CLASS IX MATHS CHAPTER REAL NUMBERS

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1 Previous knowledge question Ques. Define natural numbers? CLASS IX MATHS CHAPTER REAL NUMBERS counting numbers are known as natural numbers. Thus,,3,4,. etc. are natural numbers. Ques. Define whole numbers? All natural numbers together with zero are whole numbers. Thus 0,,,3,4. etc. are whole numbers. Ques.3 Define integers? All natural numbers, 0 and negatives of natural numbers form the collection of all integers. Thus..-5,-4,-3,-,-,0,,,3,4,5 etc. are integers. Ques.4 Define Rational numbers? The numbers of the form q p, where p and q are integers and q 0 are Rational numbers. Ques.5 Give an eample of Rational Number? 3, etc. 4 NCERT Solutions Ques. Is zero a rational number? Can you write it in the form, where p and q are integers and q 0? Yes. 0 etc. Also denominator q can be taken as a negative integer Ques. Find si rational numbers between 3 and 4. Let us take the 3 and 4 as rational numbers with denominator 6 7 Then the 6 rational numbers between 3 and 4 are

2 Ques.3 Find five rational numbers between and. The five rational numbers between and are Ques.4 State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number. (i) True, since the collection of whole number contains all the natural numbers. (ii) False, for eample is not a whole number. (iii) False, for eample is a rational number but not a whole number. Ques.5 State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form, Where m is a natural number. (iii) Every real number is an irrational number. (i) True, since collection of real numbers is made up of rational and irrational numbers. (ii) False, no negative number can be the square root of any natural number. (iii)false, For eample is real but not irrational. Ques.6 Are the square roots of all positive integers irrational? If not, give an eample of the square root of a number that is a rational number. No, For eample is a rational number. Ques.7 Show how can be represented on the number line. Consider a unit square OABC and transfer in onto the number line making sure that the verte O coincides with zero. Then OB Construct BD of unit length perpendicular to OB. Then OD Construct DE of unit length perpendicular to OD. Then OE

3 Construct EF of unit length perpendicular to OE. Then OF Using a compass, with centre O and radius OF, draw an arc which intersects the number line in the point R. Then R corresponds to. Representation of Ques.8 Write the following in decimal form and say what kind of decimal epansion each has:(i) (ii) (iii)4 (iv) (v) (vi) (i) It is a terminating decimal. (ii). It is a non- terminating repeating decimal. (iii) 4.5. It is a terminating decimal. (iv) is a non- terminating repeating decimal. (v) is a non- terminating repeating decimal. (vi) 0.85 is a terminating decimal. Ques.9 You know that. Can you predict what the decimal epansions of,,,, are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of carefully.]

4 Yes, it can be done as follows Ques.0 Epress the following in the form, Where p and q are integers and q 0. (i) 0.6 (ii)0.47 (iii)0.00 (i) (ii) (iii) Let () Multiplying both side of equation by () () () /3 Let () Multiplying both side of equation by () ()-() /99 Let () Multiplying both side of equation by () ()-() /999

5 Ques. Epress the following) in the form. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Let () Multiplying both sides by 0(since one digit it repeating ), we get () () - () Thus Here p, q Since goes on for ever, so there is no gap between and and hence they are equal Ques. What can the maimum number of digits be in the repeating block of digits in the decimal epansion of? Perform the division to check your answer. The maimum number of digits in the repeating block of digits in the decimal epansion of can be 6.

6 Thus, By long Division, the number of digits in the repeating block of digits in the decimal epansion of 6. The answer is verified

7 Ques.3 Look at several eamples of rational numbers in the form, Where p and q are integers with no common factors other than and having terminating decimal representations (epansions). Can you guess what property q must satisfy? The property that q must satisfy in order that the rational numbers in the form, where p and q are integers with no common factors other than, have terminating decimal representation (epansions) is that the prime factorization of q has only powers of or powers of 5 or both, i.e., q must be of the form m 5 n ; m 0,,,3,.., n 0,,,3, Ques.4 Write three numbers whose decimal epansions are non-terminating non-recurring Ques.5 Find three different irrational numbers between the rational numbers and. Thus,

8 Thus, Three different irrational numbers between the rational numbers and can be taken as Ques.6 Classify the following numbers as rational or irrational: (i) (ii) (iii) (iv) (v)

9 Thus, The decimal epansion is non- terminating non-recurring. is an irrational number. (ii) 5 is a rational number. (iii) The decimal epansion is terminating is a rational number (iv) The decimal epansion is non-terminating recurring is a rational number. (v) The decimal epansion is non-terminating non-recurring is an irrational number. Ques. 7 Visualise on the number line, using successive magnification.

10 Ques.8 Visualise on the number line, up to 4 decimal places. Ques.9 Classify the following numbers as rational or irrational: (i) (ii) (3 ) (iii) (iv) (v) π (i) - is a rational number and is an irrational number. is an irrational number. The difference of a rational number and an irrational number is irrational. (ii) (3 ) (3 ) 3 3 which is a rational number. (iii) which is a rational number. (iv) ( 0) is a rational and is an irrational number. is an irrational number The quotient of a non-zero rational number with an irrational number is irrational.

11 (v) π is a rational number and π is an irrational number. The product of a non zero with an irrational number is irrational π is an irrational number Ques.0 Simplify each of the following epressions: (i) (3 ) ( ) (ii) (3 ) (3 ) (iii) ( ) (iv) ( - )( ) (i) (3 )( ) (3 ) ( ) 3( ) Left Distributive law of multiplication over addition (3)() 3 ( ()) ( )( ) (ii) (3 ) (3- )(3 )(3 ) (3) ( ) (iii) ( ) ( )( ) ( ) ( ) 5 7 (iv) ( - )( ( ( ( ) ( 5 3 Ques. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π. This seems to contradict the fact that π is irrational. How will you resolve this contradiction? Actually is only an approimate value of π and also a non-terminating decimal.

12 Ques. Represent on the number line. Draw a line segment AB9.3 units and etend it to C such that BC unit. Find the midpoint O of AC. With O as a centre and OA as radius, draw a semicircle. Now, draw BD AC, intersecting the semicircle at D. Then BD units. With B as the centre and AD as radius,draw an arc, meeting AC produced at E. then,bebd units. Ques.3 Rationalise the denominators of the following: (i) (ii) (iii) (iv) Ques.4 Find: (i) 64 / (8 ) / 8 / 8 8 (ii) 3 /5 ( 5 ) /5 5/5 (iii)(5) /3 (5 3 ) /3 5 5 Ques.5 (i) 9 3/ (ii)3 3/ (iii) 6 3/4 (iv) 5 -/3 (i) 9 3/ (3 ) 3/ 3 3/ (ii) 3 /5 ( 5 ) /5 5/5 4 (iii)6 3/4 ( 4 ) 3/4 43/4 3 8 (iv)5 -/3 (5 3 ) -/3 5 - /5 Ques.6 simplify: (i) /3. /5 /3/5 03/5 3/5 (ii)(/3 3 ) 7 /3 37 /3 (iii) / / /4 /-/4 -/4 /4 (iv)7 /.8 / (78) / 56 / HOTS Questions Solved Ques. If a and b are rational numbers and 7 a b 7 77, We have find the value of a & b

13 u sin g we get 7 ( 7 ( ( 7) {( ) ( 7) ( a b) ( a b ab) ( a b ) ( a b)( a b) ( ) ( 7 ) ( ) ( 7 ) a b Hence a Ques. If 3 8, find the value of 9 a 7) ( 7) ( & 9 } b and & 77 b 7) 7) 7 a b 77 Given 3 8

14 HOTS QUESTIONS UNSOLVED 3 Ques. Find four rational numbers between 5 and 3 3 Let and y Then, 5 y Let d 3 and 3 y ( n ), Since n ( 5 ) 90 ( 3 3) ( 3 8) ( 3 8) ( 3 8) ( 3 8) 3 ( 3) ( 8) ( 3 8) ( 3 8) ( 3 8) ( 6) Hence So the find rational numbers between 5 3 and 3 are (d), (d), (3d), (4d) & (5d) ,,,, ,,,,

15 Hence, fine rational numbers between 5 3 and 3 are, 8 8, , 45 and Ques. Epress 0.45 as a fraction in simplest form. Let 0.45 Then And On Hence Multiplying ( ) again 000 eg subtracting by multiplaying ( ) ( ) from ( 3) ( )... eg by... we ( ) 000 ( 3) get

16 VALUE BASED QUESTIONS Ques. Aman was facing some difficulties in simplifying. His classmate Sonia gave 7 3 him a due to rational use the denominator for simplification Aman simplified the epression & thanked Sonia for this goodwill how did Amansimplified. What values does it 7 3 indicate? 7 3 ( 7 3) 7 3 ( 7 3) ( 7 3) ( 7 ) ( 3) It indicates Helpfulness, cooperativeness, knowledge p 47 Ques. Teacher asked the students can we write in form as? Mukta answered, q No, it is. Is Mukta correct? Justify her answer. Which values of Mukta are depicted 00 here? Yes

17 Let Multiplying 0 Subtracting eq ( ) Scientific temper Knowledge Curiousity ( ) from ( ) 9 by ( ) ( ) ACTIVITY Make a square root spiral by paper folding giving square roots of natural numbers from to 0.

18 CHAPTER POLYNOMIALS PREVIOUS KNOWLEDGE QUESTIONS Ques. Define constants? A symbol having a fied numerical value is called a constant For. eg -8, -6, π, etc Ques. Define Variables? A symbol which may be assigned different numerical values is known as a variable. For eg C π r, where r is the radius of the circle &π are constants C & r are variables Ques.3 Define algebraic epressions? A combination of constants and variables, connected by some or all of the operations, -, and, is known as an algebraic epressions. Ques.4 Define Polynomials? An algebraic epression in which the variables involved have only non-negative integral powers is called a polynomial. foreg NCERT SOLUTIONS Ques. Which of the following epressions are polynomials in one variable and which are not? State reasons for your answer. (i) (ii) y (iii) 3 t (iv) y (v) 0 y 3 t 50 (i) is a polynomial in one variable. (ii) y is a polynomial in one variable. (iii) 3 t is not a polynomial since the power of the variable is not a whole number. (iv) y is not a polynomial since the power of the variable is not a whole number. (v) 0 y 3 t 50 is a polynomial in three variables.

19 Ques. Write the co-efficients of in each of the following: (i) (ii) 3 (iii) (iv) (i) Coefficient of is. (ii) Coefficient of is. (iii) Coefficient of is. (iv) Coefficient of is 0. Ques.3 Give one eample each of a binomial of degree 35, and of a monomial of degree and 4 00 Ques.4 Write the degree of each of the following polynomials: (i) (ii) 4 y (iii) 5t - (iv) 3. (i) Degree is 3. (ii) Degree is. (iii) Degree is. (iv) No degree. Ques.5 Classify the following as linear, quadratic and cubic polynomials: (i) (ii) 3 (iii) y y 4 (iv) (v) 3t (vi) r (vii) 7 3 (i) Quadratic Polynomial. (ii) Cubic Polynomial. (iii) Quadratic Polynomial. (iv) Linear Polynomial. (v) Linear Polynomial. (vi) Quadratic Polynomial (vii) Cubic polynomial. Ques.6 Find the value of the polynomial at (i) 0 (ii) - (iii) Let f() (i) when 0 f (0) 5(0) 4(0) 3 3 (ii) When - f (-) 5(-) 4(-) (iii) Value of f() at f() 5() 4()

20 Ques.7 Find p(0), p() and p() for each of the following polynomials: (i) p(y) y y (ii) p(t) t t t 3 (iii) p() 3 (iv) p() ( ) ( ) (i) p(y) y y p(0) (0) (0) p() () () And, p() () () 4 3 (ii) p(t) t t t 3 p(0) 0 (0) (0) 3 p() () () 3 4 and p() () () (iii) p() 3 p(0) (0) 3 0 p() () 3 and p() () 3 8 (iv) p() ( ) ( ) p(0) (0 )(0 ) ( )() p() ( ) ( ) (0)() 0 and, p() ( ) ( ) ()(3) 3 Ques.8 Verify whether the following are zeroes of the polynomial, indicated against them. (i) P() 3, - (ii) P() 5 -, (iii) P(),, - (iv) P() ( ) ( - ), -, (v) P(), 0 (vi) P() l m, - (vii)p() 3, -, (viii) P(), (i) P() 3, -, P is a zero of P(). (ii) P() 5 -, is not a zero of P(). (iii) P(),,- P() () P(-) (- ) - - 0,- are zeros of P(). (iv) P() ( ) ( - ), -, P(-) (- )(- ) (0)(-3)

21 0 P() ( )( ) (3)(0) 0 -, are zeros of P(). (v) P(), 0 P(0) (0) is a zero of P(). (vi) P() l m, - P l m - m m 0 - is a zero of P() (vii) P() 3, -, P P( ) 3( ) - 3( ) is a zero of P() but is not a zero of P() (viii) P(), P 0 is not a zero of P() Ques.9 Find the zero of the polynomial in each of the following cases. (i) P() 5 (ii) P() 5 (iii) P() 5 (iv) P() 3 - (v) P() 3. (i) P() 5 P() is a zero of the polynomial P() (ii) P() 5 P() is a zero of the polynomial P() (iii) P() 5

22 P() is a zero of the polynomial P() (iv) P() 3 - P() is a zero of the polynomial P(). (v) P() 3 P() is a zero of the polynomial P(). Ques.0 Find the remainder when is divided by (i) (ii) - (iii) (iv) (v) 5 Let p() (i) 0 - Remainder p(-) (-) 3 3(-) 3(-) (ii) Remainder (iii) Remainder (0) 3 3(0) 3(0) (iv) π π 0 -π Remainder (-π ) 3 3(-π ) 3(-π ) - π 3 3π - 3π (v) 5

23 Remainder - - Ques. Find the remainder when 3 a 6 a is divided by a. Let p() 3 a 6 a a 0 a Remainder (a) a(a ) 6(a) a a 3 a 3 6a a 5a Ques. Check whether 7 3 is a factor of will be a factor of 3 7.only if 7 3 divides 3 7. leaving no remainder. Let p() only if 7 3 divides leaving no remainder Let p() Remainder is not a factor of Ques.3 Determine which of the following polynomials has ( ) as a factor: (i) 3 (ii) 4 3 (iii) (iv) ( ) (i) 3 Let p() 3 The zero of is -. P(-) (-) 3 (-) (-) By factor theorem, is a factor of 3.

24 (ii) 4 3 Let p() 4 3 The zero of is -. P(-) (-) 4 (-) 3 (-) (-) By factor theorem, is not a factor of 4 3. (iii) Let p() The zero of is -. p(- ) (- I) 4 3(- ) 3 3(-) (-) By factor theorem, is not a factor of S. (iv) Let p() ( ) The zero of is -. p(-) (-) 3 - (-) - ( )(-) By factor theorem, is not a factor of ( ). Ques.4 Use the factor theorem to determine whether g() is a factor of P() in each of the following cases: (i) P() 3 - -, g() (ii) P() 3 3 3, g() (iii) P() 3-4 6, g() - 3 (i) p() 3 - -, g() g() Zero of g() is -. Now, p(-) (-) 3 (-) (-) By factor theorem, g() is a factor of p(). (ii) p() 3 3 3,g() g() Zero of g() is -. Now, p(-) (-) 3 3(-) 3(-) By factor theorem, g() is not a factor of p(). (iii) p() 3-4 6, g() - 3 g() 0

25 3 0 3 Zero of g() is 3. Now, p(3) (3) 3-4(3) By factor theorem, g() is a factor of p(). Ques.5 Find the value of k, if - is a factor of P() in each of the following cases: (i) P() k, (ii) P() k (iii) P() k -, (iv) P() k - 3 k (i) p() k If - is a factor of p(), then p() 0 By Factor Theorem () () k 0 k 0 k 0 k - (ii) p() k If - is a factor of p(), then p() 0 () k() 0 k 0 k -( ) (iii) p() k - If - is a factor of p(), then p() 0 k() - () 0 k - (iv) p() k - 3 k If - is a factor of p(), then p() 0 k() - 3() k 0 k 3 k 0 k By Factor Theorem k By Factor Theorem Ques.6 Factorise:(i) 7 (ii) 7 3(iii) (iv) (i) (3 )-(3 ) (4- ) (3 ) Let p() 7

26 Then p() q() Where q() - By trial, we find that q 0 By Factor Theorem, is a factor of q() Similarly, by trial, we find that q 0 By Factor Theorm, is a factor of q() Therefore, 7 (3 )(4 ) (ii) ( 3) ( 3) ( 3) ( ) Let p() 7 3 Then p() q() Where q() By trial, we find that q(-3) (-3)

27 0 By Factor Theorem, (-3), i.e. ( 3) is a factor of q() similarly, by trial, we find that q 0 By Factor Theorem, - Therefore, is a factor of q() 7 3 ( 3) (3) ( 3) ( ) (iii) ( 3) ( 3) ( 3) (3 ) Let p() Then p() 6 6q() Where q() By Factor Theorem, -, i.e., is a factor of q() Similarly, by trial, we find that By Factor Theorem, - Therefore is a factor of q()

28 ( 3) (3 ) (iv) (3 4) (3 4) (3 4) ( ) Let p() 3 4 Then p() 3 3q() Where q() - By trial, we find that q By Factor theorem, - Similarly, by trial, we find is a factor of q() q(-) (-) - By Factor Theorem, -(-), i.e. () is a factor of q(). Therefore (3 4)( ) Ques.7 Factorise: (i) 3 (ii) (iii) (iv) y 3 y y - (i) 3

29 Let p() 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) (ii) Let p() Synthetic Division ( 4 5) ( ) But ( 5) ( 5) ( ) ( 5) ( )()(-5) (iii) Let P() By trial, we find that P(-) (-) 3 3(-) 3(-) By synthetic Division ( 0 ) ( ) But ( 0 ) 0-0 ( 0) ( 0) ( ) ( 0) ( ) ( -) ( -0) (iv) y 3 y y - Let p(y) y 3 y y By trial, we find that P() () 3 () 0 By Factor Theorem, (y ) is a factor of p(y)

30 Now, y 3 y y (y 3y ) (y ) But,y 3y y y y y(y ) (y ) (y )(y ) Ques.9 Use suitable identities to find the following products: (i) ( 4) ( 0) (ii) ( 8)( 0) (iii) (3 4) (3 5) (iv) (y ) (y - ) (v) (3 ) (3 ) (i) ( 4) ( 0) ( 4)( 0) (4 0) (4)(0) Using Identity IV 440 (ii) ( 8)( 0) ( 8)( 0) ( 8)[ (-0) (8) ( 0) Using Identify IV 80 (iii) (3 4) (3 5) (3 4) (3 5) (3 4)(3 (-5)] (3) (4 (-5) (3) (4) (-5) Using Identity IV (iv) (y ) (y - ) (y ) (y - ) (z ) (z - ), Where y z (z) - Using Identity III z - (y ) - Substituting the value of z y 4 - (v) (3 ) (3 ) (3) - () 9 4 Using Identity III Ques.0 Evaluate the following products without multiplying directly: (i) (ii) (iii)

31 (i) (00 3) (00 7) (00) (3 7) (00 (3)(7) (ii) (90 5) (90 6) (90) (5 6)(90) (5)(6) Using Identity IV Another Method: (00-5) (00-4) {00 (- 5)}{00 (- 4)} (00) {(-5) (-4)}(00) (-5)(-4) (iii) (00 4) (00-4) (00) - (4) Using Identity III Ques.0 Factorise the following using appropriate identities: (i) 9 6y y (ii) 4y - 4y (iii) - (i) 9 6y y 9 6y y (3) (3)(y) (y) (3 y) (3 y)(3 y) (Using Identity I) (ii) 4y - 4y 4y - 4y (y) - (y)() () (y- ) (y- )(y - ) (Using Identity II) (iii) - - () - (Using Identity II) Ques. Epand each of the following using suitable identities: (i) ( y 4z) (ii) ( - y z) (iii) (- 3y ) (v) (3a - 7b - c) (v) (- 5y-3z) (vi) (i) ( y 4z) ( y 4z) ( y 4z)

32 () (y) (4z) ()(y) (y)(4z) (4z)()... Using Identity V 4y 6z 4y 6yz 8z (ii) ( y z) ( (-y) z) () (- y) (z) ()(- y) (- y)(z) (z)()... Using Identity V 4 y z - 4y -yz 4z (iii) (- 3y z) {(- ) 3y z)} (- ) (3y) (z) (- )(3y) (3y)(z) (z)(- )... Using Identity V 4 9y 4z y yz - 8z (iv) (3a - 7b - c) {3a (-7b) (-c)} (3a) (-7b) (- c) (3a)(- 7b) (- 7b)(- c) (- c)(3a) 9a 49b c - 4ab 4bc 6ca (v) (- 5y - 3z) {(- ) 5y (- 3z)} (- ) (5y) (- 3z) (- )(5y) (5y)(- 3z) (- 3z)(- ) 4 5y 9z - 0y - 30yz z (vi) Ques. Factorise: (i) 4 9y 6z 7y - 4yz - 6z (ii) y 8z y 4 yz - 8z. (i) 4 9y 6z y 4 yz - 6z 4 9y 6z y - 4yz - 6z () (3y) (- 4z) ()(3y) (3y)(- 4z) (- 4z)(- ) { 3y (- 4z)} ( 3y - 4z) ( 3y- 4z)( 3y - 4z) (ii) y 8z y 4 yz - 8z y 8z y 4 yz - 8z (- ) y ( z) (- )y y( z)y ( z)(- ) (- y z) (- y z)( - y z) Ques.3 Write the following cubes in epanded form: (i)() 3 (ii) (a - 3b) 3 (iii) (iv)

33 (i)( ) 3 () 3 () 3 3()()( ) Using Identify VI 8 3 6( ) (ii) (a 3b) 3 (a) 3 (3b) 3 3(a)(3b)(a 3b) Using Identify VII 8a 3 7b 3 8ab(a 3b) 8a 3 7b 3 36a b 54ab (iii) (iv) Using Identity VII y Ques.4 Evaluate the following using suitable:(i) (99) 3 (ii) (0) 3 (iii) (998) 3 (i) (99) 3 (00 -) 3 (00) 3 () 3-3(00)()(00 -)...Using Identity VII (00 - ) (ii) (0) 3 (00 ) 3 (00) 3 () 3 3(00)()(00 )...Using Identity VI (600(00 ) (iii) (998) 3 (000 - ) 3 (000) 3 - () - 3(000)()(000 - )... Using Identity VII (000 - ) Ques.5 Factorise each of the following: (i) 8a 3 b 3 a b 6ab (ii) 8a 3 -b 3 - a b 6ab

34 (iii) 7-5a 3-35a 5a (iv) 64a 3-7b 3 44a b 08ab (v) 7p 3 - (i) 8a 3 b 3 a b 6ab (a) 3 (b) 3 3(a)(b) (a b) (a b) 3 (a b)(a b)(a b) (ii) 8a 3 - b 3 - a b 6ab (a) 3 - (b) 3-3(a)(b) (a - b) (a - b) 3... (Using Identity VII) (a - b)(a - b)(a - b) (iii) 7-5a 3-35a 5a 7-5a 3-35a 5a (3) 3 - (5a) 3-3(3)(5a)(3-5a) (3-5a) 3 (3-5a)(3-5a)(3-5a) (iv) 64a 3-7b 3-44a 08ab (4a) 3 - (3b) 3-3(4a)(3b) (4a - 3b) (4a - 3b) 3...(Using Identity VII) (4a - 3b)(4a - 3b)(4a - 3b) (v) 7p 3 - (3p) (Using Identity VII) Ques.6 Verify: (i) 3 y 3 ( y)( -y y ) (ii) 3 - y 3 ( - y)( y y ). (i) We know that ( y) 3 3 y 3 3y( y)... Using Identity VI 3 y 3 ( y) 3-3y( y) 3 y 3 ( y)[( y) - 3y) 3 y 3 ( y)( y y - 3y)... (Using Identity I) 3 y 3 ( y) ( - y y ) (ii) We know that ( - y) y 3-3y( - y) (Using Identity VII) 3 - y 3 ( - y) 3 3y( - y) 3 - y 3 ( - y)[( - y) 3y) 3 - y 3 ( - y)( - y y 3y)... (Using Identity I) 3 y 3 ( - y) ( y y )

35 Ques.7 Factorise each of the following: (i) 7y 3 I5z 3 (ii) 64m 3-343n 3. (i) 7y 3 5z 3 7y 3 I5z 3 (3y) 3 (5z) 3 (3y 5z){(3y) - (3y)(5z) (5z) } (3y 5z)(9y - 5yz 5z ) (ii) 64m 3-343n 3 64m 3 343n 3 (4m) 3 - (7n) 3 (4m - 7n) {(4m) (4m)(7n) (7n) } (4m - 7n)(l6m 8mn 49n ). Ques.8 Factorise : 7 3 y 3 z 3-9yz. 7 3 y 3 z 3-9yz (3) 3 (y) 3 (z) 3-3(3)(y)(z) {(3) (y) (z) - (3)(y) - (y)(z) - (z)(3)} (Using Identity VIII) (3 y z) (9 y z - 3y - yz - 3z). Ques.9 Verify that 3 y 3 z 3-3yz ( y z)[( - y) (y - z) (z - ) ]. L.H.S. 3 y 3 z 3-3yz ( y z) ( y z - y - yz - z)...(using Identity VIII) ( y z){( y z -y - yz - z)} ( y z)( y z - y -yz - z) ( y z){( - y y ) (y - yz z ) (z -z )} II) ( y z)[( - y) (y - z) (z - ) ]...(Using Identity Ques.30 If y z O, show that 3 y 3 z 3 3yz We know that 3 y 3 z 3-3yz ( y z)( y z - y - yz - z) (Using Identity VIII) (0) ( y z - y - yz - z) ( y z 0 ) 0 3 y 3 z 3 3yz Ques.3 Without actually calculating the cubes, find the value of each of the following: (i) (- ) 3 (7) 3 (5) 3 (ii) (8) 3 (-5) 3 (-3) 3. (i) (-) 3 (7) 3 (5) 3 (-) 3 (7) 3 (5) 3 0 3(-)(7)(5)...( (- ) (7) (5) 0) Using identity VIII

36 -60 (ii) (8) 3 (-5) 3 (-3) 3 0 3(8)(-5)(-3)... ( (8)(-5)(-3) 0) - Using identity VIII 6380 Ques.3 Give possible epressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : 5a - 35a (ii) Area: 35y 3y -. (i) 5a - 35a 5a - 0a - 5a 5a(5a - 4) - 3(5a - 4) (5a - 4)(5a - 3) The possible epressions for the length and breadth of the rectangle are 5a - 3 and 5a 4 (ii) 35y 3y - 35y 8y 5y - 7y (5y 4) - 3(5y 4) (5y 4)(7y-3) The possible epressions for the length and breadth of the rectangle are 7y - 3 and 5y 4. Ques.33 What are the possible epressions for the dimensions of the cuboids whose volumes are given below? (i) Volume: 3 - (ii) Volume :ky 8ky - 0k. (i)3 3( - 4) The possible epressions for the dimensions of the cuboid are 3, and -4. (ii) ky 8ky - 0k 4k(3y y - 5) 4k(3y 5y-3y-5) 4k{y(3y 5) - (3y 5)} 4k(3y 5)(y- ) The possible epressions for the dimensions of the cuboid are 4k, 3y 5 and y - HOTS QUESTIONS SOLVED 5 Ques. Factorise: 4

37 5 4 4 ( ) 4 ( 4) [ 4 0 ] [ ] [ 6( 4 ) ( 4 ) ] [( 4 )( 6 ) ] ( 4 )( 6 ) Ques. Find the value of p and q so that ( )&( ) Here 3 ( ) 0 p q Since is the factor of ( ) 4 ( ) If is a factor By factor theorem, 3 ( ) 0( ) p( ) q p q 0 3 p q 0 p q 3... ( ) Also 3 are the factors of 0 p q ( ), then ( ) ( ) is a factor of ( ), then ( ) 3 ( ) 0( ) p( ) q 0 0 If 0 0 p q 0 p q... ( ) New adding ( ) &( ) p q 3 p q 3p p 3 New put p 7 in eg ( ) 7 q 3 q 3 4 q 8

38 HOTS QUESTIONS SOLVED Ques. Factorise: ( ) 6 6 y If 4 4 5, evaluation Given 5 Squaring both sides, we get ( ) ( ) ( )( )( ) ( )( ) [ ] ( )( ) [ ]( )( ) [ ] ( )( ) ( )( ) [ ] ( )( )( )( ) sin y y y y y y y b ab a b a b a b ab a b a b a g U y y y y y y b a b a b a y y y y ( ) ( ) [ ] ( ) ( ) Hence sides both squaring again ab b a b a

39 VALUE BASED QUESTIONS Ques. The number of trees planted on van-mahotsav in a region was 03 X 98. Find the number of trees planted without actual multiplication. Whichvalues of the people living in his region are depicted here? 03 X 98 (00 3) (00 ) Environmental care Social Happiness Ques. If the were requested for fencing a rectangular field is 30m and the length eceeds its breadth by 5m, then find the area of rectangular field. Which values are depicted here. Let the length of the rectangular field be lm and breadth by bm Perimeter (lb) 30 lb 5....() Also we are given that l b5 or l b 5..() squaring eq. ()& (), then subtracting resultant of eq. () from () we get [( l b) ( l b) ] 5 5 [( l b) ( l b) ] [( l b) ( l b) ] ( 5 5)( 5 5) [ l] [ b] 0 0 4lb 00 lb 50 Therefore the area is 50m Knowledge Curiosity Activity

40 Interpret geometrically the factors of quadratic equation b c paper slips [take b 5, C 6] using square grids, strips

41 Previous Knowledge Questions CHAPTER 5 INTRODUCTION TO EUCLID S GEOMETRY Ques. Define point? A point is an eact position or location on a plane surface. Ques. Define line? A line isa straight path that is endless. Ques.3 Define straight line? A straight line is a line which lies enenlywith a point on itself. NCERT SOLUTIONS Introduction to Euclid s Geometry Ques. Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can pass through a single point. (ii) There are an infinite number of lines which pass through two distinct points. (iii) A terminated line can be produced indefinitely on both the sides. (iv) If two circles are equal, then their radii are equal. (v) In following figures,if AB PQ and PQ XY, then AB XY. (i) False. Infinite number of lines pass through a single point. (ii) False. There is unique line that passes through two distinct points (iii) True. Postulate. (iv) True. If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide therefore, their radii will coincide. (v) True. The first Aiom of Euclid. Ques. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? (i) Parallel lines (ii) Perpendicular Lines (iii) line segment (iv) radius of a circle (v) square. (i) Parallel lines. Lines which do not intersect each other anywhere are called parallel lines. (ii) Perpendicular lines. Two lines which are at a right angle to each other are called

42 perpendicular lines. (iii) Line segment. It is a terminated line. (iv) Radius. The length of the line-segment joining the centre of a circle to any point on its circumference is called its radius. (v) Square. All the sides are equal in length. Ques.3 Consider two Postulates given below: a. Given any two distinct points A and B, there eists a third point C which is in between A and B. b. There eist atleast three points that are not on the same line. Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid s Postulates? Eplain. a. Yes! These postulates contain two undefined terms: Point and Line. b. Yes! These postulates are consistent because they deal with two different situations (i) say that given two points A and B, there is a point C lying on the line in between them, (ii) say that given two points A and B, we can take another point C not lying on the line through A and B. These 'postulates' do not follow from Euclid's postulate however, they follow from Aiom 5.. Ques.4 If a point C lies between two points A and B such that AC BC, then prove that AC AB. Eplain by drawing the figure. AC BC AC AC BC AC Equals are added to equally AC AB BC AC coincides with AB AC Ques.5 In previous question, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point. In the previous question we proved that, AC () ( C be a point which is lies between A and B) Suppose D be mid point of A and B, AD () From () and (),

43 AC AD Every line segment has one and only one mid-point. Ques.6 In the following figure, if AC BD, then prove that AB CD. AC BD Given.() AC AB BC Point B lies between A and C..() BD BC CD Point C lies between B and D..(3) Substituting () and (3) in (), we get AB BC BC CD AB CD Subtracting equals from equals Ques.7 Why is aiom5, in the list of Euclid s aioms, considered a universal truth? (Note that the question is not about the fifth postulate.) Ans This is true for anything in any part of the world, this is a universal truth. Ques.8 How would you rewrite Euclid s fifth postulate so that it would be easier to understand? Two distinct intersecting lines cannot be parallel to the same line. Ques.9 Does Euclid s Fifth postulate imply the eistence of parallel lines? Eplain. If a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid s fifth postulate the line will not meet on this side of l. Net, we know that the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are,

44 therefore parallel. HOTS OUESTIONS SOLVED Ques. If a point 0 lies between two points P and R such that PO OR then prove that PQ PR From fig PO OR PR..() And PO OR (given).() Using () & () PO PO PR PO PR PQ PR Ques. Does Euclid s fifth postulateimply the eistence of parallel lines? Eample. If straight linelfalls on two straight lines m and nsuch that the sum of interior angles on the same side of lis 80 0, then by Euclids fifth postulates the lines will not meet on this side of l. Also the sum of interior angles on other side of l will be 80 0, they will not meet on the other side also. l and m never meet They are parallel

45 HOTS QUESTIONS UNSOLVED Ques. Prove that every line segment has one and only one midpoint. Let us prove this statement by contradiction method let us assume that the line segment PT has two midpoints R and S. PR PT PS PT ( R and S arethe midpoint according toassumption) PR PS But this is possible only if R and S coincide. VALUE BASED QUESTIONS Ques. Teacher holding two sticks AB & CD of equal length in her hands & marked their midpoints M & N respectively. She then asked the students whether AM is equal to ND or not Aprajitaanswered yes. Is Aprajita correct? State the aniom of Eudidsthat support her answer. Which values of Aprajita are depicted here? Ans - Yes Things which are halves of the same things are equal to one another. Curiousity, knowledge, truthfulness Ques. In a society, the number of persons using CNG instead of petrol for their vehicles has increased by 5 and now the number is 5. From a linear equation to find the original number of persons using CNG & solve it using Euclid s aiom. Which values are depicted in the questions? [Using Euclid s third postulates] 0 Environmental care Responsible citizens Futuristic ACTIVITY

46 To verify that for every line landfor everypoint A hot lying on l, thereeist a unique line m passing through A and parallel to l.