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TERM -1
CHAPTER:1
NUMBER SYSTEM
EXPECTED LEARNING OUTCOMES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Identify the Natural numbers, Whole numbers, rational numbers and Irrational numbers.
Understand the relationship between different types of systems of number.
Identify the decimal form of the rational and Irrational numbers
Represent the irrational numbers on the number line.
Represent the decimal numbers having more than one digit in the decimal part by
Magnification
Learn how to operate two rational numbers or two irrational numbers or one rational and one
irrational.
Learn to simplify expressions containing surds by using the operations addition, subtraction,
multiplication and division.
Learn how to rationalize the denominator and using it to simplify the expressions.
Learn how to simplify the expressions by using Laws of Exponents where the exponents are
real numbers.
CONCEPT MAP
Number System.vue
THREE LEVELS OF GRADED EXERCICES
LEVEL -1
1. How many integers are there between two successive integers?
2.
Find the value of
√
√
√ is always a rational number only if is a ………………..
4. Express 0. ̅ in the form where
Z and
5. Find two rational numbers between 0.5 and 0.55
6. Find two irrational numbers between 1.1 and 1.11
7. Represent √ on the number line.
8. Find the value of
9. Express 1.252525……. in the form where
10. Visualise 4.̅̅̅̅ on a number line up to 4 decimal places.
11. Find four different irrational numbers between
LEVEL- 2
12. Simplify :
√
13. Find a rational number between 0.02 and 0.2
14. Evaluate :
√
√
15. Simplify:
16. Simplify:- (a) (√
(b) (11+√
√
√ )
17. Find the value of
18. Find two irrational numbers between √ and 2
19. Find the value of
√ , then find
20. If
21. Represent √
on the number line.
√ then find the value of
23. Express 0.3 ̅ in the form where
22. If
24. Find the value of
25. If
⁄
√
√
√
√
√
⁄
√
√
⁄
determine the rational numbers
√
√
√
26. Find and
if
27. If
√ , then find the value of
28. If
√ , find the value of √
29. If
√ find the value of
√
LEVEL 3
30. Divide√
√ by √
√
31. Find an irrational number between √ and √
32. Find an irrational number between √ and √
33. Simplify:-
√
√
√
√
√
34. Simplify:-
√
√
35. If
, then find the value of
√
36. If
, find the value of
37. If
then show that
then find √
38. If √
39. Simplify :
40. If
41.If
√
√ √
√ √
√ √
√ √
√
and
and
√
√
√
√
√ √
√ √
√ √
√ √
then find the value of
√ √
√ √
, find the value of
√ √
.
√
VALUE BASED QUESTIONS
1. In a school, 5 out of every 7 children participated in ‗SAVE WILDLIFE‘ campaign organized by
the school authorities.
(i) What fraction of the students participated in the campaign?
(ii) Find what type of decimal expansion it has?
(iii) What value do the participating students possess?
2. A shopkeeper sells its items at the rate of Rs.
and Rs. Per item respectively. Which of the two is
a better deal for the customer? If the shopkeeper suggests Rs. Per item to be the better price, then
what moral value does it depicts?
3. Two students Ravi and Raj were quarrelling with each other on the issue: ―Whether ‗
number or an irrational number‖. Ravi said that ‗
isa rational
is a rational number as it is equal to
. Raj
argued that ‗ is an irrational number as it is a non-terminating non-recurring decimal.
(i) Who is correct? Ravi or Raj
(ii) Comment on the behavior of Ravi and Raj.
Error Analysis and Remediation
Sl.No.
1
2
3
4
5
6
ERRORS
REMEDIATION
Identifying the decimal expansion of Give more examples like
irrational numbers
1.010010001…,12.121121112…for irrational
1.2345…..is irrational
numbers. And make them understand how to
identify them.
Representing √ ,√ , … on a number Give more practice to start always with a
perpendicular line at the square root of the
line
preceding integer
Simply doing the magnification without Practice the magnification problems by using the
the rules
step by step rules
Operations of radicals
Clearing basic ideas of operations of like and
unlike variables
√ +√ =√
√ +√ = 2√
√ +√ =√
Simplification by using rationalization
Creating necessary ideas in simplification by
using L C M and other basic operations
Simplification by using exponential Practice more examples by using the laws
laws
Question Bank
1. The product of any two irrational numbers is :
(a) always an irrational number (b) always a rational number (c) always an integer
(d) sometimes rational, sometimes irrational
2. Every rational number is :
(a) a natural number (b) an integer (c) a real number (d) a whole number
3. A rational number between and is
(a)
(b)
(c)
(d)
4. Which of the following is an irrational number?
(a) √
(b) √
(c) √
(d) √
5. The number 0.̅̅̅̅ is equal to
(a)
(b)
(c)
(d)
6. . The rationalizing factor of denominator in
(a) √ +2√
(b) √
7. The value of
√
√
is
√
√
is
(a)
(d)
8. Insert four rational numbers between and
9. Express the number 0.2̅̅̅̅ in the form where p and q are integers, q
10 . Represent geometrically the number √
11. If a and b are rational numbers and
12. If
13. If √
√
√
√
√
and
√
√
√
√
√
on the number line.
√
√
, then find the value of
√
then find the value of √
√
14. Arrange the following in the ascending order.
√ ,√ ,√
15. Express 0.00323232….. in the form
√ , then find the values of a and b.
CHAPTER-2
POLYNOMIALS
EXPECTED LEARNING OUTCOMES
1. Identify the types of polynomials and their degree.
2. Identify the polynomials with reference to the number of terms.
3. Learn to find the value of a polynomial at a given point.
4. Factorise the polynomials using algebraic identities.
5. Learn the use of Remainder theorem and factor theorem to factorise the given polynomials.
CONCEPT MAP
.~.~New Map.vue rpu.vue
THREE LEVELS OF GRADED EXERCICES
LEVEL-1
3 4
1.The degree of the polynomial 4x -x +3x is ---2. Factorisation of x3+1=------3. If p(x)=2x3-3x then p(1)=
4. Zero of the polynomial 3x—5 is -----5. When p(x) =2x 2-x-6 is divided by x-2 then the remainder is-----
Ans1)4 2)(x+1)(x2+x+1) 3) 2-3=-1 4.)x=5/3
5. )p(2)
LEVEL- 2
2
2
1. Factorise a +b +2ab+2bc+2ac.
2. Factorise
3. Verify if 2 and 0 are zeroes of the polynomial
4. Evaluate : 999
3
5. Expand: (3a+5b)
6. If
3.
then find the value of
ANS1)(a+b+c)(a+b+c) 2)(2x+5)(3x+1)
5)27a3+135a2b+225ab2+125b3 6)36
3.)yes
4)997002999
LEVEL -3
1. If (x+3) and(x-3) are both factors of ax2+5x+b then show that a=b.
2. If x=
2
√
then find the value of x -4x+1.
3. Find the value of a if x+6 is a factor of
4. Factorise
5. Find the remainder when
is divided by x-a.
6. If ( x − 4) is a factor of the polynomial 2 x 2 + Ax + 12 and ( x − 5) is a factor of the
polynomial x 3 − 7 x 2 + 11 x + B , then what is the value of ( A − 2 B )?
ANS; 2)0 3)a=-232
4)(x+1)(x-1)(x+2) 5)5a 6)A=-11,B=-5
VALUE BASED QUESTIONS
1. If the perimeter of two rectangular fields are (a+b)2 and (a2 +ab+b2)units .A property dealer
sells these fields by using a=5 and b=2.If the dealer suggests the perimeter of the field
(a+b)2units is better for the customer. Is it true? What value is depicted by the customer?
2.If a teacher divides a material of volume x3+6x2+12x+8 cubic units among three students of his
class equally .is it possible to find the quantity of material each gets? What value is depicted by
the teacher?
3. In a restaurant the owner says you divide x3-3x2-x+6 by x-3 and pay that money. If the owner
does not return any balance then what is the amount paid and remaining balance If the owner
returns the balance then what value is depicted by him.?
ANS;1)yes,honesty 2)x+2, impartial
3)x2-1, 3, honesty
Error Analysis and Remediation
SLNO COMMON ERRORS
1.
4X2 +2X .Degree is 4
2.
(x-y)2
3.
(2x)2 is written as 2x2
4.
x2-5x+6 is factorized as (x-6)(x+1)
5.
To find the value of the polynomial x2+2x
if x=-2 the student writes it as
-22+2x(-2)=-4+-4=-8
is written as x2-y2
REMEDIATION
Correction- Degree is 2 and coefficient of x2 is
4.concept of degree and coefficient must be
made very clear to the students
Correction (x-y)2=x2-2xy+y2
Students must be thoroughly drilled in
Algebraic identities
Correction (2x)2 =4x2 importance of brackets
must be emphasized
Correction x2-5x+6=x2-3x-2x+6 = (x-3)(x-2)
Product of -6 and1 is -6 and not 6. Practice
exercises must be worked out in class
Correction (-2)2 +2(-2)=4+-4=0
Importance of brackets must be emphasized.
Question Bank
1. On factorising x2+ 8x + 15, we get :
(a) (x + 3) (x – 5) (b) (x – 3) (x + 5)
(c) (x + 3) (x + 5 )
(d) (x – 3) (x – 5)
2
2. On dividing x – 2x – 15 by (x – 5), the quotient is (x + 3) and remainder is 0. Which of the
following statements is true?
(a) x2– 2x – 15 is a multiple of (x – 5)
(b) x2– 2x – 15 is a factor of (x – 5)
(c) (x + 3) is a factor of (x – 5)
(d) (x + 3) is a multiple of (x – 5)
3. The value of the polynomial 3x + 2x2– 4 at x = 0 is :
(a) 2 (b) 3 (c) – 4
(d) 4
4. On factorising x2– 3x – 4, we get :
(a) (x – 4) (x + 1) (b) (x – 4) (x – 1) (c) (x + 4) (x – 1) (d) (x + 4) (x + 1)
5. If p(x) = x + 3, then p(x) + p(–x) is equal to :
(a) 3 (b) 2x (c) 0
(d) 6
2
6. If x + kx + 6 = (x + 2) (x + 3) for all x, then the value of kis:
(a) 1 (b) – 1 (c) 5 (d) 3
7. Writethe degree of the polynomial 4x4 + 0x5?
8. The coefficient of x in the expansion of (x + 3)3is :
(a) 1 (b) 9 (c) 18 (d) 27
2
9. Find the zeroes of the polynomial p(x) = x – 5x + 6.
3
6
2
10. For the polynomial 3x + 10x -5x – 25 write:
(i) The degree of the polynomial
3
(ii) The coefficient of x
(iii) The coefficient of x6
(iv) The constant term
11. Check whether the polynomial 3x – 1 is a factor of 9x3– 3x2 + 3x – 1.
12. Using factor theorem, show that (2x + 1) is a factor of 2x3+ 3x2– 11x – 6.
13. Check whether (x + 1) is a factor of x3 + x + x2+ 1.
ANS 1)c 2)a 3)c 4)a 5)d 6)c 7)4 8)a 9)3,2 10a)6 b)3 c)10 d)-25
Projects
1. Verify the following identities
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a + b) (a – b) = a2 – b2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b)3 = a3 + b3 + 3ab (a + b)
(a – b)3 = a3 – b3 – 3ab (a – b)
( a3 + b3)= (a + b) (a2 – ab + b2)
(a3 – b3)= (a – b) (a2 + ab + b2)
a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
2. Verify factor theorem for at least 5 different polynomials.
3. Verify Remainder theorem for at least 5 different polynomials.
Practicals
1. Representation of a quadratic polynomial
Procedure

To represent this we need 1 square tile representing x2 , 5 algebra tiles representing x and 6
algebra tiles representing

By splitting the middle term of the given polynomial we get the expression x2 -3x -2x + 6.
Place a square tile of dimension 10x10 representing x2 .

Subtract 3 tiles of dimension 10 x 1 each to any side of the tile x2.The area of new shape
formed represents x2 -3x.

Add 6 tiles of dimension 1 x 1 each to get 2 tiles of dimension 10 x 1 each to the side adjacent
to the previous side. The area of new shape formed represents x2 -3x+6.

Subtract 2 tiles of dimension 10 x 1 each to complete the rectangle. The area of new shape
formed represents x2 -3x+6-2x.
2. Verifying algebraic identities
To prove the algebraic identity (a-b)3 = a3 -3a2b+3ab2-b3 using unit cubes.
Take any suitable value for a and b.
Let a=3 and b=1.
To represent (a-b)3 make a cube of dimension
(a-b) x (a-b) x (a-b) i.e. 2x2x2 cubic units as shown below.
To represent (a)3 make a cube of dimension a x a x a. i.e. 3x3x3 cubic units as shown below.
To represent 3ab2 make 3 cuboids of dimension a x b x b i.e. 3x1x1 cubic units as shown below.
To represent a3 + 3ab2 , join the cube and the cuboids formed in steps 2 and 3 as shown below.
To represent a3 + 3ab2- 3a2b extract from the shape formed in the previous step 3 cuboids of
dimension 3x3x1 to get the shape shown below.
To represent a3 + 3ab2- 3a2b-b3 extract from the shape formed in the previous step 1 cube of
dimension 1x1x1 .The shape shown below will be obtained.
Arrange the unit cubes left to make a cube of dimension2x2x2 cubic units.
Observe the following
3
•The number of unit cubes in a = …27…..
•The number of unit cubes in 3ab2 =…9……
•The number of unit cubes in 3a2b=…27……
•The number of unit cubes in b 3 =…1……
CBSE Study Material For Class 9
Mathematics
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