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Countdown Class 7th Mathematics Chapter 10 Solution

Countdown Class 7th Mathematics Chapter 10 Solution
Countdown Class 7th Mathematics Solution

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Countdown Class 7th Mathematics Chapter 10 Solution

  1. 1. COUNTDOWN CLASS 7 ֍ Algebraic Expressions ֍ Countdown Maths Class 7 Chapter No: 10 Exercise 10a (Solution)
  2. 2. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 1 | P a g e My Email Address is: adilaslam5959@gmail.com Notes By Adil Aslam Algebraic Expressions The Number 4 can be written as the factors: 2 × 2 = 22 We read it as two squared or the power of two. 8 can be written as: 2 × 2 × 2 = 23(𝑡𝑤𝑜 𝑡𝑜 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑝𝑜𝑤𝑒𝑟) 16 can be written as: 2 × 2 × 2 × 2 = 24 (𝑡𝑤𝑜 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑝𝑜𝑤𝑒𝑟) Look at the number: 24 2 is called the base and 4 is called the exponent.  The exponent indicates the number of times the base occurs as a factor. Examples: 3 × 3 = 32 Where 3 is called base and 2 is called the exponent. 5 × 5 × 5 = 53 Where 5 is called base and 3 is called the exponent. In the same way we can use letters instead of numbers to express the base of exponent. Examples: 𝑥 × 𝑥 × 𝑥 = 𝑥3 Where 𝑥 is called the base and 3 is called the exponent. 𝑎 × 𝑎 × 𝑎 × 𝑎 = 𝑎4
  3. 3. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 2 | P a g e My Email Address is: adilaslam5959@gmail.com Where 𝑎 is called the base and 4 is called the exponent. Variable, Coefficient and Constants Look at the expression 2𝑥 Here 2𝑥 = 2. 𝑥 If 𝑥 = 2, then value of 2𝑥 will be 2 × 2 = 4 When 𝑥 = 3 then value of 2𝑥 will be 2 × 3 = 6 and so on. Since the value of 2𝑥 changes or varies according to the value given to ‘x’, so x is called a variable.  A variable is a symbol used to represent one or more values. Variable also called unknowns.  A number that is placed before the variable is called coefficient. Thus in 2𝑥, 2 is the coefficient. Examples: 𝑥 + 𝑦 + 3𝑧 Here 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 are variables and 3 is a coefficient of variable z.  A numeral without any variable is called a constant. Examples: 3, 5, 6 etc are constants. Algebraic Expressions An algebraic expression consists of a single term or terms connected by operations of addition, subtraction, multiplication and division is called an algebraic expression. Examples: 12, 𝑥, 3𝑥, 𝑥 − 2𝑦, 𝑥2 − 𝑦𝑧, −2 𝑥 + 𝑦, 3𝑦 + 7𝑧 + 5 7 etc. Terms An algebraic term is either a numeral, a variable or product of a numeral and one or more variables.
  4. 4. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 3 | P a g e My Email Address is: adilaslam5959@gmail.com Examples: 4, 2𝑥, 𝑥𝑦, 3𝑥2 𝑦, 6𝑥𝑦𝑧, 𝑥 𝑦 etc are terms. 𝟑𝒙 + 𝟓 = 𝟐𝟎  Here 3 is a Coefficient of Variable 𝑥.  Here 𝑥 is a Variable (mean change able) also called Unknown.  Here 5 is a Constant value (mean fix value).  Here + and = are Symbols/Operators/Signs.  Here 3x and 5 are a terms.  Here 3x + 5 is an Algebraic Expression.  Here 3x + 5 = 20 is an Equation. Polynomials A polynomial is an algebraic expression consisting of one or more terms, in each of which exponent of the variable is zero or a positive integer. Examples: 2, −𝑥, 3𝑥, 𝑥2 , 3𝑥 − 4, 1 2 𝑥3 , 𝑥2 + 3𝑥 − 1 are all polynomials. Questions: Which of the following expressions are polynomials? 1. 𝑥3 (Yes, it is a polynomial because all the terms having zero or positive exponent) 2. 𝑥¯2 (No, it is not a polynomial because exponent of x(base) is negative) 3. 1 𝑦 (No, it is not a polynomial because 1 𝑦 = 𝑦¯1 , exponent is negative) 4. 𝑥3 + 𝑥¯2 + 1 (No, it is not a polynomial because one term having negative exponent which is (𝑥¯2 )) 5. 𝑦 2 + 6𝑥 (yes it is a polynomial because all the terms having zero or positive exponent) 6. 𝑥 𝑦 + 5𝑥 (No, it is not a polynomial because 𝑥 𝑦 = 𝑥𝑦¯1 , exponent is negative)
  5. 5. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 4 | P a g e My Email Address is: adilaslam5959@gmail.com Degree of the Polynomials Degree of the polynomial is the degree of the highest degree of a part(term) in a polynomial. Examples: 1. 𝑥 + 1 (Polynomial having degree 1) 2. 𝑥2 + 𝑥 (Polynomial having degree 2) 3. 𝑥3 + 𝑥𝑦 − 1 (Polynomial having degree 2) 4. 𝑥2 𝑦2 + 𝑥3 + 𝑦2 − 5 (Polynomial having degree 4 because term 𝑥2 𝑦2 = sum of the exponents is equal to 4) 5. 2𝑥3 𝑦2 (Polynomial having degree 5) 6. 2.3 + 0.2𝑥 (Polynomial having degree 1) Types of Polynomials Linear Polynomial: Polynomial having degree one is called linear polynomial. Examples: 𝑥, 3𝑥, 4𝑥 + 1, 𝑥 + 𝑧 etc. Quadratic Polynomial: Polynomial having degree two is called linear polynomial. Examples: 𝑥2 , 𝑥2 − 3, 𝑥𝑦 + 1 etc. Cubic Polynomial: Polynomial having degree three is called linear polynomial. Examples: 𝑥³ , 𝑦3 + 2, 𝑥3 + 𝑧² etc. Variables in a Polynomials Polynomials in one Variable: Polynomial having only one type of variable is called polynomial in one variable.
  6. 6. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 5 | P a g e My Email Address is: adilaslam5959@gmail.com Examples: 1. 𝑥3 + 4 (Polynomial having only one variable which is “x”) 2. 𝑥3 + 𝑥 − 1 (Polynomial having only one variable which is “x”) 3. 𝑦4 + 𝑦 (Polynomial having only one variable which is “y”) 4. 𝑧 + 1 (Polynomial having only one variable which is “z”) Polynomial in two Variables: Polynomial containing two type of variables is called polynomial in two variables. Examples: 1. 𝑥2 + 𝑦 + 1 (Polynomial having two variables which are x and y) 2. 𝑥2 𝑧 + 8 (Polynomial having two variables which are x and z) 3. 𝑥3 𝑧 + 𝑥𝑧 (Polynomial having two variables which are x and z) Polynomial in three Variables: Polynomial containing three type of variables is called polynomial in two variables. Examples: 1. 𝑥 + 𝑦 + 𝑧 (Polynomial having three variables which are x, y and z) 2. 𝑥2 𝑦𝑧 + 𝑥𝑦2 𝑧 + 𝑥𝑦𝑧2 (Polynomial having three variables which are x, y and z) Monomial: A polynomial consisting of a single term is called a monomial. Examples: 𝑥, 3𝑥, 𝑥2 𝑦, −5𝑥𝑦, 𝑥 𝑦 etc. Binomial: A polynomial with two terms is called a binomial. Examples: 2𝑥 + 3, 𝑥2 − 3𝑦, 𝑥𝑦 + 2𝑧, 𝑥 2 + 5𝑦 etc. Trinomial: A polynomial with three terms is called a trinomial.
  7. 7. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 6 | P a g e My Email Address is: adilaslam5959@gmail.com Examples: 𝑥 + 𝑦 − 𝑧, 𝑥2 + 3𝑧 − 6, 2𝑥2 + 3𝑦 + 1, 𝑦 2 + 3𝑧 − 4𝑥 etc. Expression in Order Expressions Increasing Order Decreasing Order 𝑥 + 2𝑥2 + 5 5 + 𝑥 + 2𝑥2 2𝑥2 + 𝑥 + 5 −𝑎𝑏2 + 𝑎3 + 3𝑏3 + 2𝑎2 𝑏 3𝑏3 − 𝑎𝑏2 + 2𝑎2 𝑏 + 𝑎3 𝑎3 + 2𝑎2 𝑏 − 𝑎𝑏2 + 3𝑏3 Like and Unlike Terms Like Terms: Terms containing the same variables and same corresponding exponents are called likes terms. Examples: 2𝑥𝑧 𝑎𝑛𝑑 − 5𝑥𝑧 both terms having same variables which is “𝑥𝑦” and also having same exponent. 3𝑥2 𝑦 𝑎𝑛𝑑 − 1 2 𝑥²𝑦 both terms having same variables which is “𝑥2 𝑦” Unlike Terms: Terms containing different variables or same variables but different corresponding exponents are called unlike terms. Examples: 2𝑥𝑦 and 2𝑧𝑥 are unlike terms. 𝑥2 𝑦, 𝑥²𝑦⁴ and 𝑥𝑦2 are unlike terms because all the variables having different exponents.
  8. 8. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 7 | P a g e My Email Address is: adilaslam5959@gmail.com Exercise 10a 1. Add: I. 2𝑎 + 3𝑏 4𝑎 + 7b Solution: 2𝑎 + 3𝑏 4𝑎 + 7b 6𝑎 + 10𝑏 II. 3𝑎 − 4𝑏 2𝑎 + 5𝑏 Solution: 3𝑎 − 4𝑏 2𝑎 + 5𝑏 5𝑎 + 𝑏 III. 3𝑎 + 6𝑏 𝑎 − 7𝑏 Solution: 3𝑎 + 6𝑏 𝑎 − 7𝑏 4𝑎 − 𝑏 IV. −𝑎 − 4𝑏 3𝑎 + 𝑏 Solution: −𝑎 − 4𝑏 3𝑎 + 𝑏 ___________ ___________ ___________ ___________ 2𝑎 − 3𝑏
  9. 9. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 8 | P a g e My Email Address is: adilaslam5959@gmail.com V. 𝑥2 + 5𝑦 −3𝑥2 − 2𝑦 Solution: 𝑥2 + 5𝑦 −3𝑥2 − 2𝑦 −2𝑥2 + 3𝑦 VI. −3𝑎2 − 5𝑎𝑏 + 2𝑏2 7𝑎2 + 2𝑎𝑏 − 𝑏² Solution: −3𝑎2 − 5𝑎𝑏 + 2𝑏2 7𝑎2 + 2𝑎𝑏 − 𝑏² 4𝑎2 − 3𝑎𝑏 + 𝑏² VII. 4𝑥𝑦 − 8 𝑥𝑦 + 6 Solution: 4𝑥𝑦 − 8 𝑥𝑦 + 6 5𝑥𝑦 − 2 VIII. −4𝑎2 + 6𝑎 − 3 2𝑎2 − 3𝑎 + 7 3𝑎2 + 4𝑎 − 2 Solution: _____________ __ _____________________ _ __________ __________ __
  10. 10. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 9 | P a g e My Email Address is: adilaslam5959@gmail.com −4𝑎2 + 6𝑎 − 3 2𝑎2 − 3𝑎 + 7 3𝑎2 + 4𝑎 − 2 𝑎2 + 7𝑎 + 2 IX. 2𝑎2 − 3𝑎𝑏 + 4𝑏2 −7𝑎2 + 5𝑎𝑏 − 3𝑏2 −3𝑎2 + 4𝑎𝑏 − 52 Solution: 2𝑎2 − 3𝑎𝑏 + 4𝑏2 −7𝑎2 + 5𝑎𝑏 − 3𝑏2 −3𝑎2 + 4𝑎𝑏 − 52 −8𝑎2 + 6𝑎𝑏 − 4𝑏² X. −5𝑥2 + 6𝑥𝑦 − 8𝑦2 2𝑥2 + 7𝑥𝑦 + 3𝑦2 −𝑥2 − 3𝑥𝑦 − 4𝑦2 Solution: −5𝑥2 + 6𝑥𝑦 − 8𝑦2 2𝑥2 + 7𝑥𝑦 + 3𝑦2 −𝑥2 − 3𝑥𝑦 − 4𝑦2 −4𝑥2 + 10𝑥𝑦 − 9𝑦² XI. 4𝑥2 − 5𝑥𝑦 − 6𝑦2 , 10𝑥𝑦 − 6𝑥2 + 3𝑦2 , 3𝑦2 − 4𝑥2 + 2𝑥𝑦 Solution: __________________ ____ _____________________ ____ _____________________ ____
  11. 11. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 10 | P a g e My Email Address is: adilaslam5959@gmail.com 4𝑥2 − 5𝑥𝑦 − 6𝑦2 −6𝑥2 + 10𝑥𝑦 + 3𝑦2 −4𝑥2 + 2𝑥𝑦 + 3𝑦2 −6𝑥2 + 7𝑥𝑦 + 0𝑦2 = −6𝑥2 + 7𝑥𝑦 XII. 𝑥2 − 2𝑥𝑦 + 𝑦2 , 2𝑥𝑦 + 𝑥2 + 𝑦2 , 4𝑥2 − 𝑥𝑦 + 𝑦2 , 𝑥2 − 𝑦² Solution: 𝑥2 − 2𝑥𝑦 + 𝑦2 𝑥² + 2𝑥𝑦 + 𝑦2 4𝑥2 − 𝑥𝑦 + 𝑦2 𝑥2 + 0𝑥𝑦 − 𝑦² 7𝑥2 − 𝑥𝑦 + 2𝑦² 2. If 𝐴 = 2𝑥 − 3𝑦 + 4𝑧, 𝐵 = 5𝑥 − 6𝑦 + 7𝑧, 𝐶 = −𝑥 − 𝑦 + 𝑧, find 𝐴 + 𝐵 + 𝐶. Solution: 𝐴 + 𝐵 + 𝐶 … … … … … . (1) Now putting the values of 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 in equation in (1) = (2𝑥 − 3𝑦 + 4𝑧) + (5𝑥 − 6𝑦 + 7𝑧) + (−𝑥 − 𝑦 + 𝑧) = 2𝑥 − 3𝑦 + 4𝑧 + 5𝑥 − 6𝑦 + 7𝑧 − 𝑥 − 𝑦 + 𝑧 = (2𝑥 + 5𝑥 − 𝑥) + (−3𝑦 − 6𝑦 − 𝑦) + (4𝑧 + 7𝑧 + 𝑧) = 6𝑥 − 10𝑦 + 12𝑧 3. If 𝑋 = 2𝑥2 − 3𝑦𝑥 + 4𝑦2 , 𝑌 = 𝑥2 + 2𝑥𝑦 − 3𝑦2 , 𝑍 = −4𝑥2 + 5𝑥𝑦 + 𝑦2 , find the value of 𝑋 + 𝑌 + 𝑍. Solution: 𝑋 + 𝑌 + 𝑍 … … … … … . (1) Now putting the values of 𝑋, 𝑌 𝑎𝑛𝑑 𝑍 in equation in (1) = (2𝑥2 − 3𝑦𝑥 + 4𝑦2) + (𝑥2 + 2𝑥𝑦 − 3𝑦2) + (−4𝑥2 + 5𝑥𝑦 + 𝑦2 ) _____________________ ____ _________________ ________
  12. 12. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 11 | P a g e My Email Address is: adilaslam5959@gmail.com = 2𝑥2 − 3𝑦𝑥 + 4𝑦2 + 𝑥2 + 2𝑥𝑦 − 3𝑦2 − 4𝑥2 + 5𝑥𝑦 + 𝑦2 = (2𝑥2 + 𝑥2 − 4𝑥²) + (−3𝑥𝑦 + 2𝑥𝑦 + 5𝑥𝑦) + (4𝑦2 − 3𝑦² + 𝑦²) = −𝑥2 + 4𝑥𝑦 + 2𝑦² 4. Subtract: I. 𝑎 − 𝑏 + 𝑐 from 2𝑎 + 𝑏 − 𝑐 Solution: Way No.1 (2𝑎 + 𝑏 − 𝑐) − (𝑎 − 𝑏 + 𝑐) = 2𝑎 + 𝑏 − 𝑐 − 𝑎 + 𝑏 − 𝑐 = (2𝑎 − 𝑎) + (𝑏 + 𝑏) + (−𝑐 − 𝑐) = 𝑎 + 2𝑏 − 2𝑐 II. 3𝑎 − 2𝑏 − 4𝑐 from 2𝑎 + 3𝑏 + 𝑐 Solution: (2𝑎 + 3𝑏 + 𝑐) − (3𝑎 − 2𝑏 − 4𝑐 ) = 2𝑎 + 3𝑏 + 𝑐 − 3𝑎 + 2𝑏 + 4𝑐 = (2𝑎 − 3𝑎) + (3𝑏 + 2𝑏) + (𝑐 + 4𝑐) = −𝑎 + 5𝑏 + 5𝑐 III. 7𝑎2 − 8𝑎𝑏 − 𝑏2 from − 2𝑎2 + 3𝑎𝑏 − 2𝑏2 Solution: (−2𝑎2 + 3𝑎𝑏 − 2𝑏2) − (7𝑎2 − 8𝑎𝑏 − 𝑏2 ) = −2𝑎2 + 3𝑎𝑏 − 2𝑏2 − 7𝑎2 + 8𝑎𝑏 + 𝑏2 = (−2𝑎2 − 7𝑎2) + (3𝑎𝑏 + 8𝑎𝑏) + (−2𝑏2 + 𝑏2) = −9𝑎2 + 11𝑎𝑏 − 𝑏2 Way No.2 2𝑎 + 𝑏 − 𝑐 𝑎 − 𝑏 + 𝑐 𝑎 + 2𝑏 − 2𝑐 _______________− + −
  13. 13. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 12 | P a g e My Email Address is: adilaslam5959@gmail.com IV. 𝑥2 − 𝑦2 + 𝑧2 + 2𝑥𝑦 from 𝑥2 + 𝑦2 + 𝑧2 + 2𝑥𝑦 Solution: (𝑥2 + 𝑦2 + 𝑧2 + 2𝑥𝑦) − (𝑥2 − 𝑦2 + 𝑧2 + 2𝑥𝑦) = 𝑥2 + 𝑦2 + 𝑧2 + 2𝑥𝑦 − 𝑥2 + 𝑦2 − 𝑧2 − 2𝑥𝑦 = (𝑥2 − 𝑥2) + (𝑦2 + 𝑦2) + (𝑧2 − 𝑧2) + (2𝑥𝑦 − 2𝑥𝑦) = 0𝑥2 + 2𝑦2 + 0𝑧2 + 0𝑥𝑦 = 2𝑦² V. 𝑝4 − 2𝑝3 − 3𝑝2 − 4𝑝 − 5 𝑓𝑟𝑜𝑚 − 𝑝4 + 5𝑝3 + 4𝑝2 + 3𝑝 + 2 Solution: (−𝑝4 + 5𝑝3 + 4𝑝2 + 3𝑝 + 2) − (𝑝4 − 2𝑝3 − 3𝑝2 − 4𝑝 − 5) = −𝑝4 + 5𝑝3 + 4𝑝2 + 3𝑝 + 2 − 𝑝4 + 2𝑝3 + 3𝑝2 + 4𝑝 + 5 = (−𝑝4 − 𝑝4) + (5𝑝3 + 2𝑝3) + (4𝑝2 + 3𝑝2) + (3𝑝 + 4𝑝) + (2 + 5) = −2𝑝4 + 7𝑝3 + 7𝑝2 + 7𝑝 + 7 VI. 2𝑥2 + 3 𝑓𝑟𝑜𝑚 𝑥2 + 3𝑥 − 2 Solution: (𝑥2 + 3𝑥 − 2) − (2𝑥2 + 3 ) = 𝑥2 + 3 − 2 − 2𝑥2 − 3 = (𝑥2 − 2𝑥2) + (3𝑥) + (−2 − 3) = −𝑥2 + 3𝑥 − 5 5. The sum of two polynomials is 4𝑎𝑥 + 3𝑏𝑦 + 2𝑐𝑧. If one of them is 5𝑎𝑥 + 𝑏𝑦 − 𝑐𝑧, find the other. Solution: (4𝑎𝑥 + 3𝑏𝑦 + 2𝑐𝑧) − (5𝑎𝑥 + 𝑏𝑦 − 𝑐𝑧) = 4𝑎𝑥 + 3𝑏𝑦 + 2𝑐𝑧 − 5𝑎𝑥 − 𝑏𝑦 + 𝑐𝑧 = (4𝑎𝑥 − 5𝑎𝑥) + (3𝑏𝑦 − 𝑏𝑦) + (2𝑐𝑧 + 𝑐𝑧)
  14. 14. Instructor: Adil Aslam Subject: Mathematics Class 7 Chapter: 10 13 | P a g e My Email Address is: adilaslam5959@gmail.com = −𝑎𝑥 + 2𝑏𝑦 + 3𝑐𝑧 6. What should be subtracted from 4𝑎3 + 3𝑎2 − 𝑎 − 5 to give a remainder of 4. Solution: If we subtract 4𝑎3 + 3𝑎2 − 𝑎 − 9 then remainder is 4. (4𝑎3 + 3𝑎2 − 𝑎 − 5) − (4𝑎3 + 3𝑎2 − 𝑎 − 9) = 4𝑎3 + 3𝑎2 − 𝑎 − 5 − 4𝑎3 − 3𝑎2 + 𝑎 + 9 = (4𝑎3 − 4𝑎3) + (3𝑎2 − 3𝑎2) + (−𝑎 + 𝑎) + (−5 + 9) = 0𝑎3 + 0𝑎2 + 0𝑎 + 4 = 4 (which is remainder)  Best of Luck 
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Countdown Class 7th Mathematics Chapter 10 Solution Countdown Class 7th Mathematics Solution

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