{3} + 2x^{2} + 3x + b are tangent with the tangent line passing through the point (2,15), find the values of constants a, b and the equation of the tangent line.'

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Q.42

'Practice 39: x² = x + 3, that is x² - x - 3 = 0 has two solutions α, β (α < β), and from the relationship between the solutions and the coefficients we have α + β = 1, αβ = -3. Prove it. Also, prove that the recurrence formula is a_{n+2} - (α + β)a_{n+1} + αβa_{n} = 0. Finally, find a_{n}.'

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Q.43

'29- (2/3) x + 11/3\n30- (4 x^2 + 10 x + 5)'

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Q.44

'Please demonstrate that an equation with real coefficients of odd degree has at least one real solution.'

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Q.45

'A math problem set 62'

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Q.46

'Find the value of the following expression.'

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Q.47

'(3) From the sum of two numbers α+β=-4 and the product αβ=13, find the quadratic equation and its solutions.'

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Q.48

'Classify the number of solutions of the equation sin ^{2} \\theta-\\cos \\theta+a=0 (0 ≤θ<2π) based on the value of the constant a.'

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Q.49

'Let the first term be a and the common difference be d, then the 10th term is 1 and the 16th term is 5, so a+9d=1, a+15d=5. Solving these equations gives a=-5, d=2/3. Let Sn denote the sum of the terms from the first term to the nth term. Therefore, S30=1/2*30{2*(-5)+(30-1)*2/3}=140, and S14=1/2*14{2*(-5)+(14-1)*2/3}=-28/3. Hence, S=S30-S14=140-(-28/3)=448/3'

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Q.50

'Practice 38: Transform the recurrence relation into a_{n+2} + 4a_{n+1} = -4(a_{n+1} + 4a_{n}). Therefore, the sequence {a_{n+1} + 4a_{n}} has an initial term of a_{2} + 4a_{1} = 9, a common ratio of -4, prove it is a geometric sequence. Also, demonstrate that a_{n+1} + 4a_{n} = 9·(-4)^{n-1}. Finally, find the value of a_{n}.'

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Q.51

'For the quadratic equation ax2+bx+c=0 a x^{2}+b x+c=0 with two solutions alpha,\eta \\alpha, \eta , we have alpha+\eta=fracba \\alpha+\eta=-\\frac{b}{a} and alpha\eta=fracca \\alpha \eta=\\frac{c}{a} .'

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Q.52

'Let D1D_{1}, D2D_{2}, and D3D_{3} be the discriminants of the three equations, respectively. Determine the range of values for a that make each discriminant have complex roots. Use the discriminant results based on the equations.'

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Q.53

'Three real numbers a, b, c form an arithmetic progression in the order a, b, c, and a geometric progression in the order b, c, a. When the product of a, b, and c is 125, find the values of a, b, and c.'

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Q.54

'From the equation of C2, we have (x-3)^2 + (y-a)^2 = a^2 - 4a + 5. Find the conditions for this equation to intersect the line y=x+1 at two distinct points.'

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Q.55

'Math II Practice Booklet page 61'

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Q.56

'x^{2}+y^{2}=10\n(3) y=2 x-8\n5 x^{2}-32 x+54=0\nLet the discriminant of this quadratic equation be D\nfrac{D}{4}=(-16)^{2}-5 cdot 54=-14\nSince D<0, this quadratic equation has no real solutions. Therefore, Circle (A) and Line (3) do not have any intersection points.'

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Q.57

'y = bx - a² / 4'

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Q.58

'When a = 1, the equation for C₂ is x^2-6x+y^2-2y+8=0. Now, let k be a constant and consider the following equation: k(x^2+y^2-4)+x^2-6x+y^2-2y+8=0. Find the conditions for this to form a line.'

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Q.59

'The condition for both solutions to be greater than 4 is D>0 and (α-4)+ (β-4)>0 and (α-4)(β-4)>0'

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Q.60

'Since the midpoint of the line segment PQ is (3+p)/2, (4+q)/2 lies on the line ℓ, therefore'

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Q.61

'A cubic equation with repeated roots'

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Q.62

'Solve the following 4th degree equation: x^{4}=4'

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Q.63

'Prove that at least one of \ a, b, c \ is 1 when \ a+b+c=1, \\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=1 \.'

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Q.64

'Please explain the solutions to the equation (x-3)^{2}(x+2)=0 and its repeated root.'

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Q.65

'Let Q(x)Q(x) be a quadratic polynomial. The polynomial P(x)P(x) cannot divide Q(x)Q(x), but {P(x)P(x)}^2 can be divided by Q(x)Q(x). Prove that the quadratic equation Q(x)=0Q(x)=0 has a repeated root.'

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Q.66

'Since the real number is positive, it follows that x2>0x-2>0 and 3x>03-x>0. Therefore, 2<x<32<x<3. And since 2log4(3x)=log22(3x)2=log2(3x)2\\log_{4}(3-x)=\\log_{2^{2}}(3-x)^{2}=\\log_{2}(3-x).'

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Q.67

'Interpret this as solving for b in the 2nd degree equation'

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Q.68

'Using the relationship between the roots and coefficients, find the following value.'

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Q.69

'Important Example 27 | Solutions of Two Equations'

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Q.70

'For vegetable A, each contains 8g of nutrient x₁, 4g of nutrient x₂, and 2g of nutrient x₃, while for vegetable B, each contains 4g of nutrient x₁, 6g of nutrient x₂, and 6g of nutrient x₃. Selecting some of each of these two types of vegetables to mix and make vegetable juice. The goal is to have nutrient x₁ at least 42g, nutrient x₂ at least 48g, and nutrient x₃ at least 30g in the selected vegetables. When making juice with as few vegetables of type A and B as possible, the combination of the number of vegetable A, a, and vegetable B, b, is'

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Q.71

'For all natural numbers n, derive cn + 1 by utilizing an + bn + cn = 1.'

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Q.72

'Lake 37 book p. 119 Finding the equation of the circle in the form x^2+y^2+lx+my+n=0. The circle passes through point A(8,5), so 8^2+5^2+8l+5m+n=0; passes through point B(1,-2), so 1^2+(-2)^2+l-2m+n=0; passes through point C(9,2), so 9^2+2^2+9l+2m+n=0. Simplifying gives 8l+5m+n=-89, l-2m+n=-5, 9l+2m+n=-85. Solving these equations gives l=-8, m=-4, n=-5. Therefore, the required equation is x^2+y^2-8x-4y-5=0. Another approach is that the circumcenter of triangle ABC is the center of the desired circle. The equation of the perpendicular bisector of AB is y-3/2=-1(x-9/2), hence y=-x+6. It can also be verified by substituting x=y=0 into 4(x+5)^2+(y-4)^2=r^2. From (1)-(2) ÷ 7 we get l+m=-12, from (1)-(3) we get l-3m=-4, thus 4m=-16, etc.'

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Q.73

'Example 4 | Three numbers forming an arithmetic progression\nThere are three numbers forming an arithmetic progression, with a sum of 18 and a product of 162. Find these three numbers.'

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Q.74

'Let the volume of a rectangular parallelepiped be denoted by V, where V = x y z is derived from equations (2), (3), (4), x, y, z are the roots of the cubic equation t^3 - 5 t^2 + 8 t - V = 0. The condition for the existence of positive numbers x, y, z is that equation (5) has three positive roots.'

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Q.75

'Example 42 | Equation of a Line Passing Through a Fixed Point\nLet k be a constant. The line (2k+1)x+(k-4)y-7k+1=0 passes through a fixed point regardless of the value of k. The coordinates of that fixed point are denoted by A□. Also, when the slope of this line is 1/3, the value of k is represented by B□.\n[Fukuoka University]'

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Q.76

'a³ - a² - b = 0 or 9a + 27b - 1 = 0 where a ≠ 1/3'

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Q.77

'The base is a positive number that is not equal to 1.'

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Q.78

'Find the condition for one of the equations to have complex roots.'

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Q.79

'Comprehensive Exercise'

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Q.80

'Find the quadratic equation using the sum and product of two numbers.'

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Q.81

'Example 18 The value of the symmetric equation (2)\nFor the two roots alpha,\eta \\alpha, \eta of the 2nd degree equation 2x2+4x+3=0 2 x^{2}+4 x+3=0 , find the values of the following expressions.\n(1) alpha5+\eta5\\alpha^{5}+\eta^{5}\n(2) (alpha1)4+(\eta1)4(\\alpha-1)^{4}+(\eta-1)^{4}'

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Q.82

'Example 38 Recurrence Relation Among Adjacent 3 Elements (2)'

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Q.83

'Show the solutions and discriminant of the quadratic equation ax² + bx + c = 0 with real coefficients.'

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Q.84

'Find the equation when the line passing through the intersection points of 2x - y - 1 = 0 and x + 5y - 17 = 0 becomes parallel to 4x + 3y - 6 = 0.'

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Q.85

'(1) Find the quadratic equation from the sum of two numbers α+β=7 and the product αβ=3, and solve for the roots.'

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Q.86

'Solve the following problem.'

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Q.87

'(1) Find the roots of a quadratic equation using the sum and product of two numbers.'

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Q.88

'Example 17 | Value of Symmetric Expressions (1)\nSecond-degree equation x^{2}+3x+4=0\n(1) \\alpha^{2}\eta+\\alpha\eta^{2}\n(4) \\alpha^{3}+\eta^{3}\nLet the two solutions of the equation be \\alpha, \eta, then find the values of the following expressions.\n(2) \\alpha^{2}+\eta^{2}\n(3) (\\alpha-\eta)^{2}\n(5) \\frac{\eta}{\\alpha}+\\frac{\\alpha}{\eta}\n(6) \\frac{\eta}{\\alpha-1}+\\frac{\\alpha}{\eta-1}'

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Q.89

'In (2), the relationship between the roots and coefficients is α+β=-p and αβ=q. In x²+qx+p=0, the relationship between the roots and coefficients is α(β-2)+β(α-2)=-q, α(β-2)+β(α-2)=p, and 2αβ-2(α+β)=-q. Therefore, 2q+2p=-q, which implies 2p+3q=0. From (2), we get αβ+αβ-2(α+β)+4=p, and from (1), we get q(q+2p+4)=p, thus p=-3/2q. Substituting (6) into (5) and simplifying, we get 4q²-11q=0, which leads to q(4q-11)=0. Solving this gives q=0 and 11/4. When q=0, from (6) we find p=0. In this case, α=0 and β=0, which contradicts the assumption that α and β are not equal. When q=11/4, from (6) we find p=-33/8.'

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Q.90

'Let the discriminants of the equations (1) and (2) be denoted as D1 and D2.'

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Q.91

'Simplify the equation to'

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Q.92

'Since the point (1,2) lies on the line (3), we have a+2b=1'

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Q.93

'Find the range of possible values for y to satisfy y=-2x+3 for x within the range -3 ≤ x ≤ 2.'

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Q.94

'Prove the following equation:\n\na^3 + b^3 + c^3 = -3(a + b)(b + c)(c + a) \nwhere a + b + c = 0.'

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Q.95

'Show the relationship between the solutions of a cubic equation and the coefficients.'

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Q.96

'Determine the value of the constant k that satisfies the following conditions:\n(1) One solution is twice the other solution\n(2) One solution is squared of the other solution'

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Q.97

'For real numbers a, b, let f(x) = x^3 - 3ax + b. Let M be the maximum value of |f(x)| for -1≤x≤1.'

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Q.98

'Let the coordinates of point P be (a, b). The x-coordinate of the points where the line with slope m passing through point P intersects curve C is the real solution of the equation x^3 - x = m(x-a) + b. When this equation has three distinct real solutions, the line ℓ intersects curve C at three distinct points.'

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Q.99

'Practice'

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Q.00

'Find the value of k that satisfies the following conditions.'

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Q.01

'Find the equation of the line passing through two distinct points \\( (x_{1}, y_{1}), (x_{2}, y_{2}) \\).'

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Q.02

'Assuming the given sequence is an arithmetic progression with the first term as 5 and common difference as -7. If the nth term of this arithmetic progression is -1010, then 5+(n-1)×(-7)=-1010. Solving this equation gives 7n=1022, which means n=146 (a natural number). Therefore, the given sequence can be an arithmetic progression. Additionally, -1010 is the 146th term.'

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Q.03

'Given mathematical text converted to multiple languages.'

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Q.04

'Practice 39⇒This book p.91\\ From the relationship between the solutions and coefficients of a cubic equation \\ α+β+γ=2, \\αβ+βγ+γα=0, αβγ=4\\'

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Q.05

'When the equation x2+y24kx+(6k2)y+14k28k+1=0x^{2}+y^{2}-4 k x+(6 k-2) y+14 k^{2}-8 k+1=0 represents a circle\n(1) Find the range of values for the constant kk.\n(2) When kk varies within this range, find the trajectory of the center of the circle.'

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Q.06

'For all values of x, y, and z that satisfy x-2y+z=4 and 2x+y-3z=-7, the constants a, b, and c need to be determined such that ax^2+2by^2+3cz^2=18 holds true.'

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Q.07

'Find the values of constants a, b, and c that satisfy the equations x - 2y + z = 4 and 2x + y - 3z = -7 for all values of x, y, and z that satisfy those equations.'

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Q.08

'Determine the values of constants a, b, c so that the equation 3x^2-2x-1=a(x+1)^2+b(x+1)+c is an identity in terms of x.'

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Q.09

'Find the number of distinct real solutions to the following cubic equations.'

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Q.10

'Confirm logarithmic equations and conditions of exponents'

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Q.11

'When the cubic equation x3(a+2)x+2(a2)=0 x^{3}-(a+2) x+2(a-2)=0 has a double root, find the value of the constant a a .'

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Q.12

'Find the equations of the following lines.'

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Q.13

'Find the two numbers that have the sum and product as follows:'

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Q.14

'Develop 52: Proof problem regarding the solutions of a quadratic equation'

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Q.15

'Determine the types of solutions for the following quadratic equations. Note that k in (4) is a constant.'

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Q.16

'Find the conditions under which the given polynomial P(x) = 5x^3 - 4x^2 + ax - 2 is divisible by x = 2 and x = -1.'

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Q.17

'Determine the types of solutions for the following quadratic equations. Note that k in (4) is a constant. (1) x^{2}-5x+3=0 (2) 4x^{2}+28x+49=0 (3) 13x^{2}-12x+3=0 (4) x^{2}+6x+3k=0'

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Q.18

'When the quadratic equation x2+2(a+3)xa+3=0 x^{2}+2(a+3) x-a+3=0 has two different solutions both greater than 1, find the range of values for the constant aa.'

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Q.19

'For the quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 with two solutions alpha,\eta \\alpha, \eta and discriminant D D :\n1. alpha,\eta \\alpha, \eta are two distinct positive solutions LongleftrightarrowD>0 \\Longleftrightarrow D > 0 and alpha+\eta>0 \\alpha + \eta > 0 and alpha\eta>0 \\alpha \eta > 0 \n2. alpha,\eta \\alpha, \eta are two distinct negative solutions LongleftrightarrowD>0 \\Longleftrightarrow D > 0 and alpha+\eta<0 \\alpha + \eta < 0 and alpha\eta>0 \\alpha \eta > 0 \n3. alpha,\eta \\alpha, \eta are solutions with opposite signs quadLongleftrightarrowalpha\eta<0 \\quad \\Longleftrightarrow \\alpha \eta < 0 '

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Q.20

'Consider the signs of the differences \\\alpha-k, \eta-k\ of the real roots \\\alpha, \eta\ of a quadratic equation and a real number \k\\n\nFocus on the signs of the sum \\( (\\alpha-k)+(\eta-k) \\) and the product \\( (\\alpha-k)(\eta-k) \\)'

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Q.21

'Find the number of distinct real solutions for the following cubic equations.'

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Q.22

'Determine the range of values for the constant mm such that the quadratic equation x2+2mx+6m=0x^{2}+2mx+6-m=0 has two distinct real roots both greater than 1.'

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Q.23

'Determine the range of values for the constant m so that the quadratic equation x2+2(m1)x+2m25m3=0 x^2 + 2(m-1)x + 2m^2 - 5m - 3 = 0 satisfies the following conditions: (1) Has two positive roots. (2) Has two different negative roots. (3) Has roots with opposite signs.'

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Q.24

'Let a, b be constants. Find the values of a and b when the polynomial x^3-x^2+ax+b is divisible by the polynomial x^2+x+1.'

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Q.25

'Find the first term and the common ratio of a geometric series such that the sum of the first three terms is -7 and the sum of terms from the third to the fifth is -63.'

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Q.26

'Higher degree equation: Find the value of the constant aa and the other root of the equation x3ax2+(3a1)x24=0x^3-ax^2+(3a-1)x-24=0, given that one of the roots is x=2x=2.'

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Q.27

'Identity with Condition'

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Q.28

'For the quadratic equation 4x2+4(m+2)x+9m=04 x^{2}+4(m+2) x+9 m=0, answer the following questions.\n(1) Determine the range of values for the constant mm when the equation has two complex solutions.\n(2) Find the values of the constant mm and the repeated root when the equation has a repeated root.'

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Q.29

'Let α and β be the two solutions of the quadratic equation x^{2}-3x+4=0. Find the values of the following expressions:'

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Q.30

'Show the formula for solving the quadratic equation ax^2 + bx + c = 0 and find its roots.'

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Q.31

'Solve the following equations for 0 ≤ θ < 2π: (1) 2cos²θ - √3sinθ + 1 = 0 (2) 2sin²θ + cosθ - 2 = 0'

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Q.32

'Determine the values of the constants a, b, and c such that the equation x^2+2x-1=a(x+3)^2+b(x+3)+c holds as an identity with respect to x.'

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Q.33

'If the three solutions of the cubic equation x3+x2+x+3=0x^{3}+x^{2}+x+3=0 are denoted as α,β,γα, β, γ, find the values of α2+β2+γ2α^{2}+β^{2}+γ^{2} and α3+β3+γ3α^{3}+β^{3}+γ^{3}.'

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Q.34

'When the cubic equation x3+(a+1)x2a=0x^{3} + (a+1)x^{2} - a = 0 has a repeated root, find the value of the constant aa.'

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Q.35

'Find the sum and product of the two solutions to the following quadratic equations.\n(1) x24x3=0 x^{2}-4 x-3=0 \n(2)2x23x+6=0 2 x^{2}-3 x+6=0 \n(3) 3x2=54x 3 x^{2}=5-4 x '

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Q.36

'Find the equations of the following lines:'

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Q.37

'Basic 62: Solving Higher Degree Equations (2) - Utilizing the Factor Theorem'

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Q.38

'Solution: Using the formula x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = -3, c = -3. Answer: x = 3 or x = -1.'

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Q.39

''

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Q.40

'Determine the types of solutions for the following quadratic equations. Here, k in equation (4) is a constant.'

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Q.41

'Find the solution and answer for 2x^{2}+4x-1=0.'

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Q.42

'176 (1) y = 4x - 9 (2) y = -2x, y = 6x - 16'

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Q.43

'Show the relationship between the solutions of a quadratic equation and its coefficients. Let the solutions of the quadratic equation be α and β for ax^2+bx+c=0, then using the formula for solutions, demonstrate the following relationships:\n\n1. Sum of solutions α+β\n2. Product of solutions αβ'

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Q.44

'Find the general term of the sequence defined by the recurrence relation a1=1,an+1=2an+3na_{1}=1, a_{n+1}=2a_n+3^n.'

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Q.45

'When S_{2}=2 S_{1} \\quad-\\frac{1}{6}(m+3)^{3}=9 \\text{, that means} \\quad(m+3)^{3}=-54 \\text{, since } m \\text{ is a real number} \\quad m=-3-3 \\sqrt[3]{2}'

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Q.46

'If the two solutions of the quadratic equation x^2 + 2x - 4 = 0 are α and β, then what is the quadratic equation with solutions α + 2 and β + 2?'

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Q.47

'Find the values of x and y when the identity (k-1) x + (3-2k) y + 4k-7 = 0 holds for all values of k.'

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Q.48

'Find the range of values for the constant aa when the quadratic equation x2ax+4=0x^{2}-a x+4=0 has two distinct solutions, both of which are less than 3.'

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Q.49

'Extension 53: Integer solutions of quadratic equations (using the relationship between solutions and coefficients)'

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Q.50

'Solve the high degree equation x^{3}-4 x^{2}+2 x+4=0.'

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Q.51

'Basic Example 62 Determine Coefficients of Degree 64 Polynomial (1) ... Conditions for Real Solutions 3rd degree equation x3+ax217x+b=0x^{3}+ax^{2}-17x+b=0 has -1 and -3 as solutions. (1) Find the values of constants aa and bb. (2) Find the other solutions to this equation.'

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Q.52

'When the quadratic equation x22ax+3a2=0 x^{2}-2 a x+3 a-2=0 has two distinct positive solutions, find the range of values for the constant a a .'

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Q.53

'Consider the polynomial P(x)=x^{3}-2 x^{2}+qx+2r. When the solutions of the cubic equation P(x)=0 are -2 and two natural numbers α, β(α<β), find the values of α, β, q, and r.'

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Q.54

'When the quadratic equation x^2+2mx+15=0 has the following roots, find the value of the constant m and the two roots.'

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Q.55

'Find the formula for the solutions of the quadratic equation ax^{2}+bx+c=0.'

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Q.56

'Find the value of the constant m m and the two solutions when the two solutions of the quadratic equation 3x2+6x+m=0 3 x^{2}+6 x+m=0 satisfy the following conditions:'

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Q.57

'Find the general term of the sequence {an} determined by the 25 conditions. (1) a1=1, an+1=2an-3 (2) a1=1, 2an+1-an+2=0'

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Q.58

'Consider the cubic equation x3+(a5)x2+(a+8)x6a4=0x^{3}+(a-5)x^{2}+(a+8)x-6a-4=0.'

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Q.59

"Find information about 'high-degree equations' based on the following table."

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Q.60

'\ 67 a=1,10 \'

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Q.61

'Solve the following quadratic equations.'

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Q.62

'Find the value of the constant m and the two solutions of the quadratic equation 3x2+6x+m=03x^{2}+6x+m=0 such that it satisfies the following conditions: (1) One solution is three times the other. (2) The ratio of the two solutions is 2:3.'

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Q.63

'Solution to a quadratic equation'

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Q.64

'Find the equation of a circle passing through three points.'

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Q.65

'Development 69: Solution of Higher Order Equations (3)'

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Q.66

'(5) \ x= \\pm 2, \\pm \\frac{1}{2} i\'

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Q.67

'Proving the equation A=B in 3 ways\n\nThe equation A=B may have conditions attached to it, but basically it is an identity. There are three styles of proving the equation as follows:\n\n(1) Comparing both sides, transforming the more complex side to derive the simpler side.\n\nA=⋯⋯ Transformation ⋯⋯ = B\n(or B =⋯⋯ Transformation ⋯⋯ = A)\n\nTherefore A = B\n\n(2) Transforming both sides separately to obtain the same expression C.\n\nA=⋯⋯ Transformation ⋯⋯ = C\n\nB =⋯⋯ Transformation ⋯⋯ = C\n\nTherefore A = B\n\n(3) Transforming A - B to show A - B = 0.\n\nA - B =⋯⋯ Transformation ⋯⋯ = 0\n\nTherefore A = B'

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Q.68

'Let TR be real numbers, and let the equation x ^ {3}-2 x ^ {2} + ax + b = 0 have x = 2 + i as a root. Find the values of a, b and all the roots of the equation.'

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Q.69

'Find the range of values for the constant aa when the equation aleft(x2x+1right)=1+2x2x2a\\left(x^{2}-x+1\\right)=1+2 x-2 x^{2} has real solutions.'

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Q.70

'Find the equations of the following lines:\n(1) Passing through the point (3, 0) with a slope of 2\n(2) Passing through the point (-1, 4) with a slope of -3\n(3) Passing through the point (3, 2) and perpendicular to the x-axis\n(4) Passing through the point (1, -2) and parallel to the x-axis'

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Q.71

'The function y=4x+4x2x+121xy=4^{x}+4^{-x}-2^{x+1}-2^{1-x} takes the minimum value bb at x=ax=a. Find the value of a+b|a+b|.'

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Q.72

'46 (1) 6x^2 + x - 12 = 0 (2) 4x^2 - 12x + 7 = 0 (3) 3x^2 - 4x + 3 = 0'

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Q.73

'Determine the values of constants a, b, and c such that the following equation is an identity in x: (1) (a+b-3) x^{2} + (2a-b) x + 3b - c = 0'

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Q.74

'Investigate the conditions for a cubic equation to have repeated roots'

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Q.75

'Substitute the third equation into the first equation to obtain the following equation: a^{2} + (-7a + 25)^{2} = 25. Simplifying, we get the following quadratic equation: a^{2} - 7a + 12 = 0. Therefore, we get the following solutions: (a - 3)(a - 4) = 0, so a = 3, 4. Substituting these values into the third equation, we get the following: when a = 3, b = 4; when a = 4, b = -3. Thus, the equations of the tangents are as follows: 3x + 4y = 25, 4x - 3y = 25'

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Q.76

'Basic 43: Value of two solutions of a symmetric equation'

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Q.77

'When the maximum value of the function f(x) = a x^3 + 3 a x^2 + b(-1 ≤ x ≤ 2) is 10, and the minimum value is -10, find the values of constants a, b.'

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Q.78

'When S_{2}=2 S_{1}, \\frac{1}{6}(m+3)^{3}=9, that is (m+3)^{3}=54. Since m is a real number, m=-3+3 \\sqrt[3]{2}'

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Q.79

'When k=0, there is one real solution; when k=-1, there is a repeated root; when -1<k<0 or 0<k, there are two distinct real solutions; when k<-1, there are two distinct imaginary solutions.'

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Q.80

'Basic 42: Sum and product of the two solutions to a quadratic equation'

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Q.81

'Let k be a constant. Determine the types of solutions of the equation kx^2 + 4x - 4 = 0.'

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Q.82

'Find the values of m such that the lines l1 and l2 are parallel or perpendicular.'

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Q.83

'Find the number of distinct real solutions to the following cubic equations:\n(1) -x^{3}+3x^{2}-1=0\n(2) x^{3}-3x^{2}+3x+1=0'

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Q.84

'Find the values of x and y such that (k+2)x-(1-k)y-k-5=0 holds for any value of k.'

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Q.85

'Solve the following equations and inequalities.'

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Q.86

'The condition for having only imaginary solutions is'

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Q.87

'Find the solutions of the quadratic equation x^2=k. Here, k is any real number.'

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Q.88

'Let the first term of a geometric sequence, with a common ratio r, be a, with the second term being 4 and the sum of the terms from the first to the third being 21. Therefore, we have a= and common ratio r=.'

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Q.89

"Considering q, r as real numbers, let's look at the polynomial P(x)=x^{3}-2 x^{2}+q x+2 r. If the solutions of the 333rd equation P(x)=0 are -2 and two natural numbers \\( \\alpha, \eta(\\alpha<\eta) \\), find \ \\alpha, \eta \ and \ q, r \. [Similar to center test]"

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Q.90

'Solve the following equations and inequalities.'

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Q.91

'The solution to the example 3x^2+3x+1=0 is'

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Q.92

'Development 54: Range of existence of solutions of a quadratic equation (2)'

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Q.93

'Solve the following equations:'

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Q.94

'A and B solved the same quadratic equation in terms of x. A mistakenly obtained the coefficient of x² as 26-2/3, with a solution of 1. B mistakenly obtained the constant term as -1/3, with a solution of 1/2. Find the solutions to the original correct quadratic equation.'

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Q.95

'Find the range of values for the constant aa when the cubic equation x33a2x+4a=0x^{3}-3 a^{2} x+4 a=0 has three distinct real roots.'

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Q.96

'Consider the quadratic equation x2+(m+1)x+m1=0x^{2}+(m+1)x+m-1=0.'

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Q.97

'Standard 65: Determining coefficients of higher order equations (2) - Conditions for imaginary solutions'

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Q.98

'Find the value of the constant a a when the cubic equation x3(a+2)x+2(a2)=0 x^{3}-(a+2) x+2(a-2)=0 has a double root.'

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Q.99

'Standard 49: Range of existence of solutions of a quadratic equation (1)'

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Q.00

'Find the first term and common ratio of a geometric sequence such that the sum from the 3rd term to the 5th term is -63 and the sum from the 1st term to the 3rd term is -7.'

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Q.01

'Basic 41: Conditions for a quadratic equation to have complex roots, repeated roots'

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Q.02

'Let the three solutions of the cubic equation x32x+1=0 x^{3}-2 x+1=0 be alpha,\eta,gamma \\alpha, \eta, \\gamma . Find the value of the following expressions.'

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Q.03

'Extension Study - Development 192 The number of real solutions of a cubic equation (3) Utilizing extreme values'

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Q.04

'58 divided by, in order of remainders (1) x^2+2x-6, -10 (2) x^2-5x+4, 3'

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Q.05

'Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \ that represents the sum \ S_{n} \ from the first term to the nth term.'

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Q.06

"When the function f(x) represented by a polynomial satisfies f'(x)-f(x)=x²+1, f(x) is a □ degree function, and f(x)= ."

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Q.07

'Standard 40: Discrimination of types of solutions of quadratic equations (2)'

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Q.08

'By rearranging the equation x-2y+6=0, we can express it as y=\\frac{1}{2}x+3, thus representing a line with a slope of \\frac{1}{2} and a y-intercept of 3.'

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Q.09

"Determine the range of the constant 'm' so that the quadratic equation x2+2(m1)x+2m25m3=0x^{2}+2(m-1)x+2m^{2}-5m-3=0 satisfies the following conditions: (1) has two positive roots, (2) has two distinct negative roots, (3) has roots of different signs."

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Q.10

'Expansion 66: Relationship between the solutions of a cubic equation and its coefficients'

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Q.11

'Chapter 3 High Degree Equations - 49\nEX Let a, b, c, d be real constants. The polynomial P(x) = ax^{3} + bx^{2} + cx + d leaves a remainder of 29 when divided by x^{2}-1 and a remainder of 3x + 4 when divided by x^{2}+1. In this case, a=->□, b=-1□, c=d=√□. [Shonan University]\nLet Q(x) be the quotient when P(x) is divided by x^{2}-1, and let R(x) be the quotient when P(x) is divided by x^{2}+1. Then, the following equations hold.\n\nP(x) = (x+1)(x-1)Q(x) + x + 2\nP(x) = (x^{2}+1)R(x) + 3x + 4\nP(1) = 3, P(-1) = 1, P(i) = 4 + 3i'

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Q.12

'Chapter 7 Exponential and Logarithmic Functions'

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Q.13

'Find the equation of the line passing through two different points (x1, y1) and (x2, y2).'

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Q.14

'Find the sum and product of the two roots of the following quadratic equations.'

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Q.15

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Q.16

'Find two numbers whose sum is 2 and product is -4.'

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Q.17

'Find the range of values for the constant aa when the cubic equation x33a2x+4a=0x^{3}-3a^{2}x+4a=0 has three distinct real roots.'

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Q.18

'Developing 68: Conditions for a cubic equation to have three distinct real roots'

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Q.19

'Prove that the equation a^{2}+b^{2}=c^{2}-2 a b holds true when a+b+c=0.'

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Q.20

'When the equation \\(a\\left(x^{2}-x+1\\right)=1+2x-2x^{2}\\) has real solutions, find the range of values for the constant \a\.'

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Q.21

'Regarding the underlined part g, last year the Ministry of Land, Infrastructure, Transport and Tourism also conducted a subsidy grant solicitation aimed at promoting the adoption of next-generation vehicles. Choose the correct combination of the following statements X・Y regarding next-generation vehicles as either true or false.'

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Q.22

'Choose the appropriate expression to represent the distance moved by block A with respect to block C, and provide the symbol.'

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Q.23

'1 (1) \y=mx-2m+2 \\n(2) \u=\\frac{m-1}{m}, v=1-m \\n(3) \y=\\frac{1}{x-1}+1 \, Figure omitted'

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Q.24

'Prove that the following equations have at least one real number solution in the given range.'

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Q.25

'Let a be a real number, find the number of real solutions of the equation f(g(x))+f(x)-|f(g(x))-f(x)|=a.'

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Q.26

'Please remove the denominator and solve the following equation:\n(2x-3)(x^{2}-3x+1)=0'

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Q.27

'Find a fifth-degree polynomial f(x) that satisfies conditions (A) and (B) simultaneously.'

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Q.28

'Let a, b be real numbers, and suppose that the cubic equation x^3+ax^2+bx+1=0 has an imaginary root α. Show that the conjugate complex number of α, denoted by α¯, is also a root of this equation. Express the third root β and the coefficients a, b in terms of α and α¯.'

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Q.29

'Find the velocity, acceleration, position, and distance traveled (linear motion).'

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Q.30

'Prove that when a > 1, the two solutions of the equation a x^2 − 2 x + a = 0 (1) are denoted as α and β, and the two solutions of the equation x^2 − 2 a x + 1 = 0 (2) are denoted as γ and δ. Let A(α), B(β), C(γ), D(δ) and prove that the four points A, B, C, D lie on a common circle.'

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Q.31

'Fundamentals 8: Algebraic solutions for irrational equations and inequalities'

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Q.32

'For a point (a,b)(a, b) on the hyperbola x24y2=4x^{2}-4 y^{2}=4 with a tangent line having a slope mm, answer the following questions. Assume b0b \neq 0.\n(1) Find the relationship between a,b,ma, b, m.\n(2) Let the distance between a point on this hyperbola and the line y=2xy=2x be denoted as dd. Find the minimum value of dd. Also, determine the coordinates of the point on the curve that provides the minimum value of dd.[Kanagawa University]'

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Q.33

'What geometric shape is formed by the set of points that satisfy the following equations?'

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Q.34

'Solve the equation \ \\frac{1}{x} + \\frac{1}{x-1} + \\frac{1}{x-2} + \\frac{1}{x-3} = 0 \.'

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Q.35

'20 (1) \ |\\alpha|^{2} \\n(2) Omitted (3) Maximum value when \ a=b \ is \ \\frac{1}{2} ; a=1, \\quad b=3 \ and minimum value is \ \\frac{3}{10} \'

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Q.36

'Solve the following equations and inequalities.'

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Q.37

'Let two complex numbers w and z (z ≠ 2) satisfy w = iz/(z-2).\n[Hirosaki University]\n(1) When point z moves on the circumference of a circle with radius 2 centered at the origin, what shape does point w trace out?\n(2) When point z moves on the imaginary axis, what shape does point w trace out?\n(3) When point w moves on the real axis, what shape does point z trace out?'

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Q.38

'Radioactive substances like radium decrease in mass at a rate proportional to the mass at each instant. Express the mass x as a function of time t with the proportionality constant k (k > 0) and initial mass A. Additionally, for radium, it takes 1600 years for the mass to halve. Approximately what percentage of the initial amount is left after 800 years? Round to the nearest whole number.'

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Q.39

'Solve the equation.\nFind the solution.'

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Q.40

'Solve the inequality \ \\log _{2} 256 x > 3 \\log _{2 x} x \. Let \\\log _{2} x = a \.'

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Q.41

'Number of real solutions of an equation'

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Q.42

'Consider complex numbers z that satisfy conditions (A) and (B) simultaneously. (A) z + i/z is real (B) The imaginary part of z is positive. (1) Let |z|=r, express z in terms of r. (2) Find the z for which the imaginary part of z is maximum.'

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Q.43

'Let a ≠ 0. For the function f(x) = 2ax - 5a^2, find the value of the constant a such that f^{-1}(x) and f(x) are equal.'

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Q.44

'Assuming the existence of a sequence {a_{n}} and its sum from the first term to the nth term'

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Q.45

'Solve the following quadratic equations:\n(1) x23x+2=0 x^{2}-3 x+2=0 \n(2) 2x23x35=0 2 x^{2}-3 x-35=0 \n(3) 12x2+16x3=0 12 x^{2}+16 x-3=0 \n(4) 14x219x3=0 14 x^{2}-19 x-3=0 \n(5) 5x23=0 5 x^{2}-3=0 \n(6) (2x+1)29=0 (2 x+1)^{2}-9=0 '

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Q.46

'Prove the relationship between the roots and coefficients of the following quadratic equation. For a quadratic equation ax² + bx + c = 0, let the two roots be α and β. Then, α + β = -b/a and αβ = c/a.'

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Q.47

'As a sprinter in the 100m race, Tarou decided to focus on (1) and figure out the best stride and pitch to improve his time.'

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Q.48

'Solve the following quadratic equations.'

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Q.49

'Find the range of values for the constant a that satisfy the given conditions for the two quadratic equations x2x+a=0x^{2}-x+a=0 and x2+2ax3a+4=0x^{2}+2ax-3a+4=0.'

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Q.50

"To determine the range in which the solutions of a quadratic equation exist, let's consider the graph that satisfies the following conditions:"

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Q.51

'Let a and p be constants. Find the real solutions of the following equations in x.'

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Q.52

'Solve the following system of simultaneous equations.'

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Q.53

'How many ways are there to divide 12 different books as follows?'

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Q.54

'(Example) In the case of 2x + 3y = 33 (where x and y are natural numbers), using y ≥ 1, we have 2x = 3(11 - y) ≤ 30, hence 1 ≤ x ≤ 15.'

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Q.55

'If the two distinct real solutions of the quadratic equation x2a2x4a+2=0x^{2}-a^{2}x-4a+2=0 are denoted as alpha\\alpha and \eta\eta, and satisfy 1<alpha<2<\eta1<\\alpha<2<\eta, determine the range of values of the constant aa.'

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Q.56

'When the equation ax2+bx+1=0ax^{2}+bx+1=0 has two solutions 2,3-2,3, find the values of constants a,ba, b.'

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Q.57

'For the quadratic equation \ x^{2}-a^{2} x-4 a+2=0 \ with two distinct real solutions \ \\alpha, \eta \ where \ 1 < \\alpha < 2 < \eta \, determine the range of values for the constant \ a \.'

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Q.58

'Determine the number of real solutions of the quadratic equation 2x2+3x+k=02 x^{2}+3 x+k=0.'

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Q.59

'Solve the following system of simultaneous equations.'

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Q.60

'Chapter 1\nNumbers and Expressions\n23\nExample\n(1) Find the expression that when summed with 2x^2-3x+1 results in x^2+2x.'

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Q.61

'What is the range of the constant a when one real solution of the quadratic equation 2x^{2}-3ax+a+1=0 is in the range 0<x<1 and the other real solution is in the range 4<x<6?'

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Q.62

'Solve the following quadratic equations.'

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Q.63

'What is the range of existence of the roots of a quadratic equation?'

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Q.64

'Find the number of real solutions for the following quadratic equations.'

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Q.65

'Given line segments of length a and b, find the positive solution to the quadratic equation x^{2}-a x-b^{2}=0 and draw a line segment with that length.'

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Q.66

'Find the range of values for the constant a a such that the quadratic equation x2+axa2+a1=0 x^{2}+a x-a^{2}+a-1=0 has two distinct real solutions within the range 3<x<3 -3<x<3 .'

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Q.67

'Determine the range of values for the constant a a when the quadratic equation x22ax+a+6=0 x^{2}-2 a x+a+6=0 satisfies the following conditions: (1) Has positive and negative roots. (2) Has two distinct negative roots.'

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Q.68

'In all permutations formed by the 8 letters of YOKOHAMA, find the number of permutations that contain at least one of the sequences AO or OA.'

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Q.69

'Transform the given mathematical expression into a different form.'

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Q.70

'Chapter 2 Sets and Propositions\n(2) Solve the following equation\n\\[(p q+6)+(3 p+q) \\sqrt{2}=8+7 \\sqrt{2}\\]\nwhere p and q are rational numbers.'

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Q.71

'How many ways are there for 4 men and 5 women to line up in a row with the following conditions? (1) All 4 men are adjacent (2) Men are not adjacent to each other'

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Q.72

'Please state the converse, contrapositive, and inverse of the proposition.'

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Q.73

'Determining coefficients from maximum and minimum values (3)'

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Q.74

'Let PR denote all real numbers. Choose the appropriate term to fill in the square (∎) from below (ア)–(工). (1) x=2 is a ∎ for x^2-5x+6=0. (2) ac=bc is a ∎ for a=b. (3) a=b is a ∎ for a^2+b^2=2ab. (ア) Necessary and sufficient condition (イ) Necessary but not sufficient condition (ウ) Sufficient but not necessary condition (工) Neither necessary nor sufficient condition'

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Q.75

'54 (2), (3); (2) maximum at x=2 is 7, minimum at x=0 is 3; (3) maximum at x=2 is 5, minimum at x=-1.5 is -13'

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Q.76

'What is the range of values for kk for which the quadratic equations x2+x+k=0x^{2}+x+k=0, x2+kx+1=0x^{2}+k x+1=0 both have real number solutions?'

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Q.77

'Given x ≥ 0, y ≥ 0, and 2x+y=8, find the maximum and minimum values of xy.'

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Q.78

'Use the quadratic formula to solve the following 2nd-degree equations.'

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Q.79

'Find the linear function based on the following conditions to maximize profit.\n\n(1) When x is 250, y is 300.\n(2) When x is 300, y is 250.\n(3) Also holds when x = 350, y = 200.\n\nFurthermore, using revenue xy and expenses 120y + 5000, let z be the profit and find the value of x that maximizes z and the maximum profit at that time.'

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Q.80

'Determine the truth values of the following propositions.'

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Q.81

'Solve the following equations.'

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Q.82

'If one of the solutions of the quadratic equation x2+(a+4)x+a3=0 x^{2}+(a+4) x+a-3=0 is a a , find the other solution.'

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Q.83

'(Example) 1) When 1/x + 1/y + 1/z = 1 (0 < x < y < z), we use 1/z < 1/y < 1/x to get 1 = 1/x + 1/y + 1/z < 3/x, hence x = 1,2.'

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Q.84

'Solve the following inequality in terms of x. Where a is a constant. \\[ x^{2}-\\left(a^{2}+a\\right) x+a^{3} \\leqq 0 \\]'

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Q.85

'Solve the following system of simultaneous equations.'

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Q.86

'Applications of Quadratic Equations'

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Q.87

'At a certain school, it was decided to completely drain the water from the pool for cleaning. However, it is assumed that a constant amount is drained per minute with a pump. Let the remaining volume of water in the pool after t minutes of drainage be V m³.'

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Q.88

'Solve the following equations.'

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Q.89

'Solve the following system of simultaneous equations.'

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Q.90

'At a party with 5 participants, where each person prepares a gift and a drawing is held to distribute all gifts. The number of ways in which two specific people, A and B, receive the gifts they prepared, while the remaining three people receive gifts other than the ones they prepared is denoted by A. The number of ways in which only one person receives the gift they prepared is denoted by B.'

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Q.91

'Basic Example 30 Number of Integer Solutions (Using Combinations with Repetition)'

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Q.92

'Find the range of existence of solutions for a quadratic equation with x < 2 and x > 2.'

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Q.93

'61 a=2, b=-5 or a=-2, b=3'

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Q.94

'Solve the inequality for x. Here, a is a constant. \ x^{2}-3 a x+2 a^{2}+a-1>0 \'

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Q.95

"Please provide 'Linear Diophantine Equations' and its page."

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Q.96

'Find the coordinates of the two points of intersection of the two parabolas y=x^2-x+1 and y=-x^2-x+3.'

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Q.97

'31 (1) \ x=6,-2 \\n(2) \ x \\leqq-5, \\quad \\frac{1}{5} \\leqq x \'

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Q.98

'Find the largest 4-digit natural number that leaves a remainder of 2 when divided by 11 and a remainder of 5 when divided by 6.'

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Q.99

'Solve equations and inequalities involving absolute value: (1) x=c|x|=c, (2) x<c|x|<c, (3) x>c|x|>c'

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Q.00

'Consider the proposition p⇒q (where p is the hypothesis and q is the conclusion). Let P be the set of all elements satisfying condition p, and let Q be the set of all elements satisfying condition q. The proposition p⇒q being true is equivalent to P ⊆ Q. Please determine the truth value of this proposition.'

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Q.01

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Q.02

'For the function f(x)=x2+2xa2+5f(x)=x^{2}+2x-a^{2}+5, find the range of values of aa that satisfy the following conditions:'

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Q.03

'Let a and b be real numbers, and consider the parabolas represented by the functions y=4 x^{2}-8 x+5 and y=-2(x+a)^{2}+b. Find the values of the constants a and b when the vertices of these parabolas coincide.'

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Q.04

'When the quadratic equation x22mx+2(m+4)=0x^2 - 2mx + 2(m + 4) = 0 has equal roots, find the value of the constant mm and the equal roots at that time.'

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Q.05

'Solve the following equations and inequalities.'

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Q.06

'At x=3/2, the minimum value is 3/2, and there is no maximum value.'

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Q.07

'Use the formula for solving a quadratic equation ax^{2}+bx+c=0 to solve the equation.'

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Q.08

'Find the value of the constant k when the parabola y = x^2 + (2k-3)x-6k cuts a segment of length 5 from the x-axis.'

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Q.09

'There is a lottery where three dice are rolled at once. There are multiple venues for the lottery, each with different winning conditions.'

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Q.10

'Find the range of values for the constant m when the quadratic equation x^2 + 3x + m - 1 = 0 has no real solutions.'

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Q.11

'For the quadratic equation x2+mx+9=0x^{2}+mx+9=0, find the range of values for the constant mm when it has the following:'

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Q.12

'For example, there are many integer solutions to the equation x+y=10. Find any three integer solutions to this equation.'

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Q.13

'Solve the equation x2+3x5=0x^{2}+3x-5=0.'

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Q.14

'When the solutions of the quadratic equation x2+2mxm+2=0 x^{2}+2 m x-m+2=0 are as follows, find the range of the constant m m . (1) Having two distinct real solutions. (2) Having real solutions. (3) Having no real solutions.'

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Q.15

"Chapter 5 Quadratic Equations and Quadratic Inequalities\nLet h meters be the height above the ground of a ball thrown directly upwards at a certain velocity x seconds after the launch. When the value of h is given by h=-5x^2+40x, at what range of x values is the ball's height between 35 meters above the ground and 65 meters above the ground?"

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Q.16

'Find the values of \ a, b \ such that \ P=4 \ for the point \ x, y \ with values \\( (2, 1) \\).'

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Q.17

'Solve the following equations. 1. Fundamental 86 - Solving quadratic equations using factorization. 2. Fundamental 87 - Solving quadratic equations using the formula for roots. 3. Fundamental 88 - Conditions for having real solutions (1)'

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Q.18

'Solve the equation x^2+3x-4=0.'

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Q.19

'Determine the range of values for the constant aa so that the quadratic equation x2(a4)x+a1=0x^{2}-(a-4)x+a-1=0 satisfies the following conditions: (1) has two distinct negative roots. (2) has a positive root and a negative root.'

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Q.20

'24 (1) Inverse: If at least one of x, y is a negative number, then x+y=-3, False Converse: If x≥0 and y≥0, then x+y≠-3, Contrapositive: If x+y≠-3, then x≥0 and y≥0.'

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Q.21

'2 (1) 5 expressions(2)(ア)2 expressions, constant term 2 y^{2} + 5 y - 12(1)2 expressions, constant term 6 x^{2} - 6 x - 12(ら)2 expressions, constant term -12'

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Q.22

'When the roots of the quadratic equation x2+mx+9=0 x^{2}+m x+9=0 have the following characteristics, find the range of values for the constant m m .'

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Q.23

'The condition for a graph to always be above the x-axis is that the graph is a concave down parabola and does not have any points of intersection with the x-axis. Therefore, the discriminant of the quadratic equation mx^2 + 3x + m = 0 is denoted as D.'

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Q.24

'Solve the following equations and inequalities. (1) |(√14-2)x+2|=4 (2) 3|x-1|≤4 (3) x+|3x-2|=3'

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Q.25

'Translate the given text into multiple languages.'

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Q.26

'Find the value of the constant mm and the repeated root of the quadratic equation x22mx+2(m+4)=0x^{2}-2mx+2(m+4)=0 when it has repeated roots.'

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Q.27

'When the quadratic equation x2+5x+7m=0x^{2}+5x+7-m=0 has two distinct real roots, find the range of values for the constant mm.'

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Q.28

'For the function f(x) = x^2 - 2ax - a + 6 in mathematics I TR, the range of values of the constant a for which f(x) > 0 for all real numbers x is from A to T. Also, the range of values of a for which f(x) ≥ 0 always holds for -1 ≤ x ≤ 1 is from ウ to エ.'

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Q.29

'In two inequalities, first replace > with = and then solve the quadratic equation. To solve the quadratic inequality x^2-6x+3>0, first solve the equation x^2-6x+3=0. Using the formula for roots, x=(-(-3) ± √((-3)^2-1*3))/1=3 ± √6.\n\nThe solution to x^2-6x+3>0 is to find the range of x values on the graph of y=x^2-6x+3 where y>0, which is x<3-√6, 3+√6<x.'

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Q.30

'(2) The two inequalities x2+mx+m+2<0x^{2}+mx+m+2<0 have solutions'

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Q.31

'At x=-1, the maximum value of 71(2) is 5, there is no minimum value.'

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Q.32

'(1) Solve the equation 2 x^{2}+x-1=0 \\\\\\\n(2) Solve for x=-1, \\ \\frac{1}{2} \\\\\\\n(3) Given \\theta=60^{\\circ}, \\ 180^{\\circ} \\\\\\\n'

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Q.33

'The classification of solutions of a quadratic equation can be determined by examining the sign of the discriminant D=b24ac D = b^{2} - 4ac .'

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Q.34

'Find the range of constants a that satisfy the following conditions for the equations x^2+ax+a+3=0 (1) and x^2-2ax+8a=0 (2):'

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Q.35

'Find the intersection point of the two lines 2x + 3y = 7 (1) and 4x + 11y = 19 (2), and the equation of the line passing through the point (5,4).'

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Q.36

'Find the values of a,b,ca, b, c when the 3rd-degree function f(x)=x3+ax2+bx+cf(x)=x^{3}+a x^{2}+b x+c satisfies (x2)fprime(x)=3f(x)(x-2)f^{prime}(x)=3 f(x).'

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Q.37

'Solve the following quadratic equations.'

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Q.38

'(1) Find the solutions of the equation 2x^2 - 2√6x + 3 = 0.'

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Q.39

'Chapter 1 Sequences'

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Q.40

'Determine the number of solutions to the equation 4cos²x-2cosx-1=a within the range -π < x ≤ π.'

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Q.41

'Find the value of a that satisfies the given equations.'

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Q.42

'Prove that the equation ax+by+cz=0ax+by+cz=0 holds true when (2) fracxbc=fracyca=fraczab\\frac{x}{b-c}=\\frac{y}{c-a}=\\frac{z}{a-b}.'

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Q.43

'Find the values of k for which x = -1, -4 and x = 1, 4 are solutions.'

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Q.44

'The solutions to the cubic equation x3+ax221x+b=0x^{3}+a x^{2}-21 x+b=0 are 1,3,c1,3, c. Find the values of constants aa, b,cb, c.'

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Q.45

'Find the value of a that satisfies the following equation. a=2'

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Q.46

'Regarding the lines l: 2x - y + 3 = 0, m: 3x - 2y - 1 = 0, answer the following questions.'

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Q.47

'Determine the range of the constant a so that the two quadratic equations 9x^{2}+6ax+4=0 (1) and x^{2}+2ax+3a=0 (2) satisfy the following conditions. (1) Both have complex roots (2) At least one has complex roots (3) Only (1) has complex roots'

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Q.48

'When a = -1, x = 2, -1\nWhen a = 0, x = 0, 2\nWhen a = 8, x = -4, 2'

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Q.49

'When the equation x2+y26kx+(12k2)y+46k216k+1=0x^{2}+y^{2}-6 k x+(12 k-2) y+46 k^{2}-16 k+1=0 represents a circle,'

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Q.50

'Find the range of values for c so that the equation x^3-6x+c=0 has two distinct positive roots and one negative root.'

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Q.51

'Determine the following for the sequence {\ \\left\\{a_{n}\\right\\} \} where the sum of the first \ n \ terms, denoted as {\ S_{n} \}, satisfies the relation {\ S_{n}=-2 a_{n}+4 n \}:\n(1) Find the first term {\ a_{1} \}.\n(2) Find the relation between {\ a_{n} \} and {\ a_{n+1} \}.\n(3) Find the general term of the sequence {\ \\left\\{a_{n}\\right\\} \}.'

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Q.52

'Find all real numbers aa such that the cubic equation x3+(a+2)x24a=0x^{3}+(a+2)x^{2}-4a=0 has exactly 2 real roots.'

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Q.53

'Use the formula for solving the quadratic equation ax^2 + bx + c = 0 to find the solutions.'

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Q.54

'After reading the conversation between Hanako and Taro about the problem, answer the following question.'

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Q.55

'Determine the values of constants a and b that satisfy the following conditions:\n(1) When x-1 is a factor of x^3-3x^2+a, the remainder is 2.\n(2) When 2x+1 is a factor of 2x^3-3x^2+ax+6, it divides evenly.\n(3) When x+2 divides x^3+ax^2-5x+b, the remainder is 8, and when x+1 is a factor.'

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Q.56

'Transform the given text into multiple languages.'

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Q.57

'Determine the value of the real number k so that the equation (1 + i)x^{2} + (k + i)x + 3 + 3ki = 0 has real solutions. Find these real solutions.'

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Q.58

'The two numbers that sum to -2 and multiply to 3 are the solutions to the quadratic equation x^2 + 2x + 3 = 0.'

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Q.59

'Find the value of the constant a such that the equation 2x^3 - (3a + 1)x^2 + 2ax + 4 has two distinct real roots.'

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Q.60

'Solve the following system of equations.'

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Q.61

'Chapter 2\nComplex Numbers and Equations\nShow that for the two solutions α, β of the quadratic equation 2x^2 + 4x + 3 = 0, the following holds:\n1. (α-1)(β-1)=9/2\n2. (α-1)^3 + (β-1)^3 = -10'

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Q.62

'If the quadratic equation x2+4ax+8a220a+25=0x^{2}+4 a x+8 a^{2}-20 a+25=0 has real roots, then a=textconstanta=\\{ \\text{constant} \\} and the solutions are x=textconstantx=\\{ \\text{constant} \\}.'

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Q.63

'Create a quadratic equation with the sum as \ p \ and the product as \ q \.'

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Q.64

'Find the values of real numbers x, y that satisfy the following system of equations.'

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Q.65

'For the quadratic equation x^2 - 2ax + a + 6 = 0, find the range of values for the constant a that satisfy the following conditions: (1) Having one solution greater than 1 and one solution less than 1. (2) All solutions are greater than or equal to 2.'

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Q.66

'Create a quadratic equation with the roots α and β.'

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Q.67

'Find the values of the constants p, q, and r such that the equation p x^2 + q y^2 + r z^2 = 12 holds for all real numbers x, y, z satisfying the system of equations 2x + y - 3z = 3 and 3x + 2y - z = 2.'

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Q.68

'When the equation x4x3+ax2+bx+6=0 x^{4}-x^{3}+a x^{2}+b x+6=0 has roots at x=1,3 x=-1,3 , find the values of constants a,b a, b . Also, find the other roots in this case.'

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Q.69

'The sum and product of the two roots of the quadratic equation x2+ax+b=0 x^{2}+a x+b=0 are the roots of the quadratic equation x2+bx+a=0 x^{2}+b x+a=0 . Determine the values of constants a,b a, b .'

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Q.70

'The profit per day is (a x + 3 y) 10,000 yen. If we let ax + 3 y = l ....5), then (5) represents a straight line with a slope of -a/3 and a y-intercept of l/3.'

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Q.71

"Let's consider the maximum and minimum values of -3x+y under the same conditions as A (let's call this C). If we express -3x+y=k, then y=3x+k。"

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Q.72

'Let A be a polynomial. When x^6 - 6x^3 + 5x^2 - 4x + 10 is divided by A, the quotient is A and the remainder is 5x^2 - 4x + 1. Find the polynomial A.'

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Q.73

'Determine the values of constants a, b, c, and d so that the following equations hold true for x:'

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Q.74

'Find the range of values for the constant p such that the equation x^3-3p^2x+8p=0 has three distinct real solutions.'

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Q.75

'For the quadratic equation in terms of x, 8x^2-4x-a=0, with two solutions being sin θ and cos θ, find the value of the constant a and the 2 solutions of the equation. [Similar to Keio University] From the relationship between the solutions of the quadratic equation and the coefficients, sin θ+cos θ=-\\frac{-4}{8}=\\frac{1}{2}, sin θ cos θ=-\\frac{a}{8}. Squaring both sides of (1) gives sin^2 θ+2sin θ cos θ+cos^2 θ=\\frac{1}{4}, hence 1+2sin θ cos θ=\\frac{1}{4}, which implies sin θ cos θ=-\\frac{3}{8}. Substituting this into (2) gives -\\frac{a}{8}=-\\frac{3}{8}, thus a=3. Therefore, the given quadratic equation is 8x^2-4x-3=0. Solving this equation, the two solutions are x=(1±√7)/4.'

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Q.76

'Basic Column Question 18 Determining the Coefficients of an Identity'

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Q.77

'Use recurrence relation to find the general term of the sequence.'

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Q.78

'Given the quadratic equation 3x22x4=03 x^{2}-2 x-4=0 with two roots alpha,\eta\\alpha, \eta, find the value of the following expressions.'

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Q.79

'The cubic curve y = ax^3 + bx^2 + cx + d touches the x-axis at x = 2 and the equation of the tangent line at the origin is y = -2x. Find the values of the constants a, b, c, and d.'

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Q.80

'When (x + y) / 2 = (y + z) / 5 = (z + x) / 7 (not equal to 0), find the value of (xy + yz + zx) / (x^2 + y^2 + z^2).'

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Q.81

'The quartic equation x^4+ax^3+7x^2+bx+26=0 has two common roots with the quadratic equation x^2+2x+2=0 and the sum of the common roots is 37. [Tokushima Bunri University] (1) Find the values of the real constants a, b. (2) Find the remaining roots of the quartic equation.'

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Q.82

'Since this line passes through the point (2, -4), -4 = 2(a-1) • 2 - a^2. Simplifying, we get a^2 - 4a = 0. Therefore, a(a-4) = 0, which means a = 0, 4. Hence, the equation of the tangent line we seek is y = -2x when a = 0, and y = 6x - 16 when a = 4.'

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Q.83

'Find the range of possible values for x given that x, y, z are real numbers satisfying x+y+z=0 and x^2-x-1=yz. Determine the maximum and minimum values of x^3+y^3+z^3 and the corresponding values of x.'

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Q.84

'(3) For a geometric sequence where the first term a and common ratio r are both real numbers, if the sum from the first term to the nth term is Sn, and when Sn=3 and Sn=27. Find the values of a, r.'

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Q.85

'In the mathematics \ \\mathbb{I} \ EX\\nLet the two solutions of the quadratic equation \\( 2 x^{2}-2(2 a-1) x-a=0 \\) be \ \\sin \\theta, \\cos \\theta \. Find the positive constant \ a \ and the values of \ \\sin \\theta, \\cos \\theta \. Given that \ 0 \\leqq \\theta \\leqq \\pi \.'

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Q.86

'(1) x214x+45=0 x^{2}-14 x+45=0 '

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Q.87

'Let a be a constant. Prove that at least one of the two quadratic equations x^{2}+2ax+3a=0 and 3x^{2}-2(a-3)x+(a-3)=0 has real roots.'

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Q.88

'Regardless of the constant a, the circle C1 passes through the fixed point A. Find the coordinates of this fixed point A.'

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Q.89

'Example 76: Equation of a line passing through a fixed point'

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Q.90

'Question 77'

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Q.91

'Determine the types of solutions for the following quadratic equations:'

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Q.92

'Solve the system of equations x^2-3xy+y^2=19, x+y=2.'

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Q.93

'When the equation x2+y2+6px2py+28p+6=0 x^{2}+y^{2}+6 p x-2 p y+28 p+6=0 represents a circle, find the range of values for the constant <