ADDIIONAL PROBLEMS ON AXIOMS<POSTULATES

Transcription

1 CHAPTER-3 UNIT-1 ADDIIONAL PROBLEMS ON AXIOMS<POSTULATES 1. Choose the correct option: i. If a=60 and b=a, then b=60 by... a) Axiom 1 b) Axiom 2 c) Axiom 3 d) Axiom 4 [a] 2. What is the angle between the hour`s hand and minute`s hand of a clock at (i) 1.40 hours, (ii) 2.15 hours? (Use 1 0 =60 minutes) 1 hour=360 0 m 1 hour the minuts and complete 1 revolutions=1 circle 1 circle= hours= hours=360/12= minutes is also = munites=30/5= hrs =32 minuts -22 6= hrs= 4 minuts 4 6= How much would hour`s hand have moved from its position ta 12 noon when the time is 4.24 p.m.? noon at 4.24 pm Number of minutes the hour hand moved = 24 minutes 1 minute = minuts = 24 6 = Let AB be a line segment and let C be the midpoint of AB to D such that B lies between A and D. Prove that AD+BD=2CD. A a C a B D In the above figure AC=BC=a AB s extended to D Now AD+BD = AC+CB+BD+BD = a + a +BD+BD = 2a + 2BD = 2(a+BD) = 2(CB+BD) = 2CD

2 5. Let AB and CD be two lines intersecting at a point O. Prove that the ray bisecting BOD. Prove that the extension of OX to the left of O bisects AOC. A D a Y O a X C The lines AB and CD intersect at O Let OX is the bisects of BOD Let BOX = OBX = a 0 BOX = AOY = a 0 And DOX = COY = a 0 AOY = COY OY bisects AOC B 6. Let OX be a ray and let OA and OB be two rays on the same side of OX and OB. Let OC be the bisects of AOB. Prove that AOX + XOB = 2 XOC. In the above figure OC is the angle bisecter of AOB AOC = BOC Now XOA + XOB = XOA + XOA + AOC + BOC = 2 XOA + AOC + AOC = 2 XOA + 2 AOC = 2 ( XOA + 2 AOC) XOA + XOB = 2 (XOC)

3 7. Let OA and OB be two rays and let OX be a ray between OA and OB such that AOX > XOB. Let OC be the bisector of AOB. Prove that AOB XOB = 2 COX B X C O In the along side figure AOX > XOB Ray OC the angle bisecting of AOB AOC = BOC Now AOX = XOB = AOC + COX -( BOC - COX) = AOC + COX ( BOC + COX) = COX + COX = 2 COX AOX XOB = 2 COX

4 8. Let OA, OB, OC, be three rays such that OC lies between OA and OB. Suppose the bisectors of AOC and COB are perpendicular to each other. Prove that B, O, A are collinear. Y C X b a B b a A In the figure OC lies in between the rays OA and OB. OX and OY are the bisectors of AOC and COB AOX = COX = a 0 and BOY = COY = b 0 It is given OX OY COX + COY = 90 0 a 0 + b 0 = 90 0 AOX = BOY a 0 + b 0 = 90 0 AOY + COX + COY + BOY = =180 0 BOA is straight angle BOA is a straight line Hence B, O, A are collinear. 9. In the adjoining figure, AB DE. Prove that ABC + DCB + = B A D E C Given: In the given figure AB DE

5 To Prove: ABC DCB + CDE = 180 Construction: Draw a PQ through C parallel to AB and DE. Proof: AB PC and BC is transversal ABC + BCP = (1) (Seen of interior angles on its side) lllly CDE + DCQ = (2) Seen of interior angles on its side) Adding (1) and (2) we get ABC + CDE + BCP + DCQ = (3) Now BCP + DCQ = 180 DCB...(4) Substituting in (4) and (3) ABC + CDE DCB = ABC + CDE - DCB = ABC + CDE DCB = Consider two parallel lines and a traversal. Among the measurement of triangles formed, how many distinct numbers are there? P a L b A b a B a H b C b a D Q

6 In the figure AB CD and PQ is Traversal PQ intersects AB in L and CD in H respectively ALP = BLM = a 0 LHC = DMQ = a 0 PLB = ALM = b 0 V.O.A V.O.A V.O.A PLB = LMD = b 0 Corresponding angles LMD = CHQ = b 0 V.O.A There are only two distinct angles a 0 and b 0 are these. CHAPTER-1 UNIT-2 ADDITIONAL PROBLEM ON SQUARES, SQUARE ROOTS, CUBES AND CUBE ROOTS 1. Match the numbers in the column A with their squares in the column B. A B Answers (1) 5 (2) 8 (3) 2 (4) -6 (5) -22 (6) 12 (a) 25 (b) 144 (c) 36 (d) 484 (e) 64 (f) 4 (g) Choose the correct option. (1) (a) (2) (e) (3) (f) (4) (c) (5) (d) (6) (b) a) The number of perfect squares from 1 to 500 is a) 1 b) 16 c) 22 d) 25 [c] b) The last digit of a perfect square can never be

7 a) 1 b)3 c) 5 d) 9 [b] c) If a number ends in 5 zeros, its square ends in a) 5 zeros b) 8 zeros c) 10 zeros d) 12 zeros [c] d) Which could be the remainder among the following when a perfect square is divided by 8? a) 1 b) 3 c) 5 d) 7 [a] e) The 6 th triangular number is a) 6 b) 10 c) 21 d) 28 [c] 3. Consider all integers from -10 to 5 and the square of each of them. How many distance numbers do you get? (-10) 2 =100, -9 2 =81...Answer: Write the digit in units place when the following numbers are squared. 4, 5, 9, 24, 17, 76, 34, 52, 33, 2319, 18, 3458, Sl No. n n 2 Digit in the Unit place

8 5. Write all numbers from 400 to 425 which end in 2, 3, 7, 8. Check if any of these is a perfect square None of the above numbers are perfect squares. 6. Find the sum of digits of ( ) 2. Answer: Suppose x 2 +y 2 =z 2. If x=4 and y=3 find z substituting =16+9= =25 3=5 If x=5 and z=13 find y substituting 5 2 +y 2 = y 2 =169 y 2 =169-25=144 y=12 If y=15 and z=17 find x substituting x =17 2 x =289 x 2 = =64 x=8 8. A sum of Rs is equally distributed among several people. Each gets as many ropees as the number of persons. How much does each one get? 2304 is a perfect square of 48 therefore 48 get RS 48 each. 9. Define a new addition * on the set of all natural numbers by m*n=m 2 +n 2 1) Is N closed under *? Ans:Yes it is closed 2) Is * Commutative on N? Yes it is commutative 3) Is * Associative? Ans: Yes it is associative. 4) Is there an identity element in N with respect to *? Ans: No. 10. Find all perfect squares from 1 to 500, each of which is a sum of two perfect squares: Find all perfect squares from 1 to 500, each of which is a sum of perfect squares.

9 =9+16=25 25 is a perfect square = = = Suppose the area of a square field is 7396 m 2 Find its perimeter: Area of a square field =a 2 =7396 Each side = 7396 =86m Perimeter =4a=4x86=344m. 12. Can 101 be written as a difference of two perfect squares? Ans:1010=a 2 -b 2 for some integers of a and b. So possibilities are 1. Both a and b are odd 2. Both a and b are even that is a 2 -b 2 is divisible by 4. But 1010 is not therefore it cannot be expressed as the difference of two squares. 13. What are the remainders when a percfect cube is divided by 7? Ans: 0,1,6 Divide 27 by 7, 64 by 7 34 by 7 and can confirm. 14. What is the least perfect square which leaves the remainder 1 when divided by 7 as well as 11? Ans: (34) 2 =1156 when divided by 7 and 11 leaves the remainder Find two smallest perfect squares whose product is a perfect cube 4 2 X16 2 =16 2

10 CHAPTER-2 UNIT-2 ADDITIONAL PROBLEMS ON FACTORISATION 1. Choose the correct answer: a) 4a+12b is equal to a) 4a b) 12b c) 4(a+3b) d)3a [c] b) The product of two numbers is positive and their sum negative only when a) both are positive b) both are negative c) one positive the other negative d)one them is equal to zero c) Factorising x 2 +6x+8, we get a) (x+1)(x+8) b) (x+8)(x+2) c) (x+10)(x-2) d) (x+4)(x+2) [d] d) The denominator of an algebraic fraction should not be a) 1 b) 0 c) 4 d) 7 [b] e) If the sum of two integers is -2 and their product is -24, the numbers are a) 6 and 4 b) -6 and 4 c) -6 and -4 d) 6 and -4 f) The difference (0.7) 2 -(0.3) 2 simplifies to a) 0.4 b) 0.04 c) 0.49 d) 0.56 [a] 2. Factorise the following: i) x 2 +6x+9=(x+3) 2 (a+b) 2 =a 2 +b 2 +2ab [b]

11 ii) 1-8x+16x 2 rearranging 16x 2-8x+1=(4x-1) 2 (a+b) 2 =a 2 +b 2 +2ab iii) 4x 2-81y 2 = (2x+9y)(2x-9y) a 2 -b 2 =(a+b)(a-b) iv) 4a 2 +4ab+b 2 =(2a+b) 2 (a+b) 2 =a 2 +b 2 +2ab v) a 2 b 2 +c 2 d 2 -a 2 c 2 -b 2 d 2 rearranging a 2 b 2- a 2 c 2 + c 2 d 2 - b 2 d 2 = a 2 (b 2 -c 2 )-d 2 (b 2 -c 2 )=(a 2 -d 2 )(b 2 -c 2 ) 3. Factorise the following: (splitting the middle term) (i) x 2 +7x+12 (iii) x 2-3x-18 (ii) x 2 +x-12 (iv) x 2 +4x-21 (v) x 2-4x-192 (vi) x 4-5x 2 +4 (vii) x 4-13x 2 y 2 +36y 4 4. Factorise the following: (i) 2x 2 +7x+6 6 2=12, 12 can be split as 4 3=12 and 4+3=7 Therefore the Ans: is (2x+3)(x+2) (ii) 3x 2-17x =60 Factors required 12 5= =17 Ans: (3x-5)(x-4) (iii) 6x 2-5x =-84 Factors required 7 12= =5 Ans: (x-2)(6x+7)

12 (iv) 4x 2 +12xy=5y 2 4 5=20 Factors required 10 2= =12 Ans: (2x+y)(2x+5) (v) 4x 4-5x =4 Factors required 4 1=4 4+1=5 Ans: 4x 4-4x 2 -x x 2 (x 2-1)-1(x 2-1) (x 2-1)(4x 2 1) (x+1)(x-1)(2x-11)(2x-1) 5. Factorise the following: (i) x 8 -y 8 this can be written as (x 4 ) 2 -(y 4 ) 2 applying a 2 -b 2 =(a+b)(a-b) We get (x 4 +y 4 )(x 4 -y 4 ) (x 4 -y 4 )=(x 2 ) 2 -(y 2 ) 2 =(x 2 -y 2 )(x 2 +y 2 ) =(x+y)(x-y)(x 2 +y 2 ) Ans: (x 4 +y 4 )(x 2 +y 2 )(x+y)(x-y) (ii) a 12 x 4 -a 4 x 12 =a 4 x 4 (a 8 -x 8 ) (a 8 -x 8 )=(a 4 +x 4 )(a 2 +x 2 )(a+x)(a-x) Ans: =a 4 x 4 (a 4 +x 4 )(a 2 +x 2 )(a+x)(a-x) (iii) x 4 +x 2 +1=(x 2 +x+1) (x 2 -x+1) this is of the form x 2 +px+q=(x+a)(x+b) Where a.b=q a+b=p (iv) x 4 +5x 2 +9=(x 2 +x+3)(x 2 -x+3) (as the previous problem) 6. Factorise x 4 +4y 4. Use this to prove that is a composite number. X 4 +4y 4 =(x 2 +2xy+2y 2 )(x 2-2xy+2y 2 ) Applying to = (2) 4 = ( )( ) simplification shows that it is a composite number.

13 CHAPTER-3 UNIT-2 ADDITIONAL PROBLEMS ON THEOREMS ON TRIANGLES 1. Fill in the blacks to make the following statement true: a) Sum of the angles a triangle is...(180 0 ) b) An exterior angle of a triangle is equal to the sum of... opposite angles. (interior) c) An exterior angle of a triangle is always... than either of the interior opposite angles. (greater) d) A triangle cannot have more than... right angle. (one) e) A triangle cannot have more than... obtuse angle. (one) 2. Choose the correct answer from the given alternatives: a) In a triangle ABC, A = 80 0 and AB = AC, then B is... a b c d [a] b) In right angled triangle, A is right angle and B = 35 0, then C is... a b c d [b] c) In a triangle, ABC, B = C = 45 0, then the triangle is... a. right triangle b. acute angled triangle c. obtuse angle triangle d. equilateral triangle [a] d) In an equilateral triangle, each exterior angle is... a b c d [c]

14 e) Sum of the three exterior angle of a triangle is... a. two right angles b. three right angles c. one right angles d. four right angles [d] 3. In a triangle ABC, B = Find A + C? A + B + C = 180 A + C + B =180 A + C + 70 = 180 x x A + C = = B C 4. In atriangle ABC, A = and AB = AC. Find Fing. B and C. A + B + C =180 0 A + x + x = 180 A + 2x = x = = 70 x = 70 = B = 35 0 C = If three angles of triangle are in the ratio 2:3:4, determine three angles. 2x + 3x + 5x =180 10x = 180 x = =18 6. The angles of a triangle are RRnged in ascending order of magnitude. If the difference between two consecutive angles is 15 0, find the three angles? A 110

15 x + x 15 + x + 30= 180 3x + 45 =180 3x = = 135 x = = 450 x = 45 0 x + 15 = = 60 0 x + 30 = = The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle. A + B = C A + B + C = 180 C + C =180 2 C =180 C= = In a triangle ABC, if 2 A = 3 B=6 C, Determine A, B and C. 2 A = 6 C A = 3 C 3 B = 6 C B = 2 C A + B = C=180 3 C + 2 C + C =180 6 C = The angles of a triangle x 40 0, x 20 0 and 1 2 x Find the value of x. A + C + C = 180 x-40+x-20+x+15=180

16 2 1 2 x-45=180 5x = 225 x= = In a triangle ABC, A- B=15 0 and B- C=30 0, find A, B and C. A+ B=15 A=15+ B B- C=30 B-30=C A+ B= C= C+B+B-30=180 3B-15=180 3B=180+15=195 B= = In a triangle ABC, A- B=15 0 and B- C=30 0,find A, B and C. A=15+B=15+65=80 0 C= B-30=65-30=35 0 A=80 0, B=65 0, C= The sum of two angles of a triangle is 80 0 and their difference is Find the angles of the triangle. A+B=80 A-B=20 2A=100 A=50 B=30 C= In a triangle ABC, B=60 0 and C=80 0. Suppose the bisector of Band C meet at I. Find BIC?

17 A 40 I B C 14. In a triangle, each of the smaller angles is half the largest angle. Find the angles. B= 1 2 A and C=1 2 A A+ B+ C=180 A+ 1 2 A+1 2 A=180 2A=180 A= =900 A=90 0 B=45 0 C= In a triangle, each of the bigger is twice the third angle, Find the angles. Q=2 P and R=2 P P+ Q+ R=180 0 P+2 P+2 Q= P=180 0 P= =360 P=36 0 Q=72 0 R= In atriangle ABC, B=50 0 and A=60 0. Suppose BC is extended to D. Find ACD.

18 A o B C D Ext.: ACD = Sum of interior opposite angles Ext.: ACD = A+ B= Ext.: ACD = In an Isosceles triangle, the vertex angle is twice the sum of the base angles. Find the angles of the tringle. A 4a B a a C A+ B+ C=180 4a+a+a=180 6a=180 a=30 0 A=4a=4 30=120 0 B=a=30 0 C=a=30 0

19 CHAPTER-1 UNIT-3 RATIONAL NUMBERS EXERCISE Identify the property in the following statements: (i) 2+(3+4)=(2+3)+4 Ans: Associative property of addition. (ii) 2.8=8.2 Ans: Commutative property of multiplication. (iii) 8.(6+5)=(8.6)+(8.5) Ans: Distributive property. 2. Find the additive inverses of the following integers: (i) 6 Ans: -6 is the additive inverse of 6. (ii) 9 Ans: -9 is the additive inverse of 9. (iii) 123 Ans: -123 is the additive inverse of 123. (iv) -76 Ans: 76 is the additive inverse of -76. (v) -85 Ans: -85 is the additive inverse of 85. (vi) 1000 Ans: is the additive inverse of Find the integer m in the following: (i) m+6=8 Ans: m=6-8 m=2 (ii) m+25=15 Ans: m=15-25 m=-10

20 (iii) m-40=26 Ans: m= m=+14 (iv) m+28=-49 Ans: m= m = Write in the following in increasing order: 21,-8,26,85,38,-333,-210,0,2011 Ans: -333,-210,-26,-8,0,21,33,85, Write the following in decreasing order: 85,210,-58,2011,- 1024,528,364,-10000,12 Ans: 2011,528,364,210,85,12,-58,-1024, EXERCISE Write down ten rational numbers which are equivalent to 5 and the 7 denominator not exceeding 80. Ans: 5 2 = = = = = = = = = =55 77

21 2. Write down 15 rational numbers which are equivalent to 11 and the 5 numerator not exceeding 180. Ans: 11 2 = , = 33, = , = 55, = , = 77, = , = 99, = , = 121, = , = 143, = , = 165, = Write down the ten positive numbers such that the sum of numerator and denominator of each is 11. Write them in decreasing order. Ans: Number : 10 Decreasing order : 10 1, 9 2, 8 2, 7 4, 6 5, 5 6, 4 7, 3 8, 2 9, , 9 2, 8 3, 7 4, 6 5, 5 6, 4 7, 3 8, 2 9, Write down the ten positive numbers such that the numerator and denominator for each them is -2. Write them in increasing order. Ans: Increasing order : = = 0.811, 8 = 0.80, 3 = 0.6, = Is 3 a rational number? If so, how do you write it in a form 2 conforming to the definition of a rational number (that is, the denominator as a positive integer)? Ans: 3 3 is not a rational number. It should be written as to be rational 2 2 number.

22 6. Earlier you have studied decimals 0.9,0.8. Can you write these as rational numbers? Ans: Yes, we can write decimals like 0.9, 0.8 as rational numbers. Ex.: 0.9 = 9, 0.8 = 8 18, 1.8 = EXERCISE Name the property indicated in the following: (i) = 430 Ans: Closure property of addition. (ii) = Ans: Closure property of multiplication. (iii) 5+0 = 0+5 =5 Ans: 0 is the additive identity. (iv) = 8 9 Ans: 1 is the multiplicative identity. (v) = 0 Ans: Additive inverse. (vi) = 1 Ans: Multiplication inverse. 2. Check the commutative property of addition for the following pairs: (i) , 3 4 Ans: a+b = b+a

23 = (ii) 8, a + b =b+a = = = 351 Proved (iii) 7, a+b=b+a = ( 162 ) 162 +( 133 ) = = 171 Proved 3. Check the commutative property of multiplication for the following pairs: (i) 22, Ans: a b = b a 22 3 = = = Proved (ii) 7, a b = b a 7 25 = = Proved

24 (iii) 8, Ans: a b = b a 8 17 = = Proved 4. Check the distributive property for the following triples of rational numbers: (i) 1 8, 1 9, 1 10 Ans:a(b+c) = ab+ac = = = = = (ii) 4, 6, Ans: a = 4 b = a(b+c) = ab + ac c = = = = =

25 46 = (iii) 3 13, 0, 8 7 Ans: = = = = Proved 5. Find the additive inverse of each of the following numbers: (i) 8 5, 6 10, 3 Ans: 8 5 = 8, = = 3 8, = = Find the multiplicative inverse of each of the following numbers: 2, 6, 8, 19, Ans: 2 1 = = =

26 19 = = EXERCISE Represent the following rational numbers on the number line: (i) 8 5 B A AB represents 8 5 (ii) 3 8 A B 0 AB represents (iii) 2 7 A B 0 AB represents

27 (iv) 12 5 A B AB represents (v) A B AB represents Wrtie the following rational numbers in ascending order: 3 4, 7 12, 15 11, 22 19, , 4 5, , 13 7 Ans: Ascending order 13 7, , 4 5, 7 12, 3 4, , 22 19, Method: 3 4, 7 12, 15 11, 22 19, , 4 5, 13 7, Write 5 rational number between 2 and 3, having the same 5 5 denominators.

28 Ans: Method II: Numbers between 2 5 and 3 5 are 41, 42, 43, 44, Totally there are 30 divisions, Each division is 1 30 Between 2 5 and there are, 14, 15, 16, How many positive rational numbers less than 1 are there such that the sum of the numerator and denominator does not exceed 10? Ans: 1 9, 1 8, 1 7, 1 6, 1 5, 1 4, 1 3, 1 2, 2 7, 2 5, 2 3, 3 7, 3 5, 3 4, 4 5

29 Only 15 positive rational number are possible such that they are less then 1 and the sum of the numerator and denominator does not exceed Suppose m/n and p/q are two positive rational numbers. Where does m+p lie, with respect to m/n and p/q? n+q Ans: m n = 1 2 and p q = 3 4 m +p = 1+3 = 4 = 2 = 0.66 n +q m+p n+q lies between m n and p q 6. How many rational numbers are there strictly between 0 and 1 such that the denominator of the rational number is 80? Ans: 1, 2, 3, 4,.. 77, 78, There are 79 positive rational numbers. 7. How many rational numbers are there strictly between 0 and 1 with the property that the sum of the numerator and denominator is 70? Ans: 1, 2, 3, 4,. 33, There are 34 rational numbers. ADDITIONAL PROBLEMS ON RATIONAL NUMBERS 1. Fill in the blanks:

30 (a) The number 0 is not in the set of (Natural numbers) (b) The least number in the set of all whole number is (0) (c) The least number in the set of all even natural numbers is (2) (d) The successor of 8 in the set of all natural numbers is (9) (e) The sum of two odd integers is (even) (f) The product of two odd integers is (odd) 2. State whether the following statements are true or false. (a) The set of all even natural numbers is a finite set. [False] (b) Every non-empty subset of Z jas the smallest element. [False] (c) Every integer can be identified with a rational number. [True] (d) For each rational number, one can find the next rational number. [False] (e) There is the largest rational number. [False] (f) Every integer is either even or odd. [false] (g) Between any two rational numbers, there is an integer. [False] 3. Simplify (i) 100(100-3)-( ) =100 (97)-( ) = =-297 (ii) [20-( )]+[2011-(201-20)] =[ ]+[ ] = =40

31 4. Suppose m is an integer such that m -1 and m -2. Which is larger m or m+1? State your reasons. m+1 m+2 Ans- Let m=2 z then m = 2 = 2 m m+1 = 2+1 = 3 m [ 2 3 = 0.66] [ 3 4 = 0.75] 3 4 > Define an operation * on the set of all rational numbers Q as follows: r*s = r+s-(r s) for any two rational numbers r,s. Answer the following with justification: (i) Is Q closed under the operation *? Ans- Let r=2, s=3 2*3 = 2+3 (2*3) 2*3 = 5-6 2*3 = -1 Q Q is closed under * (ii) Is * an associative operator on Q? Let r,2 and t be three integers (r*t)*t=r*(s*t) Assume r=1, s=2 and t=3 (1*2)*3=1*(2*3) Now (1*2)*3=1*(2*3) (1*2)= =3-2=1 (1*2)*3=1*3 (1*2)*= =4-3=1 3=1

32 r*(s*t)=1*(2*3) 2*3= =5-6=-1 1*(2*3) = 1*-1 1*(2*3) = 1*-1 = 1+(-1)-(1-1) = 0+1=1 1*(2*3) = 1 (1*2)*3 = 1*(2*3) is associative on Q. (iii) Is * a commutative operation on Q? Assume 4, 5 Q 4*5=5*4 4*5=4+5-(4 5) = 9-20 = -11 5*4 = 5+4-(5 4) =9-20 = -11 4*5 = 5*4 is commutative on Q. (iv) What is a*1 for any a in Q? Now r * s =r s-(r 2) a*1 = a+1-(a 1)=a+1-a=1 a*1 = 1 (v) Find two integers a 0 and b 0 such that a*b=0. Let a*b = a+b-(a b) A*b +a b-(a b) 2 2=2+2-(2 2) =4 4=0 2*2=0 a=2 and b=2

33 6. Find the multiplicative inverse of the following rational numbers: 8, 12, 26, 13, Ans: Rational numbers Multiplicative inverse Wtrite the following in increasing order: 10 13, 20 23, 5 6, 40 43, 25 28, Ans: Taking the LCM of the denominator and computing the numerators we observe , 5 6, 20 23, 25, and Write the following in decreasing order: 21, 31, 13, 41, 51 and are in order of increasing order. Ans: Taking the LCM of the denominators and arranging the numerators in the decreasing order. 21, 13, 31, 9, and are in decreasing order. 47

34 9. a. What is the additive inverse of 0? Additive inverse of 0 is 0. b. What is the multiplicative inverse of 1? Additive inverse of 1 is 1? c. Which integers have multiplicative inverses? Every integer have multiplicative inverse. 10. In the set of all rational numbers, give 5 examples each illustrating the following properties. (i) associativity (ii) commutativity (iii) distributivity of multiplication over addition. Associativity Commutativity Distributivity 1. Let 1,2,3 Q a+(b+c)=(a+b)+c 1+5=3+3 6=6 2. Let 1,2,3 Q a (b c)=(a b) c 2 (3 4)=(2 3) =(6) 4 24=24 1,2 Q a+b=b+a 1+2=2+1 3=3 (a b)=b a 2 3=3 2 6=6 1,2,3 Q a(b+c)=ab+ac 1(2+3)= (5)=2+3 5=5 a(b+c)=ab+ac 2(2+3)= =6 8 14= Simplify the following using distributive property: (i) = = = 275 =

35 (ii) (iii) = = = = = = = = Simplify the following: (i) = = = (ii) = = = (iii) = = = = (iv) = = =

36 13. Which is the property that is there in the set of all rationals but which is not in the set of all integers? Every non-zero number is inversible but only 1 is not inversible. 14. What is the value of 1 + 1? = = = = = = = Find the value of = = = = 6 12 = Find all rational numbers each of which is equal to its reciprocal. +1 and -1 are the rational numbers equal to their reciprocal. 17. A bus shuttles between two neighbouring towns every two hours. It starts from 8am in the morning and last trip is at 6 pm. On one day the driver observed that the first trip had 30 passengers and each subsequent trip has one passenger less than the previous trip. How many passengers travelled on that day? 1 st trip 8 am -30 passengers 2 nd trip 10 am -29 passengers 3 rd trip 12 noon -28 passengers 4 th trip 2 pm -27 passengers 5 th trip 4 pm -26 passengers 6 th trip 6 pm -25 passengers 165 passengers Total number of passengers travelled on that day =165

37 18. How many rational numbers p/q are there 0 and 1 for which q<p? Not a single rational number lie between 0 and 1with q < p. When p > p then the number is greater than Find all integers such that 3n+4 is also integer. n+2 3n +4 When n = 0, = = 4 = 2 n +2 n is an integer When n = -1, is an integer. = = 1 1 = By inserting parenthesis (that is brackets), you can get several values (For example ((2 3) + 4) 5 one way of inserting parenthesis). How many such values are there? (i) (2 3) + (4 5) = = 26 (ii) 2 (3+4) 5=2 7 5=70 (iii) (2 3+4) 5=10 5=50 (iv) 2 (3+4 5)=2 23=46 There are four values. 21. Suppose p/q is a positive rational in its lowest form. Prove that 1 q + 1 p+q is also in its lowest form. Let p = 3 i.e., p=3 and q=5 q 5 Then 1 q + 1 p +q = =

38 13 40 is in the lowest form. = 8+5 = Show that for each natural number n, the fraction 14n+3 is in its lowest form. Let n=5 be a natural number Then 14 n+3 = = = n n is also in the lowest form. 23. Find all integers n for which the number (n+3)(n-1) is also an integer. Let n=-2 is an integer (n+3)(n-1)=(-2+3)(2-1) = (+1)(-3) =-3-3 is an integer.

39 CHAPTER-2 UNIT-3 LINEAR EQUATIONS IN ONE VARIABLE EXERCISE Solve the following (i) x+3=11 (ii) y-9=21 x=11-3 y=21+9 x=8 y=30 (iii) 10=z+3 (iv) 3 + x = =z x= 9 7=z z= z=7 (v) 10x=z+3 (vi) s = x= s=4 7 x=3 s=28 (vii) 3x 6 = 10 (viii) 1.6= x 1.5 3x= x= =x 2.4=x x=20 x=2.4 (ix) 8x=48+8 (x) x = x=48+8 x= 56 8 x 3 = x =

40 x=7 x= x= 8 5 (xi) x 5 = 12 x=12 5 3x (xii) = x= x=60 x= 3 x=25 (xii) 3(x+6)=24 (xiv) x 4-8=1 3x+18=24 3x=24-18 x 4 = 1+8 x 4 = 9 x=2 x=9 4 (xv) 3(x+2)-2(x-1)=7 3x+6-2x+2=7 x+8=7 x=7-8 x=1 2. Solve the equations (i) 5x=3x+24 5x-3x=24 2x=24 x= 24 2 x=12 x=36

41 (ii) 8t+5=2t-31 8t-2t= t=-36 t= 36 6 t = -6 (iii) 7x-10=4x+11 7x-4x= x=21 x = 21 3 x = 7 (iv) 4z+3=6+2z 4z-2z=6-3 2z = 3 Z= 3 2 (v) 2x-1=14-x 2x+x=14+1 3x=15 x = 15 3 x = 5 (vi) 6x+1=3(x-1)+7 6x+1=3x-3+7

42 6x-3x= x= 3 3 x = 1 (vii) 2x = x x 2x = x 4x 10 = 5 2 x = = = -25 x = -25 (viii) x 3 2 = 2x 5 5 x x = 2x 5 5 = 2x x-13 = 2x x-2x = 13 -x=13 x = -13 (ix) 3(x+1) = 12+4(x-1) 3x+3 = 12+4x-4 3x-4x = x=+5 x = -5

43 (x) 2x-5 = 3(x-5) 2x-5 = 3x-15 2x-3x= x=-10 x x = +10 (xi) 6(1-4x)+7(2+5x) = x+14+35x = 53-24x+35x = x= x=3 (xii) 3(x+6)+2(x+3) = 64 3x+18+2x+6 = 64 3x+2x = x=40 x = 40 5 x = 8 (xiii) 2m = m 2 1 2m 3 m 2 = m 3m 6 = 9 m 6 = 9 m = -9 6 m = -54 (xiv) 3 4 (x-1) = x-3 3x = x-3 3x 4 x 1 = x 4x 4 =

44 x = x = x = 9 EXERCISE If 4 is added to a number and the sum is multiplied by 3, the result is 30. Find the number. Ans: (x+4) = 30 3x+12 =30 3x = x = 18 3 x = 6 2. Find three consecutive odd numbers whose sum is 219. Ans: Let the numbers be x+x+2+x+4 x+x+2+x+x4=219 3x+6=219 3x=219-6 x= x = 71 x+2 = 71+2 = =73 x+4 = 71+4 = 75 The numbers are 71,73 and A rectangle has length which is 5cm less than twice its breadth. If the length is decreased by 5cm and breadth is increased by 2cm. The perimeter of the resulting rectangle will be 74cm. Find the length and breadth of the original rectangle.

45 Ans: Length is 5 less than 2 breadth Let the length be l Let the breadth be b l = 2b-5 New length = l - 5 = (2b-5)-5 L = 2b-10 New breadth = b+2 B=b+2 Perimeter of rectangle = 74 2(L+B) = 74 2(2b-10+b+2) = 74 2(3b-8) = 74 6b = b = 90 6 b = 15cm l = 2b-5 = 2(15)-5 = 30-5= 25cm Original length = 25cm Original breadth = 15cm 4. A number subtracted by 30 gives 14 subtracted by 3 times the number. Find the number. Let the number be x 30-x = 3x-14 -x-3x= x=-44 x = 44 4 x = Sristi s salary is same as 4 times Azar s salary. If together they earn Rs. 3,750 a month. Find their individual salaries.

46 Ans: Let Azar s salary be x Sristi s salary = 4x Sum of their salaries = 3,750 x+4x =3,750 5x = 3,750 x = x = 750 Azar s salary = 750 Sristi s salary = 4x = =3, Prakri s age is 6 times Sahil s age. After 15 years, Prakruthi will be 3 times as old as Sahil. Find their age. Ans: Let Sahil s age be x Prakruthi s age = 6x Sahil = 6x After 15 years = Sahil s age = (x+15) Prakruthi s age = (6x+15) Prakruthis age = 3 Sahil s age (6x+15) = 3 (x+15) 6x+15 = 3x+45 6x-3x = x = 30 x = 30 3 x = 10 Sahil s age = 10 years Prakruthi s age = 6x = 6 10 = 60 years

47 7. In the figure, AB is a straight line. Find. C D x+40 x+20 x A Ans: x+20+x+40+x = x+60 = x = x = x = 40 0 B 8. If 5 is subtracted from three times a number, the result is 16. Find the number. Ans: Let the number be x 3x-5 = 16 3x = 16+5 x = 21 3 x = 7 9. Find two numbers such that one of them exceeds the other by 9 and their sum is 81. Ans: Let the no. be x, the other no. be x+9 x+x+9 = 81 2x+9 = 81 2x = 81-9 x = 72 2 x = 36 x+9=36+9=45 The numbers are 36 and The length of a rectangular field is twice its breadth. If the perimeter of the field is 288m. Find the dimensions of the field.

48 Ans: Let the length of the field be l Let the breadth of the field be b Length is twice its breadth l = 26 Perimeter of the field = 288m 2(2b+b) = 288 2(l+b) = 288 2b+2b+b+b=288 6b = 288 b = b = 48m b = 48m l = 2b = 2 48 = 96m 11. Ahmed s father is thrice as old as Ahmed. After 12 years, his age will be twice that of his son. Find their present age. Ans: Let Ahmed s age be x years Let Ahmed s father age be 3x years After 12 years, Ahmed s father s age = (3x+12) Ahmed s age = (x+12) Ahmed s father s age = 2 Ahmed s age (3x+12) = 2(x+12) 3x+12 = 2x+24 3x-2x = x = 12 Ahmed s father s age = 12 years Ahmed s father s age = 3x = 3 12 = 36 years

49 12. Sajnu is 6 years older than his brother Nishu. If the sum of their ages is 28 years, what are their present age? Ans: Let Nishu s age be x Let Sanju s age be (x+6) Sum of their ages = 28 years Present age = x+6+x = 28 2x+6 = 28 2x = 28-6 x = 22 2 x = 11 Nishu s age = 11 years Sanju s age = x+6 = 11+6 = 17 years 13. Viji is twice as old as his brother Deepu. If the difference of their ages is 11 years. Find theirpresent age. Ans: Let Deepu s age be x Let Viji s age be 2x Difference between their ages = 11 years 2x-x = 11 x = 11 Deepu s age = 11 years Viji s age = 2x = 2x 11 = 22 years 14. Mrs. Joseph is 27 years ilder than her daughter Bindu. After 8 years she will be twice as old as Bindu. Find their present age.

50 Ans: Let Bindu s age be x Let Mrs. Joseph s age be = (x+2x) After 8 years, Bindu s age = (x+8) Mrs. Joseph age = (x+27+8) Mrs. Joseph s age = 2 BIndu s age (x+27+8) = 2(x+8) x+27+8 = x = -19 x = 19 x = 11 Bindu s age = 19 years Mrs. Joseph s age = x+27 = = 46 years 15. After 16 years, Leena will be three times as old as she is now. Find her present age Ans: Let Leena s age be x After 16 years, Leena s age will be x+16 x+16 = 3x 16 = 3x-x 16 = 2x 2x = 16 x = 16 2 x = 8 years

51 ADDITIONAL PROBLEM ON LINEAR EQUATIONS IN ONE VARIABLE 1. Choose the correct answer. a. The value of x in the equation 5x-35 = 0is: a) 2 b) 7 c) 8 d) 11 [b] b. If 14 is taken away from one fifth of a number, the result is 20. The equation expressing this statement is a) (x/5)-14 = 20 b) x-(14/5) = (20/5) c) x-14 = (20/5) d) x+(14/5) = 20 [a] c. If five times a number increased by 8 is 53, the number is: a) 12 b) 9 c) 11 d) 2 [b] d. The value of x in the equation 5(x-2) = 3(x-3) is: a) 2 b) ½ c) ¾ d) 0 [b] e. If the sum of two numbers is 84 and their difference is 30, the numbers are: a) -57 and 27 b) 57 and 27 c) 57 and -27 d) -57 and -27 [b] f. If the area of a rectangle whose length is its breadth is 800m 2, then length and breadth of the rectangle are: a) 60m and 20m b) 40m and 20m c) 80m and 10m d) 100m and 8m [b] g. If the sum of three consecutive odd numbers is 249, the numbers are: a) 81,83,85 b) 79,81,83 c) 103,105,107 d) 95,97,99 [a] h. If the (x+0.7x)/2 = 0.85, then x equals: a) 2 b) 1 c) -1 d) 0 [b] i. If 2x-(3x-4)=3x-5 then x equals: a) 14/9 b) 9/4 c) 3/2 d) 2/3 [b]

52 2. Solve: (i) (3x+24)/(2x+7)=2 3x+24 2x+7 = 2 Multiply both sides by (2x+7) 3x x + 7 x 2x + 7 = 2 2x + 7 3x+4x = 4x+14 3x-4x = x = -10 x = 10 x = 10 (ii) (1-9y)/(11-3y)=(5/8) 1 9y 11 3y = 5 8 By cross multiplication 8(1-9y)=5(11-3y) 8-72y=55-15y -72y+15y= y=47 -y= y= The sum of two numbers 45 and their ratio is 7:8. Find the numbers.

53 Ans: Let the numbers be x and y x+y=45 x=45-y... (1) Now x y = 7 8 Cross multiplication 8x=7y... (2) Substituting (1) in (2) 8(45-y) =7y 360-8y=7y -8y-7y= y= -360 y= = 24 Now x=45-y x=45-24 x=21 The numbers are 21 and Shona s mother is four times as old as Shona. After five years, her mother will be three times as old as Shona (at that time). What are their present ages? Ans: Let the present age of Shone be x years. her mother s age 4x years. After 5 years Shona will be (4x+5) years Her mother will be three times as old as Shona (4x+5)=3(x+5) 4x+5=3x+15 4x+3x=15-5 x=10 Present age of Shona = 10 years 5. The sum of three consecutive even numbers is 336. Find the?

54 Ans: Let the 3 consecutive even numbers x, 2+2 and x+4 Sum of three consecutive even numbers =x=x+x+x+4 3x+6=336 3x= x=330 x = =110 The three consecutive even numbers are 110,112, Two friends A and B start a joint business with a capital Rs. 60,000. If A s share is twice that of B, how much have each invested? Ans: Let the share of B Rs. X Share of A = Rs. 2x x+2x=60,000 3x=60,000 x= 60,000 3 = 20,000 A s share = 2x = 2 20,000 = Rs.40,000 B s share = x = 1 20,000=Rs.20, Which is the number when 40 is subtracted gives one third of the original number? Ans: Let the number be x x-40= 1 3 x 3(x-40)=x 3x-120=x Cross multiplication 3x-x=120 2x=120 x = = 60 The original number = 60

55 8. Find the number whose sixth exceeds its eight parts by 3. Ans: Let the number be x Sixth part of the number = x 6 Eighth part of the number = x 8 X 6 x 8 = 3 4x 3x 24 = 3 LCM=24 x=3 24 x=72 The number = A house and a garden together coast Rs.8,40,000. The price of the garden is 5 times the price of the house. Find the price of the house 12 and the garden. Ans: Price of the garden = ,40,000 = Rs. 3,50,000 Price of the house = 8,40,000-3,50,000 = Rs. 4,90, Two farmers A and B together own a stock of grocery. They agree to divide it by its value. Farmer A takes 72 bags while B takes 92 bags and Rs. 8,000 to A. What is the cost of each bag? Ans: Let the cost of each bag be Rs. X Cost of 72 bags = Rs. 72x Cost of 92 bags = Rs. 92x 72x+8000=92x x-92x= x=

56 x = Cost of each bag = Rs. 800 = A father s age is four times that of his son. After 5 years, it will be three times that of his son. How many more years will takes if father s age is to be twice that of his son? Ans: Let the age of the son be x years The father age = (4x) years After 5 years son s age = (x+5) years After 5 years faher s age = (4x+5) years (4x+5)=3(x+5) 4x+5=3x+15 4x-3x=15-5 x=10 Present age of the son = 10 years Present age of the father = 4x=4 10=40 years Let after 4 years father will be twice of his son 40+y=2(10+y) 40+y=20+2y y-2y= y=-20 y=20 After 20 years father will be twice of his son. How many more years father age is twice that of his son? y=-5=20-5=15 years 12. Find a number which when multiplied by 7 is as much above 132 as it was originally below it.

57 Ans: Let the number be x 7 twice of the number =7x 7x is as much above 132 is 7x-132 originally it was (132-x) below it Now both are equal 7x-132=132-x 7x+x= x=264 x = =33 The original number = A person buys 25 pens worth Rs. 250 each of equal cost. He wants to keep 5 pens for himself and sell the remaining to recover his money. What should be price of each pen? Ans: Number of pens bought = 25 Cost price of 25 pens = Rs.250 Price of each pen = Rs = Rs. 10 Number of pens he sold = 25-5=20 Let the selling price of each pen = Rs. X Selling price of 20 pens = Rs. 20x 20x=250 x = = 25 2 = Selling price of each pen = Rs

58 14. The sum of the digits of a two-digiy number is 12. If the new number formed by reversing the digits is greater than the original number. Check your solution. Ans: Let the digit in the unit place is y and tens place is x x+y=12 x =(12-y) The original number = (10x+y) The reversed number = (10y+x) x+y=12 -x+y=12 10y+x=10x+y+18 10y-y+x-10y=18 9y-9x=18 9(y-x)=18 y-x=2 -x+y=2 2y=14 adding both equation y = 7 x+y=12 x+7=12 x = 12-7=5 Ten s digit = 5 units digit =7 The original number = The distance between two stations is 340km. Two trains start simultaneously from these stations on parallel tracks and cross each her. The speed of one of them is greater than that if the other by 5km/hr. If the distance between two trains after 2 hours of their start is 30km. Find the speed of each train.

59 Ans: Let the speed of one train be x km/hr. The speed of the other train = (x+5) km/hr. The distance travelled by first train in 2 hours = 2x km The distance travelled by second train in 2 hours = 2(x+5) km Distance between the two trains is 2 hours = 30 km Total distance travelled = =370 km 2x+2x+10=370 4x = =360 x = = 90 Speed of the first train = 90 km/hr. Speed of the second train = (90+5) = 95 km/hr. 16. A steamer goes down stream and covers the distance between two in 4 hours while it covers the same distance up stream in 5 hours. If the speed of the steamer up stream is 2 km/hour. Find the of streamer in still water. Ans: Speed of steamer up stream = 2 km/hr. Time taken by the steamer upstream = 5 hrs. Distance travelled = 2 5 = 10 km. Speed of steamer down steam = 10 4 = 5 2 = 2 Let the speed of streamer in still water be =x km/hr. The speed of steam = y km/hr. Speed of steamer down steam = (x+y) km/hr. x+y = 2.5 x-y = 2.0 2x = 4.5 x = 2.25 Speed of steamer in still water = 2.25 km/hr.

60 17. The numerator of the rational number is less than its denominator by 3. If the numerator becomes three times and the denominator is increased by 20, the new number becomes 1/8. Find the original number. Ans: Let the original number be x y x = y-3 3x 20 +y = 1 8 3x+8=y+20 24x=y+20 24(y-3)=y+20 24y-72=y+20 24y-y= y=92 x=y-3 x=4-3=1 y= =4 The original number = The digit at the tens place of a two digit number is three times the digit at the units place. If the sum of this number and the number formed by reversing its digits is 88. Find the numbers. Ans: The digit in the units place be x The digit in the ten s place = 3x The reversed number = 10x+3x=13x Sum of the numbers = 31x+13x=44x 44x=88 x= =2 Digit in the units place = 2 Digit in the ten s place = 3x=3 2=6 The number = 62

61 19. The altitude of a triangle is five-thirds. The length of its corresponding base. If the altitude is increased by 4cm and the base decreased by 2cm, the area of the triangle would remain the same. Find the base and altitude of the triangle. Ans: Let the base of the triangle be x cms. its altitude = 5 3 x cms Area of the triangle = 1 2 b h =1 2 x 5x 3 = 5x 6 2 cm 2 Altitude is increased by 4 It becomes = 5x cms Base is decreased by 2, it becomes (x-2) cms Area of the triangle = 1 2 b h = 1 2 x 2 5x = 1 2 x 2 5x+12 3 = x 2 5x x2 6 = 5x2 +2x x 2 = 5x 2 +2x-24 2x-24=0 2x=24 x=12 base = x =12 cms

62 Altitude = 5x = 5 12 = 60 = 20cms Base of the triangle = 12 cm. Altitude of the triangle = 2cm. 20. One of the angle of a triangle is equal to the sum of the other two angles. If the ratio of the other two angles of the triangle is 4:5, find the angles of the triangle. Ans: Let the angle of the triangle be y Other two angles are 4:5, i.e., 4x, 5x y+9x=180 y=9x y+9x=180 9x+9x=180 18x= = 10 The three angle are 4x=40 0 5x=50 0 9x=90 0 Angles of the triangle 40, 50 and 90 0

63 CHAPTER-3 UNIT-3 EXERCISE CONGRUENCY OF TRIANGLES 1. Identify the corresponding sides and corresponding angles in the following congruent triangles. (a) P X (b) B R Y Z Corresponding angles Corresponding sides 1. Q = Y 1. PR=XZ 2. R = Z 2. PQ = XY 3. P = X 3. QR = YZ R A B Q P C Corresponding angles Corresponding sides 1. R = C 1. PQ=AB 2. Q = B 2. PR = AC 3. A = P 3. RQ = BC 2. Pair of congruent triangle and incomplete statements related to them are given below. Observe the figures carefully and fill up the blanks:

64 (a) In the adjoining figure if C = F, then AB=DE and BC=EF C D E A B F (b) In the adjoining figure if BC=EF, then C = F and A = D A D B C E F (c) In the adjoining figure, if AC = CE and ABC = DEC, then D= B and A= E D A C E B

65 EXERCISE In the adjoining figure PQRS in a triangle. Identify the congruent triangles formed by the diagonals. P Q O S R Ans: Data: PQRS is a rectangle O is the mid point of PR and SQ Proof: In PQR and SOR 1. PSQ RSQ 2. POQ SOR 3. POS ROQ 4. PQR PSR POS QOS (SAS postulate) The congruent triangles formed by the diagonals are POQ, SOQ, POS and QOR. 2. In the figure ABCD in a square, M,N,O and P rae the midpoints of sides AB,BC,CD and DA respectively. Identify the congruent triangles. A M B P N D O C

66 Ans: Data: ABCD is a square. M, N, O and P are the midpoints of sides AB, BC,CD and DA respectively. Proof: In APM and PDO 1. AP< BNM 2. DPO ONO 3. B = C (90 0 ) MBN NOC (SAS postulate) The congruent triangles are APM, PDO, MBN and NOC. 3. In a triangle ABC, AB = AC. Points E on AB and D on AC are such that AE = AD. Prove that triangles BCD and CBE are congruent. A E D B C Ans: Data: AB = AC AE = AD To Prove: BCD and CBE Proof: In BCD and CBE 1. BC = BC (Common side) 2. B = C (AB = AC data) AB = AC data AE = AD data 3. BE = DC From (1), (2) and (3) In BCD CBE (SAS postulate)

67 4. In the adjoining figure, the idea BA and CA have been produced such that BA = AD and CA = AE. Prove that DE BC. [Hint: Use the cincept of alternate angles] E D A B Ans: BA = AD and CA = AE To Prove: DE BC Proof: In EAD and BAC. 1. BC = BC (Common side) 2. EAD = BAC (V.O.A) 3. BS = AD = (Data) C EAD = BAC (SAS postulate) ABC = ADE (Congruent property) But they are alternate angle DE BC