This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
The document discusses quadratic equations. It begins by defining quadratic equations as polynomials of degree two that are set equal to zero. It then provides examples of identifying quadratic equations from collections of equations. Methods covered for solving quadratic equations include factoring, using the quadratic formula, and determining the nature of roots based on the discriminant. It also discusses writing quadratic equations in standard form and translating word problems into quadratic equations.
A Solutions To Exercises On Complex NumbersScott Bou
This document provides solutions to exercises involving complex numbers, including:
1) Adding, multiplying, and dividing complex numbers and writing the results in standard form (a + bi).
2) Finding conjugates, multiplicative inverses, and plotting numbers on the complex plane.
3) Determining convergence of infinite complex number series and calculating partial sums.
Problems are solved step-by-step showing work, and complex number expressions are simplified and written in standard form. Diagrams are provided when plotting on the complex plane.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This document provides a review of various algebra topics including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples to practice each topic.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
The document discusses quadratic equations. It begins by defining quadratic equations as polynomials of degree two that are set equal to zero. It then provides examples of identifying quadratic equations from collections of equations. Methods covered for solving quadratic equations include factoring, using the quadratic formula, and determining the nature of roots based on the discriminant. It also discusses writing quadratic equations in standard form and translating word problems into quadratic equations.
A Solutions To Exercises On Complex NumbersScott Bou
This document provides solutions to exercises involving complex numbers, including:
1) Adding, multiplying, and dividing complex numbers and writing the results in standard form (a + bi).
2) Finding conjugates, multiplicative inverses, and plotting numbers on the complex plane.
3) Determining convergence of infinite complex number series and calculating partial sums.
Problems are solved step-by-step showing work, and complex number expressions are simplified and written in standard form. Diagrams are provided when plotting on the complex plane.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This document provides a review of various algebra topics including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples to practice each topic.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise solutions to each problem in 3 sentences or less.
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise 3-sentence or less solutions for each of the 50 math problems.
Final Exam Name___________________________________Si.docxcharlottej5
Final Exam Name___________________________________
Silva Math 96 Spring 2020
YOU MUST SHOW ALL WORK AND BOX YOUR ANSWERS FOR CREDIT. WORK ALONE.
Solve the absolute value inequality. Write your answer
in interval notation.
1) |2x - 12 |> 2
Solve the compound inequality. Graph the solution set.
Write your answer in interval notation.
2) -4x > -8 and x + 4 > 3
Solve the three-part inequality. Write your answer in
interval notation.
3) -1 < 3x + 2 < 14
Solve the absolute value equation.
4) 4x + 9 = 2x + 7
Solve the compound inequality.
5) 3( x + 4 ) ≥ 0 or 4 ( x + 4 ) ≤ 4
Solve the inequality. Graph the solution set and write
your answer in interval notation.
6) |5k + 8| > -6
Solve the inequality graphically. Write your answer in
interval notation .
7) x + 3 ≥ 1
x-8 -6 -4 -2 2
y
8
6
4
2
x-8 -6 -4 -2 2
y
8
6
4
2
1
Graph the system of inequalities.
8) 2x + 8y ≥ -4
y < - 3
2
x + 6
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Find the determinant of the given matrix.
9) 10 5
0 -4
Use Cramer's rule to solve the system of linear
equations.
10) 6x + 5y = -12
2x - 2y = -4
Write a system that models the situation. Then solve the
system using any method. Must show work for credit.
11)A vendor sells hot dogs, bags of potato chips,
and soft drinks. A customer buys 3 hot dogs,
4 bags of potato chips, and 5 soft drinks for
$14.00. The price of a hot dog is $0.25 more
than the price of a bag of potato chips. The
cost of a soft drink is $1.25 less than the price
of two hot dogs. Find the cost of each item.
Use row reduced echelon form to solve the system.
12) x + y + z = 3
x - y + 4z = 11
5x + y + z = -9
2
Find the domain of f. Write your answer in interval
notation.
13) f(x) = 13 - 9x
If possible, simplify the expression. If any variables
exist, assume that they are positive.
14) 2x + 6 32x + 6 8x
Match to the equivalent expression.
15) 100-1/2
A) 1
1000
B) 1
10
C) 1
100
D) 1
10
Write the expression in standard form.
16) (5 + 8i) - (-3 + i)
Simplify the expression. Assume that all variables are
positive.
17) 5 t
5
z10
Solve the equation.
18) 3x + 1 = 3 + x - 4
Write the expression in standard form.
19) 3 + 3i
5 + 3i
3
Write the equation in vertex form.
20) y = x2 + 5x + 2
The graph of ax2 + bx + c is given. Use this graph to solve
ax2 + bx + c = 0, if possible.
21)
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
Solve the equation. Write complex solutions in standard
form.
22) 4x2 + 5x + 5 = 0
Graph the quadratic function by its properties.
23) f(x) = 1
3
x2 - 2x + 3
x
y
x
y
Solve the equation. Find all real solutions.
24) 2(x - 1)2 + 11(x - 1) + 12 = 0
Solve the problem.
25) The length of a table is 12 inches more than its
width. If the area of the table is 2668 square
inches, what is its length?
4
Solve the equation..
rational equation transformable to quadratic equation.pptxRizaCatli2
1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.
The document provides information on exam format and topics that need to be studied for Form 4 and Form 5 exams.
It recommends setting targets and being familiar with exam format. The main topics covered are functions, quadratic equations, trigonometry, calculus, vectors, statistics, and index numbers. Exercise and practice are strongly emphasized. Sample exam papers and questions are provided to illustrate exam structure and level of difficulty.
The document provides information about sets and operations on sets such as union, intersection, complement, difference, properties of these operations, counting theorems for finite sets, and the number of elements in power sets. It defines key terms like union, intersection, complement, difference of sets. It lists properties of union, intersection, and complement. It presents counting theorems for finite sets involving union, intersection. It states that the number of elements in the power set of a set with n elements is 2n and the number of proper subsets is 2n-2.
Diploma_Semester-II_Advanced Mathematics_Complex numberRai University
This document provides information about complex numbers. It begins by introducing complex numbers and defining them as numbers of the form a + bi, where a and b are real numbers and i = √-1. It then discusses various representations of complex numbers including Cartesian (a + bi), polar (r(cosθ + i sinθ)), and Euler (re^iθ) forms. It also covers operations on complex numbers such as addition, subtraction, multiplication, division, and powers. De Moivre's theorem relating powers of complex numbers to trigonometric functions is presented. The document concludes by stating some basic algebraic laws for complex numbers.
1. The document discusses various methods for solving different types of equations including: polynomial equations using factoring, quadratic equations using the quadratic formula, rational equations by clearing denominators, radical equations by squaring both sides, and absolute value equations geometrically.
2. Specific examples are provided to demonstrate how to solve equations involving polynomials, rationals, radicals, powers, and absolute values. Steps include factoring, using the quadratic formula, clearing denominators, isolating and squaring radicals, and raising both sides to appropriate powers.
3. The document also discusses finding zeros of rational functions and domains by determining where the numerator or denominator is equal to zero.
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
This document provides an overview of solving quadratic equations by factoring. It discusses identifying quadratic equations, rewriting them in standard form, factoring trinomials in the form x^2 + bx + c, and determining roots. Several examples of factoring trinomials and solving quadratic equations are shown. Activities include identifying quadratic equations, rewriting equations in standard form, factoring trinomials, and solving equations by factoring. The document provides resources for further learning about quadratic equations and factoring.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFYssuser2e348b
1) The document discusses theorems and proofs related to quadratic equations. It provides a necessary and sufficient condition for two quadratic equations to have a common root.
2) Several examples of solving equations that can be reduced to quadratic equations are presented. Substitutions are made to transform the equations into standard quadratic forms that can then be solved.
3) The last problem finds an expression for the sum of the reciprocals of the terms containing the roots of a quadratic equation.
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.
This document discusses complex numbers. It provides examples of solving quadratic equations using complex numbers, multiplying and dividing complex numbers, and working with powers of i. It also includes exercises involving combining, expanding, dividing, and simplifying complex number expressions. Key points covered include using i=√-1, FOIL method for multiplication, conjugate multiplication, and the cyclic pattern of powers of i.
Here are the steps to solve this equation by factorising:
1) Factorise the left hand side: 3(x - 1)(x + 2)
2) Set each factor equal to 0:
x - 1 = 0
x + 2 = 0
3) Solve for x:
x = 1
x = -2
Therefore, the solutions are x = 1 or x = -2.
MIT Math Syllabus 10-3 Lesson 7: Quadratic equationsLawrence De Vera
This document discusses different methods for solving quadratic equations:
1) Factoring - Setting each factor of the factored quadratic equation equal to zero and solving.
2) Taking square roots - Taking the square root of both sides to isolate the variable.
3) Completing the square - Adding terms to complete the quadratic into a perfect square trinomial form.
4) Quadratic formula - A general formula for solving any quadratic equation using the coefficients.
The discriminant (b^2 - 4ac) determines the nature of the solutions, with positive discriminant yielding two real solutions and negative or zero discriminant yielding non-real or repeated solutions.
This document contains 10 math questions related to trigonometric functions, sets, relations and functions, complex numbers, sequences and series, straight lines, conic sections. The questions range from proving identities and equations to finding specific values based on given information. They require various trigonometric, algebraic and geometric problem solving skills at a higher-order thinking level.
The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
More Related Content
Similar to NCERT-Solutions-for-Class-11-Maths-Chapter-5-Complex-Numbers-and-Quadratic-Equations.pdf
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise solutions to each problem in 3 sentences or less.
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise 3-sentence or less solutions for each of the 50 math problems.
Final Exam Name___________________________________Si.docxcharlottej5
Final Exam Name___________________________________
Silva Math 96 Spring 2020
YOU MUST SHOW ALL WORK AND BOX YOUR ANSWERS FOR CREDIT. WORK ALONE.
Solve the absolute value inequality. Write your answer
in interval notation.
1) |2x - 12 |> 2
Solve the compound inequality. Graph the solution set.
Write your answer in interval notation.
2) -4x > -8 and x + 4 > 3
Solve the three-part inequality. Write your answer in
interval notation.
3) -1 < 3x + 2 < 14
Solve the absolute value equation.
4) 4x + 9 = 2x + 7
Solve the compound inequality.
5) 3( x + 4 ) ≥ 0 or 4 ( x + 4 ) ≤ 4
Solve the inequality. Graph the solution set and write
your answer in interval notation.
6) |5k + 8| > -6
Solve the inequality graphically. Write your answer in
interval notation .
7) x + 3 ≥ 1
x-8 -6 -4 -2 2
y
8
6
4
2
x-8 -6 -4 -2 2
y
8
6
4
2
1
Graph the system of inequalities.
8) 2x + 8y ≥ -4
y < - 3
2
x + 6
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Find the determinant of the given matrix.
9) 10 5
0 -4
Use Cramer's rule to solve the system of linear
equations.
10) 6x + 5y = -12
2x - 2y = -4
Write a system that models the situation. Then solve the
system using any method. Must show work for credit.
11)A vendor sells hot dogs, bags of potato chips,
and soft drinks. A customer buys 3 hot dogs,
4 bags of potato chips, and 5 soft drinks for
$14.00. The price of a hot dog is $0.25 more
than the price of a bag of potato chips. The
cost of a soft drink is $1.25 less than the price
of two hot dogs. Find the cost of each item.
Use row reduced echelon form to solve the system.
12) x + y + z = 3
x - y + 4z = 11
5x + y + z = -9
2
Find the domain of f. Write your answer in interval
notation.
13) f(x) = 13 - 9x
If possible, simplify the expression. If any variables
exist, assume that they are positive.
14) 2x + 6 32x + 6 8x
Match to the equivalent expression.
15) 100-1/2
A) 1
1000
B) 1
10
C) 1
100
D) 1
10
Write the expression in standard form.
16) (5 + 8i) - (-3 + i)
Simplify the expression. Assume that all variables are
positive.
17) 5 t
5
z10
Solve the equation.
18) 3x + 1 = 3 + x - 4
Write the expression in standard form.
19) 3 + 3i
5 + 3i
3
Write the equation in vertex form.
20) y = x2 + 5x + 2
The graph of ax2 + bx + c is given. Use this graph to solve
ax2 + bx + c = 0, if possible.
21)
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
Solve the equation. Write complex solutions in standard
form.
22) 4x2 + 5x + 5 = 0
Graph the quadratic function by its properties.
23) f(x) = 1
3
x2 - 2x + 3
x
y
x
y
Solve the equation. Find all real solutions.
24) 2(x - 1)2 + 11(x - 1) + 12 = 0
Solve the problem.
25) The length of a table is 12 inches more than its
width. If the area of the table is 2668 square
inches, what is its length?
4
Solve the equation..
rational equation transformable to quadratic equation.pptxRizaCatli2
1. The document provides examples for solving quadratic equations that are not in standard form by transforming them into standard form ax2 + bx + c = 0 and then using methods like factoring or the quadratic formula.
2. It also gives examples for solving rational algebraic equations by multiplying both sides by the least common denominator to obtain a quadratic equation, transforming it into standard form, and then solving.
3. The examples cover topics like solving for the solution set, checking solutions, and using the quadratic formula to solve transformed equations.
The document provides information on exam format and topics that need to be studied for Form 4 and Form 5 exams.
It recommends setting targets and being familiar with exam format. The main topics covered are functions, quadratic equations, trigonometry, calculus, vectors, statistics, and index numbers. Exercise and practice are strongly emphasized. Sample exam papers and questions are provided to illustrate exam structure and level of difficulty.
The document provides information about sets and operations on sets such as union, intersection, complement, difference, properties of these operations, counting theorems for finite sets, and the number of elements in power sets. It defines key terms like union, intersection, complement, difference of sets. It lists properties of union, intersection, and complement. It presents counting theorems for finite sets involving union, intersection. It states that the number of elements in the power set of a set with n elements is 2n and the number of proper subsets is 2n-2.
Diploma_Semester-II_Advanced Mathematics_Complex numberRai University
This document provides information about complex numbers. It begins by introducing complex numbers and defining them as numbers of the form a + bi, where a and b are real numbers and i = √-1. It then discusses various representations of complex numbers including Cartesian (a + bi), polar (r(cosθ + i sinθ)), and Euler (re^iθ) forms. It also covers operations on complex numbers such as addition, subtraction, multiplication, division, and powers. De Moivre's theorem relating powers of complex numbers to trigonometric functions is presented. The document concludes by stating some basic algebraic laws for complex numbers.
1. The document discusses various methods for solving different types of equations including: polynomial equations using factoring, quadratic equations using the quadratic formula, rational equations by clearing denominators, radical equations by squaring both sides, and absolute value equations geometrically.
2. Specific examples are provided to demonstrate how to solve equations involving polynomials, rationals, radicals, powers, and absolute values. Steps include factoring, using the quadratic formula, clearing denominators, isolating and squaring radicals, and raising both sides to appropriate powers.
3. The document also discusses finding zeros of rational functions and domains by determining where the numerator or denominator is equal to zero.
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
This document provides an overview of solving quadratic equations by factoring. It discusses identifying quadratic equations, rewriting them in standard form, factoring trinomials in the form x^2 + bx + c, and determining roots. Several examples of factoring trinomials and solving quadratic equations are shown. Activities include identifying quadratic equations, rewriting equations in standard form, factoring trinomials, and solving equations by factoring. The document provides resources for further learning about quadratic equations and factoring.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFYssuser2e348b
1) The document discusses theorems and proofs related to quadratic equations. It provides a necessary and sufficient condition for two quadratic equations to have a common root.
2) Several examples of solving equations that can be reduced to quadratic equations are presented. Substitutions are made to transform the equations into standard quadratic forms that can then be solved.
3) The last problem finds an expression for the sum of the reciprocals of the terms containing the roots of a quadratic equation.
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.
This document discusses complex numbers. It provides examples of solving quadratic equations using complex numbers, multiplying and dividing complex numbers, and working with powers of i. It also includes exercises involving combining, expanding, dividing, and simplifying complex number expressions. Key points covered include using i=√-1, FOIL method for multiplication, conjugate multiplication, and the cyclic pattern of powers of i.
Here are the steps to solve this equation by factorising:
1) Factorise the left hand side: 3(x - 1)(x + 2)
2) Set each factor equal to 0:
x - 1 = 0
x + 2 = 0
3) Solve for x:
x = 1
x = -2
Therefore, the solutions are x = 1 or x = -2.
MIT Math Syllabus 10-3 Lesson 7: Quadratic equationsLawrence De Vera
This document discusses different methods for solving quadratic equations:
1) Factoring - Setting each factor of the factored quadratic equation equal to zero and solving.
2) Taking square roots - Taking the square root of both sides to isolate the variable.
3) Completing the square - Adding terms to complete the quadratic into a perfect square trinomial form.
4) Quadratic formula - A general formula for solving any quadratic equation using the coefficients.
The discriminant (b^2 - 4ac) determines the nature of the solutions, with positive discriminant yielding two real solutions and negative or zero discriminant yielding non-real or repeated solutions.
This document contains 10 math questions related to trigonometric functions, sets, relations and functions, complex numbers, sequences and series, straight lines, conic sections. The questions range from proving identities and equations to finding specific values based on given information. They require various trigonometric, algebraic and geometric problem solving skills at a higher-order thinking level.
The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
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Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
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Centrifugation is a powerful technique used in laboratories to separate components of a heterogeneous mixture based on their density. This process utilizes centrifugal force to rapidly spin samples, causing denser particles to migrate outward more quickly than lighter ones. As a result, distinct layers form within the sample tube, allowing for easy isolation and purification of target substances.
SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
Anti-Universe And Emergent Gravity and the Dark UniverseSérgio Sacani
Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and provide evidence for the fact that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
Embracing Deep Variability For Reproducibility and Replicability
Abstract: Reproducibility (aka determinism in some cases) constitutes a fundamental aspect in various fields of computer science, such as floating-point computations in numerical analysis and simulation, concurrency models in parallelism, reproducible builds for third parties integration and packaging, and containerization for execution environments. These concepts, while pervasive across diverse concerns, often exhibit intricate inter-dependencies, making it challenging to achieve a comprehensive understanding. In this short and vision paper we delve into the application of software engineering techniques, specifically variability management, to systematically identify and explicit points of variability that may give rise to reproducibility issues (eg language, libraries, compiler, virtual machine, OS, environment variables, etc). The primary objectives are: i) gaining insights into the variability layers and their possible interactions, ii) capturing and documenting configurations for the sake of reproducibility, and iii) exploring diverse configurations to replicate, and hence validate and ensure the robustness of results. By adopting these methodologies, we aim to address the complexities associated with reproducibility and replicability in modern software systems and environments, facilitating a more comprehensive and nuanced perspective on these critical aspects.
https://hal.science/hal-04582287
4. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Hence,
(-2 – 1/3i)3
= -22/3 – 107/27i
Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
11. 4 – 3i
Solution:
Let’s consider z = 4 – 3i
Then,
= 4 + 3i and
|z|2
= 42
+ (-3)2
= 16 + 9 = 25
Thus, the multiplicative inverse of 4 – 3i is given by z-1
12. √5 + 3i
Solution:
Let’s consider z = √5 + 3i
|z|2
= (√5)2
+ 32
= 5 + 9 = 14
Thus, the multiplicative inverse of √5 + 3i is given by z-1
13. – i
Solution:
Let’s consider z = –i
|z|2
= 12
= 1
Thus, the multiplicative inverse of –i is given by z-1
14. Express the following expression in the form of a + ib:
5. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Solution:
6. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Exercise 5.2 Page No: 108
Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.
1. z = – 1 – i √3
Solution:
2. z = -√3 + i
Solution:
7. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:
3. 1 – i
Solution:
8. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
4. – 1 + i
Solution:
5. – 1 – i
Solution:
9. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
6. – 3
Solution:
7. 3 + i
Solution:
10. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
8. i
Solution:
11. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Exercise 5.3 Page No: 109
Solve each of the following equations:
1. x2
+ 3 = 0
Solution:
Given quadratic equation,
x2
+ 3 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 1, b = 0, and c = 3
So, the discriminant of the given equation will be
D = b2
– 4ac = 02
– 4 × 1 × 3 = –12
Hence, the required solutions are:
2. 2x2
+ x + 1 = 0
Solution:
Given quadratic equation,
2x2
+ x + 1 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 2, b = 1, and c = 1
So, the discriminant of the given equation will be
D = b2
– 4ac = 12
– 4 × 2 × 1 = 1 – 8 = –7
Hence, the required solutions are:
3. x2
+ 3x + 9 = 0
Solution:
Given quadratic equation,
x2
+ 3x + 9 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 1, b = 3, and c = 9
So, the discriminant of the given equation will be
D = b2
– 4ac = 32
– 4 × 1 × 9 = 9 – 36 = –27
Hence, the required solutions are:
12. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
4. –x2
+ x – 2 = 0
Solution:
Given quadratic equation,
–x2
+ x – 2 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = –1, b = 1, and c = –2
So, the discriminant of the given equation will be
D = b2
– 4ac = 12
– 4 × (–1) × (–2) = 1 – 8 = –7
Hence, the required solutions are:
5. x2
+ 3x + 5 = 0
Solution:
Given quadratic equation,
x2
+ 3x + 5 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 1, b = 3, and c = 5
So, the discriminant of the given equation will be
D = b2
– 4ac = 32
– 4 × 1 × 5 =9 – 20 = –11
Hence, the required solutions are:
6. x2
– x + 2 = 0
Solution:
Given quadratic equation,
x2
– x + 2 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 1, b = –1, and c = 2
So, the discriminant of the given equation is
D = b2
– 4ac = (–1)2
– 4 × 1 × 2 = 1 – 8 = –7
Hence, the required solutions are
7. √2x2
+ x + √2 = 0
Solution:
13. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Given quadratic equation,
√2x2
+ x + √2 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = √2, b = 1, and c = √2
So, the discriminant of the given equation is
D = b2
– 4ac = (1)2
– 4 × √2 × √2 = 1 – 8 = –7
Hence, the required solutions are:
8. √3x2
- √2x + 3√3 = 0
Solution:
Given quadratic equation,
√3x2
- √2x + 3√3 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = √3, b = -√2, and c = 3√3
So, the discriminant of the given equation is
D = b2
– 4ac = (-√2)2
– 4 × √3 × 3√3 = 2 – 36 = –34
Hence, the required solutions are:
9. x2
+ x + 1/√2 = 0
Solution:
Given quadratic equation,
x2
+ x + 1/√2 = 0
It can be rewritten as,
√2x2
+ √2x + 1 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = √2, b = √2, and c = 1
So, the discriminant of the given equation is
D = b2
– 4ac = (√2)2
– 4 × √2 × 1 = 2 – 4√2 = 2(1 – 2√2)
Hence, the required solutions are:
14. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
10. x2
+ x/√2 + 1 = 0
Solution:
Given quadratic equation,
x2
+ x/√2 + 1 = 0
It can be rewritten as,
√2x2
+ x + √2 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = √2, b = 1, and c = √2
So, the discriminant of the given equation is
D = b2
– 4ac = (1)2
– 4 × √2 × √2 = 1 – 8 = -7
Hence, the required solutions are:
15. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Miscellaneous Exercise Page No: 112
1.
Solution:
2. For any two complex numbers z1 and z2, prove that
Re (z1z2) = Re z1 Re z2 – Im z1 Im z2
Solution:
16. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
3. Reduce to the standard form
Solution:
4.
17. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Solution:
5. Convert the following in the polar form:
18. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
(i) , (ii)
Solution:
19. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Solve each of the equation in Exercises 6 to 9.
6. 3x2
– 4x + 20/3 = 0
Solution:
Given quadratic equation, 3x2
– 4x + 20/3 = 0
It can be re-written as: 9x2
– 12x + 20 = 0
On comparing it with ax2
+ bx + c = 0, we get
a = 9, b = –12, and c = 20
So, the discriminant of the given equation will be
D = b2
– 4ac = (–12)2
– 4 × 9 × 20 = 144 – 720 = –576
Hence, the required solutions are
20. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
7. x2
– 2x + 3/2 = 0
Solution:
Given quadratic equation, x2
– 2x + 3/2 = 0
It can be re-written as 2x2
– 4x + 3 = 0
On comparing it with ax2
+ bx + c = 0, we get
a = 2, b = –4, and c = 3
So, the discriminant of the given equation will be
D = b2
– 4ac = (–4)2
– 4 × 2 × 3 = 16 – 24 = –8
Hence, the required solutions are
8. 27x2
– 10x + 1 = 0
Solution:
Given quadratic equation, 27x2
– 10x + 1 = 0
On comparing it with ax2
+ bx + c = 0, we get
a = 27, b = –10, and c = 1
So, the discriminant of the given equation will be
D = b2
– 4ac = (–10)2
– 4 × 27 × 1 = 100 – 108 = –8
Hence, the required solutions are
9. 21x2
– 28x + 10 = 0
Solution:
Given quadratic equation, 21x2
– 28x + 10 = 0
On comparing it with ax2
+ bx + c = 0, we have
a = 21, b = –28, and c = 10
So, the discriminant of the given equation will be
D = b2
– 4ac = (–28)2
– 4 × 21 × 10 = 784 – 840 = –56
Hence, the required solutions are
21. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
10. If z1 = 2 – i, z2 = 1 + i, find
Solution:
Given, z1 = 2 – i, z2 = 1 + i
11.
Solution:
22. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
12. Let z1 = 2 – i, z2 = -2 + i. Find
(i) , (ii)
Solution:
23. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
13. Find the modulus and argument of the complex number
Solution:
14. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of – 6 – 24i.
Solution:
Let’s assume z = (x – iy) (3 + 5i)
24. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
And,
(3x + 5y) – i(5x – 3y) = -6 -24i
On equating real and imaginary parts, we have
3x + 5y = -6 …… (i)
5x – 3y = 24 …… (ii)
Performing (i) x 3 + (ii) x 5, we get
(9x + 15y) + (25x – 15y) = -18 + 120
34x = 102
x = 102/34 = 3
Putting the value of x in equation (i), we get
3(3) + 5y = -6
5y = -6 – 9 = -15
y = -3
Therefore, the values of x and y are 3 and –3 respectively.
15. Find the modulus of
Solution:
16. If (x + iy)3
= u + iv, then show that
Solution:
25. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
17. If α and β are different complex numbers with |β| = 1, then find
Solution:
26. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
18. Find the number of non-zero integral solutions of the equation |1 - i|x
= 2x
Solution:
Therefore, 0 is the only integral solution of the given equation.
Hence, the number of non-zero integral solutions of the given equation is 0.
19. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
(a2
+ b2
) (c2
+ d2
) (e2
+ f2
) (g2
+ h2
) = A2
+ B2
.
Solution:
27. NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
20. If, then find the least positive integral value of m.
Solution:
Thus, the least positive integer is 1.
Therefore, the least positive integral value of m is 4 (= 4 × 1).
![NCERT Solutions Class 11 Mathematics
Chapter 5: Complex Numbers and Quadratic Equations
Exercise 5.1 Page No: 103
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.
1. (5i) (-3/5i)
Solution:
(5i) (-3/5i) = 5 x (-3/5) x i2
= -3 x -1 [i2
= -1]
= 3
Hence,
(5i) (-3/5i) = 3 + i0
2. i9
+ i19
Solution:
i9
+ i19
= (i2
)4
. i + (i2
)9
. i
= (-1)4
. i + (-1)9
.i
= 1 x i + -1 x i
= i – i
= 0
Hence,
i9
+ i19
= 0 + i0
3. i-39
Solution:
i-39
= 1/ i39
= 1/ i4 x 9 + 3
= 1/ (19
x i3
) = 1/ i3
= 1/ (-i) [i4
= 1, i3
= -I and i2
= -1]
Now, multiplying the numerator and denominator by i we get
i-39
= 1 x i / (-i x i)
= i/ 1 = i
Hence,
i-39
= 0 + i
4. 3(7 + i7) + i(7 + i7)
Solution:
3(7 + i7) + i(7 + i7) = 21 + i21 + i7 + i2
7
= 21 + i28 – 7 [i2
= -1]
= 14 + i28
Hence,
3(7 + i7) + i(7 + i7) = 14 + i28
5. (1 – i) – (–1 + i6)
Solution:
(1 – i) – (–1 + i6) = 1 – i + 1 - i6
= 2 – i7](https://image.slidesharecdn.com/ncert-solutions-for-class-11-maths-chapter-5-complex-numbers-and-quadratic-equations-230105123155-0676ea69/85/NCERT-Solutions-for-Class-11-Maths-Chapter-5-Complex-Numbers-and-Quadratic-Equations-pdf-1-320.jpg)
