Value of a polynomial at a point

This task is based on following objectives: 1. Value of a polynomial at a point 2.Concept of zero of a polynomial

Q1 If P (x) = x^4 + 2x^3 - 10x^2 - 14x + 21, then find P(1), P(-1) and P(1/2). Q2 Find the zeroes of the following polynomials: a) P(x) =3x-5 b) P(x) = 2x+7 Q3 Check whether -2 and 2 are the zeroes of the polynomial x^4 − 16 . Q4 Give examples to justify the following statements: a) A zero of a polynomial need not be 0. b) 0 may be a zero of a polynomial. c) Every linear polynomial has one and only one zero. d) A polynomial can have more than one zero.

Assignment- Polynomials Class 9

This task is based on the following concepts:
1. Definition of a polynomial, coefficient,constant term and degree
 2. Classsification of polynomials according to number of terms
3. Classification of polynomials according to degree Reference CLICK HERE!!!

Q1 Write 5 examples of expressions which are not polynomials. Justify your answers.
Q2 Give examples of the polynomials:
 a) Cubic and binomial
 b) Cubic and monomial
c) Quadratic and trinomial
 d) Quadratic and monomial
e) Linear and binomial
 f) Linear and monomial
Q3 For the polynomial p(x) = 5x^3-3x^2+2x+√2, mark the statements as true or false and justify.
 a) The degree of polynomial p(x) is 4.
b) The degree of polynomial p(x) is 3.
c) The coefficient of x^2 is 3.
d) The coefficient of x is 2
 e) The constant term is 3
 f) The number of terms is 4
Q4 Justify the following statements with examples:
a) We can have a trinomial having degree 7.
b) The degree of a binomial cannot be more than two.
 c) There is only one term of degree one in a monomial.
d) A cubic polynomial always has degree three.
 Q5 Complete the entries P(x)= 5x^7-6x^5+7x-6
 Coefficient of x^5 =
Degree of P(x) =
Constant term=
 Number of terms=

Assignment 4- Number System Class 9

Exponents

Learn about Exponents

Concept of Rationalising the denominator- Class 9

1. Pls visit the given link for details. Click here!!!

Watch this video:

Answer the following questions:

Q1

Q2 Rationalize the denominator of the following :

a) 1/(√3+ 5)

b) 1/(√2 - 5)

c) (√2+ 5)/(√2 - 5)

d) (√2+ √5)/(√2 - √5)

Proof sqrt 2 is an irrationl number

Click on the image to visit the web link.

Read the explanation of why sqrt 2 is an irrational number?

Try to write your own proof for proving sqrt 5 is an irrational number.

Prime Factorisation

It is observed that 35= 5x7

It is observed that 1092 = 2x2x3x13x7

HCF (35,1092) = 7

LCM (35,1092)= 5460

LCM (35,1092) X HCF( 35,1092) = 35 X 1092

The Fundamental theorem of Arithmetic states that every composite number can be uniquely expressed as product of prime factors, apart from the order of factors.

Find the HCF and LCM of 65 and 90. Prime factors are shown below.

Home Assignment- Real Numbers Class 10

Please do the following questions in your notebook.

Q1. State Euclid’s Division lemma.
Q2. Find the H.C.F of 24 & 18 using Euclid’s Division algorithm.
Q3. By what number should 1365 be divided to get 31as quotient and 32 as remainder?
Q4. By the Prime factorization method find the L.C.M. of 144 & 198.Also find H.C.F. using
formula.
Q5. State the Fundamental theorem of Arithmetic.
Q6. Justify 11X13X12X14+12 is a composite number.
Q7. Write the condition for p/q,q≠0,p & q are integers to have a terminating decimal expansion.
Q8. Write the condition for p/q,q≠0,p & q are integers to have a non-terminating & repeating decimal expansion.
Q9. Without actual division comment on the type on the type of decimal expansion of the given rational number
17/90
31/(5^3.2^2 )
41/1000
26/500
If the decimal expansion terminates then after how many decimal it will terminate?
Q10. Prove that √5 is an irrational number.
Q11. Prove that 3+ √5 is an irrational number.
Q12. Prove that2/√5 is an irrational number.
Q13. Show that every positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 for some integer q.

Q14. Show that any number of the form 4^n where n is a natural number can never end with the digit 0.
Q15. Fill in the empty boxes.
Q16. Show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q+1 for some integer q.
Q17. Show that every positive odd integer is of the form 4q+1or 4q+3 for some integer q.
Q18. Show that every positive even integer is of the form 4q or 4q+2 for some integer q.
Q19. Show that one and only one out of n,n+2,n+4 is divisible by 3 ,where n is any positive integer.
Q20 If H.C.F(72,120)=24 then find LCM(72,120)?
Q21. If LCM of two numbers is 2079 and their HCF is 27. If one of the numbers is 297.Find the other number.
Q22. Find the largest number which divides 2053 and 967 leaving remainder 5 and 7 respectively?
Q23. Find the greatest number which exactly divides 285 and 1249 leaving remainders 9 and 7 respectively.
Q24. Find the H.C.F of 81 and 237 and express it as a linear combination of 81 and 237.
Q25. If the H.C.f of 210 and 55 is expressible in the form 210 x 5+55 y,find y.

Irrational numbers/Rational numbers-Class9

Shade all irrational numbers in Blue colour and rational numbers in Red colour.

Practice these MCQ's... Click here!!!

Assignment 3 Number System- Class 9

Assignment 2 Number System Class 9

Do the following questions in your notebook.
1. Insert 3 rational numbers between 1/5 and 3/5 .
2. Write 5 rational numbers and 5 irrational numbers in decimal form.
3. Represent 2+√2 on number line.
4. Which of the following number lies between 2 and 3?

(a) 1+√5 (b) √3+2 (c) 1+√2 (d) √3-1
5. √7.29 is a
a) rational number between 2 and 3
b) integer
c) irrational number
d) A rational number greater than 7.

6. Which of the following is an irrational number between 2 and 3?
a) 2.357357….
b) 2.101001000….
c) 2.05131313….
d) 2.579

Assignment 1 Number System Class 9

Part 1
Write your views about the following statements:
 1. Is every natural number a rational number?
 2. Is every rational number a natural number?
3. Is every natural number a real number?
4. Is every real number a natural number?

Part 2
1. 100 rational numbers can be inserted between 2 and 7
2. Can we insert only 100 rational numbers between 2 and 7?
3. How many rational numbers can be inserted between 2 and 7?

Part 3
1. Is 2 and 5 are co primes?
2. 1.010010001… is an irrational number? True/False
3. What is rationalizing factor of (2+ sqrt3)?

Part4
 1. Give an example of an irrational number between 2 and 3.
 2. Give an example of a rational number between 2 and 3.
3. What is smallest Prime number?
4. Which whole number is not a natural number?
5. Am I right if I say “4 is smallest composite number?”
 6. Every real number is represented by a unique point on a number line. (True/False)

Guess the Number

An interactive number line tool for exploring decimal representation on Number line.

Here is the link.

Explore the following

Task : Exploration of facts

The sum or difference of a rational number and an irrational number is irrational.

The product or quotient of a non- zero rational number with an irrational number is irrational.

If we add, subtract, multiply, or divide two irrationals, the result may be rational or irrational.

To make a square root spiral

Follow the steps shown in the following picture and make a square root spiral.
Procedure:

1. Draw a line segment OP1 of unit length (Let OP1 = 2cm).
2. At P1 construct an angle of 90 degrees. Let the new ray be P1X, such that angle OP1X = 90 degrees.
3. Cut off P1P2 = 2cm on P1X.
4. Join OP2.
5. Find OP2 using Pythagoras theorem. OP2 = Sqrt (2)
6. At P2 construct an angle of 90 degrees. Let the new ray be P2Y, such that angle OP2Y= 90 degrees.
7. Cut off P2P3 = 2cm on P2Y.
8. Join OP3.
9. Find OP3 using Pythagoras theorem. OP3 = Sqrt(3).

Keep repeating, till you plot upto Sqrt(12).

Decimal expansion of rational numbers

Rational numbers are those numbers which can be expressed in the form p/q ,q is not equal to 0, p and q are integers.

The decimal expansion of rational numbers is either terminating or non terminating and recurring.

Example 1: Consider the decimal expansion of 7/8
7/8 = 0.875
its decimal expansion is terminating.
Consider the decimal expansion of 1/7.
1/7 = 0.142857142587142587....
Its decimal expansion is non terminating and repeating. Exploratory task:
Observe the decimal expansions of 1/7, 2/7, 3/7, 4/7 and 5/7 .
1/7 = 0.142857142857142857142857142857...
2/7 = 0.285714285714285714285714285714...
3/7 = 0.428571428571428571428571428571...

4/7 = 0.571428571428571428571428571428...
5/7 = 0.714285714285714285714285714285...
What specific do you notice?
Observe the following:
1/8 terminates after 3 digits.
1/8 = 0.125
2/8=1/4 terminates after 2 digits.
1/4 = 0.25
3/8 terminates after 3 digits.
3/8 = 0.375
4/8=1/2 terminates after 1 digit.
1/2 = 0.5
5/8 terminates after 3 digits.
5/8 = 0.625
6/8=3/4 terminates after 2 digits.
3/4 = 0.75
7/8 terminates after 3 digits.
7/8 = 0.875
What specific do you notice?

Rational numbers between two given rational numbers

Exploration task:
Inserting rational numbers between two given rational numbers
1. Take any two rational numbers.
2. Add them.
3. Divide the result obtained by 2.
4. Observe the number obtained.
Is the answer a rational number?
Is it between two given numbers?
Brainstorming: How many rational numbers can be inserted between two rational numbers?

Real Number System

Watch this presentation

Real Numbers - Rational and Irrational

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Useful Web links to explore further:

The Real Number System

Purple Math on this topic

Rational and Irrational Numbers

Give examples in support of following statements:

• All natural numbers are whole numbers but not vice versa
• All natural numbers are rational numbers but not vice versa
• All whole numbers are rational numbers but not vice versa
• All integers are rational numbers but not vice versa

Learning Objectives- Number Systems Class 9

• To gain the knowledge of various types of numbers viz. natural numbers, whole numbers, prime numbers, rational numbers etc.which constitutes the Real number system
• To explore the relation between various types of numbers.
• To learn to insert rational numbers between two rational numbers.
• To appreciate the fact that infinitely many rational numbers can be inserted between two given rational numbers
• To learn to represent irrational numbers like √2, √3, √5 etc on the number line.
• To find the decimal expansion of real numbers and discriminate between a rational and irrational number on this basis.
• Learn to insert irrational numbers between two rational numbers.
• To learn addition, subtraction, multiplication & division of real numbers
• To learn the method of rationalising the denominator of a rational number.
• To extend laws of exponents for negative exponents too and apply laws of exponents in calculations involving exponents.

Real Numbers (Euclid's Division Lemma)

Euclid's division lemma states that " For any two positive integers a and b, there exist integers q and r such that a=bq+r , 0≤ r< b


Useful Video on finding HCF of two given numbers using Euclid's division lemma.