ANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)

ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False: (i) False (ii) False (iii) True (iv) False (v) True 3. Write four rational numbers equivalent to given rational numbers. (i) 2/5 = 4/10 = 6/15 = 8/20 = 10/25 (ii) -5/9 = -10/18 = -15/27 = -20/36 = -25/45 4. Write the Absolute value of: (i) 5/8 = 5/8 (ii) -7/10 = 7/10 (iii) 0 = 0 Exercise: 1(B) 1. Answer True or False: (i) False (ii) False (iii) True (iv) True (v) False 2.Add: (i) 3/7 + 2/7 = 5/7 (ii) -11/13 + 9/13 = -2/13 (iii) -1/6 and 3/10 = -1/6 + 3/10 LCM of 6 and 10 is 2 6, 10 3, 5 = 2 x 3 x 5 = 30 _1/6 x 5/5 = -5/30, 3/10 X 3/3 =9/30 S0, -5/30 + 9/30 = 4/30 Ans (vii) 3 (3/4) and 4(1/3) = 15/4 + 13/3 LCM of 4 and 3 is 12 15/4 X 3/3 = 45/12, 13/3 X 4/4 = 52/12 15/4 + 13/3 = 45/12 + 52/12 = 97/12 = 8 (1/2) Ans. 3.Simplify: (i) 5/6 + 7/18 + (-11/12) LCM of 6, 8, 12 is 2 6, 8. 12 3 3 9 6

1, 3, 2 = 2 x 3x 3 x 2 = 36 5/6 x 6/6 = 30/30, 7/18 x 2/2 = 14/36 11/12 x 3/3 = 33/36 5/6 + 7/18 + (-11/12) = 30/36 + 14/36 33/36 = 30 + 14 33 = 44-33 = 11/36 Ans 36 36 4. Verify the Following: 3/4 + (-2/3) = (-2/3) + 3/4 Here we use Commutative Property of Addition i.e a/b + c/d = c/d + a/b (Two Rational Numbers can be added in any order) LHS: 3/4 + (-2/5) = (3 x 5) + (-2 x 4) 4 x 5 = (15-8)/20 = 7/20 RHS: (-2/5) + 3/4 = (-2 x 4) + (3 x 5) 4 x 5 =(-8 + 15 )/20 = 7/20 LHS = RHS (veified) 6.Additive inverse of (i) 2/3 = -2/3 (ii) 0 = 0 (iii) 17/9 = -17/9 (iv) -15/-11 = -15/11 7. Arrange and Simplify: (v) -23 = 23 1/2 + (-3/5) + 3/2 = (1/2 +3/2) 3/5 = 4/2 2/5 LCM of 2 and 5 is 10 (4 x 5) ( 2 x2) = 20 4 2 x 5 10 14/10 = 7/5 (divide by 2) = 1 (2/5) Ans (ii) 28/17 + 35/17 + (-16/17) + (-23/17) = ( 28 + 35) - (16 + 23) 17 = 63-39 17 = 24/17 = 1(7/17) Ans

(iii) 2/3 + (-3/5) + 1/6 + (-8/15) = (2/3 + 1/6) (3/5 + 8/15) = (2X2) + (1X1) - (3X3) + (8X1) 6 15 = (4 + 1) (9 + 8) = 5/6 17/15 6 15 = LCM of 6 and 15 is 30 5/6 x5/5 = 25/30, 17/15 x 2/2 = 34/30 5/6 17/6 = 25/30 34/30 = -9/30 = -3/10 (divide by 3) Ans = -3/10 8. Verify that ; - (- x) = x when x = 7/6 Sol: - (- 7/6) = 7/6 => 7/6 = 7/6 verified 9. verify: - ( x + y ) = (- x) + (- y) when x = 3/4, y = 6/7 Sol: - (3/4 + 6/7 ) = (-3/4 ) + (-6/7) - (3/4 + 6/7 ) = - (3/4 + 6/7 ) - (3x7) + (6x4) = - (3x7) + (6x4) 28 28 -(21 + 24)/28 = -(21 + 24 )/28 = -45/28 = -45/28 (verified) 10. Subtract: (i) 1/5 from 3/5 = 3/5 1/5 = (3-1)/5 = 2/5 Ans (ii) 4/9 from -1/6 (-1/6) 4/9 LCM of 6 and 9 is 18-1/6 x 3/3 = -3/18 4/9 x 2/2 = 8/18-1/6 4/9 = -3/18 8/18 = -11/18 Ans 11, Sol:

Sum of two Rational number = 5/18 2 18, 8 One of the number = 1/8 3 9 4 Other number be X 2 3 4 Given that 1/8 + X = 5/18 3 2 X = 5/18 1/8 LCM of 8 and 18 is 72 = 2 x 3 x 2 x 3 x 2 (5 x 4) (1 x 9) /72 = 20 9 / 72 X = 11/72 Ans: The other number is 11/72 12. Let the Required Number be X Given that 7/(-8) - x = -13/12 -X = -13/12 + 7/8 LCM of 12 and 8 is 24-13/12 + 7/8 = (-13 x 2) + (7x3)/24 = ( -26 + 21) / 24 -X = -5/24 => X = 5/24 Ans. 13. Let the Required number be X Given that [ 3/4 + 1/3 + 2/5] X = 1/2 -X = 1/2 - [ 3/4 + 1/3 + 2/5] LCM of 2, 4, 3 and 5 is 60 1/2 x 30/30 = 30/60, 3/4 x 15/15 = 45/60 1/3 x 20/20 = 20/60 2/5 x 12/12 = 24/60 -X = 30/60 [ 45 +20+24]/60 = 30/60 89/60 = - 59/60 => X = 59/60 Ans. Exercise 1 (c) 1. Answer True or False: (i) False (ii) False (iii) False (iv) True (v) False 2. Multiply: (i) 6/7 by 2/3 6/7 X 2/3 = 12/21 (divide by 3) 4/7 Ans

(ii) (-9)/11 x 22/63 (-9) X 22/ 11 X 63-198 / 693 (divide by 99) -2/7 Ans (iii) -12/13 X (-5)/18 (-12 X (-5) / 13 X 18 60/234 (divide by 6) 10/39 Ans (iv) 5 (1/7) X -2 (1/3) 36/7 X (-7)/3 36 X (-7) / 7 X 3 252 / (-21) -12 Ans 3. Simplify: ( -6/7 X -28/18) + (-11/13 X 65/22) [(-6) X 28] + [(-11) X 65] 7 X 18 13 X 22 4/3-5/2 (4X2) - (5X3) 3X2 8 15 6-7 / 6 Ans 4. Fill in the Blanks: (i) -18/17 (ii) -19 (iii) 6/11 (iv) 3/7 (v) -4/9, 2/9 5. Verify: (i) 4/5 X 7/9 = 7/9 X 4/5 Property : Commutative 4 x 7 = 7 x 4 5 x 9 9 x 5 28/45 = 28/45 (verified) 6. (i) [(3/4) X (1/2)] X (5/7) = 3/4 X [(1/2) X (5/7)] Property: Associativity LHS: [(3/4) X (1/2)] X (5/7) 3/8 X 5/7 15 / 56 RHS: 3/4 X [(1/2) X (5/7)] 3/4 X 5/14

15 / 56 LHS = RHS (verified) 7. (i) 2/3 X ( 4/5 + 7/8) = (2/3 X 4/5) + ( 2/3 X 7/8) Property: Distributivity of Multiplication over Addition LHS: 2/3 X ( 4/5 + 7/8) 2/3 X [(4 X 8 ) + (7 X 5)]/40 2/3 X (32 + 35)/40 2/3 X 67/40 67/60 RHS: (2/3 X 4/5) + ( 2/3 X 7/8) 8/15 + 14/24 LCM of 15 and 24 is 120 (8 x 8) + (14 X 5) 120 (64 + 70) / 120 134/120 67/60 LHS = RHS (Verified) MCQs 9. Multiplicative inverse (i) 1/6 (ii)-1/23 (iii) 20/11 (iv) -16/19 (vii) 2/3 (viii) 0 11. Divide: (i) 5/9 by 15 5/9 X 1/15 5/135 1/27 Ans. 12. Evaluate: [(5/9) (15/36)] (-5/6) [(5/9) X (36/15)] X (-6/5) 180/135 X (-6/5) -8/5 Ans 13. (c) Commutative Property 14. (b) -14 15. 0 Exercise 1(D)

1. Rational numbers Between (a) 1/5 and 1/4 Sol: we use formula 1/2 (a/b + c/d) Here a/b = 1/5 and c/d = 1/4 So, 1/2 (1/5 + 1/4) = 1/2 [( 4 + 5)/20] = 1/2 x 9/20 = 9/40 Ans. 2. Find two Rational numbers between -2 and 2 Ans: -1 and 0 3. Three rational numbers between -7/2 and -2 Let q 1, q 2, and q 3 be the three required Rational numbers q 1 = 1/2 [ -7/2-2] = -11/4 q 2 = 1/2 [-11/2-2] = -19/8 q 3 = 1/2 [-19/8-2] = -35/16 Ans: Three Rational numbers are -11/4, -19/8, and -35/16 6 Insert 10 Rational numbers between -3/11 and 8/11 Ans: -2/11, -1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 7.Insert 100 rational numbers between -3/13 and 9/13 Sol: If write -3/13 = -30/130, and 9/13 = 90/130 Therefore -29/130, -28/130, -27/130.. 87/130, 88/130 and 89/130 are the 100 Rational numbers between -3/30 and 9/130 9.True or False: (a) False (b) True (c) True Sol: Product of Two Rational Numbers = -12 One of the number = -8 Let the other number be Y Given Y x (-8) = (-12) Exercise 1(E)

Y = -12/ (-8) Y = 3/2 Ans: Other Number is 3/2 2. Sum of 3/8 and -5/12 is Reciprocal of -15/8 X 16/27 = 3/8 + (-5/12) (-15 X 16) / 8 x 27 = (-10)/9 = 3/8 5/12 Reciprocal of -10/9 is -9/10 LCM of 8 and 12 is 24 Divide the Sum by Reciprocal = (3X3) (5X2) i.e -1/24 (-9)/10 24 = -1/24 X (-10)/9 = 9-10 = 5/108 Ans 24 = -1/24 3. Sol: -15/56 Y = -5/7 = -15/56 = -5/7 x Y Y= -15/56 (-5)/7 = -15/56 X (-7)/5 Y = 3/8 Ans. 4. Sol: Perimeter of an isosceles Triangle = 10 (3/4) cm = 43/4 cm Equal side length = 2 (5/6) cm = 17/6 cm Third side be Y Perimeter = Sum of its side length So, 17/6 + 17/6 + Y = 43/4 = 34/6 + Y = 43/4 Y = 43/4-34/6 LCM of 4 and 6 is 12 Y = (43 X 3) (34 X 2) = 129 68 = 61 12 12 12 Y= 5 (1/12) Ans. 5. Sol: Area of Rectangular table top = 5 (3/8) m 2 = 43/8 m 2 Breadth = 2 (1/4) m = 9/4 m Let Length be Y m So, Y X 9/4 = 43/8 Y = 43/8 9/4 = 43/8 X 4/9 Length = 43/8 m Ans. 6. Sol: In One hour = 80 (4/5) km = 404/5 km In 4 (3/4) hours? = 19/4 hours

= 404/5 X 19/4 = 1919/5 =383(4/5) km Ans. 2. EXPONENTS Exercise ; 2 (A) 1. Exponential from: (i) (2/3) 4 (ii) (3/8) 5 (iii) (-5/7) 6 2. Express of the form p/q (i) 25/4 (ii) 64/343 (iii) -128/2187 (iv) -125/8 3. Express in power notation: (i) 9/64 = 3X3 / 8x8 = (3/8) 2 (ii) 49/25 = (7/5) 2 (iii) -8/27 = (-2/3) 3 (iv) -1/216 = (-1/6) 3 4. Find: (i) (1/3) 3 X (3/2) 2 = 1/27 X 9/4 = 1/12 Ans (iii). (-1/5) 3 X (-1/5) 2 = (-1/5) 3 + 2 = (-1/5) 5 = -1/3125 Ans (v). (-1/3) 5 (2/3) 3 = (-1/3) 5 X (3/2) 3 = -1/72 Ans. 5. Find Reciprocal (i) 125/6 (ii) (-4/3) 3 (iii) (-1/2) 4 6. Evaluate: b 2 9 (b 1) 2 if b = 1.1 (b = 11/10) = (11/10) 2-9 (11/10-1) 2 = 121/100 9 (1/10) 2 (MCQs) = 121/100-9/100 = `(121-9) / 100 = 112/100 = 1.12 Ans. 7. (b) 1 8. (c) 16/19

1. Express as a Rational Number. (i) 1/5 (ii) 64 (iii) 256/81 2. Simplify: (i) (4/9) -3 X (4/9) 11 X (4/9) -10 = (4/9) -3+11-10 = (4/9) -2 = (9/4) 2 Ans. Exercise 2 (B) (ii) (-7/11) -6 (-7/11) -2-6 (-2) =(-7/11) = (-7/11) -4 = (-11/7) 4 Ans. (iv). [(9/11) -3 X (9/11) -7 ] (9/11) -3 = (9/11) -3-7- (-3) = (9/11) -3-7+3 = (9/11) -7 = (11/9) 7 Ans. 3. Evaluate: (i) (-3/4) -2 X (-6/5) -2 =[(-3/4) X (-6/5)] -2 {using law (a/b) m X (c/d) m = (a/b X c/d) m } = (9/10) -2 = (10/9) 2 = 100/81 Ans. 4. Evaluate: (i) (3-1 4-1 ) 2 =(1/3 1/4) 2 = (1/3 X 4) 2 = (4/3) 2 =16/9 Ans. 5. Find x (i) (7/4) -3 X (7/4) -5 = (7/4) x-2 (7/4) -3-5 = (7/4) x-2 Equating powers we get -3-5 = x 2-8 = x 2 => x = -8 + 2 = -6 (The value of x is -6) 6. Find Reciprocal (ii) (-3/7) -3 (-3/7) -4 (-3/7) -3 (-4) = (-3/7) -3+4 = (-3/7) 1 Reciprocal of (-3/7) is (-7/3)

7. If 3 2x+1 9 = 27 find x i.e 3 2x+1 3 2 = 3 3 ( 9 = 3 2 and 27 = 3 3 ) Equating Powers we get 2x +1-2 = 3 => 2x 1 = 3 => 2x = 3+1 => x =2 Ans. 8. Sol: (-3/2) -3 x Y = (9/8) -2 => (-3/2) - 3 x Y = ( 3 2 / 2 3 ) -2 = (-3/2) - 3 x Y = ( 2 3 / 3 2 ) 2 Y = ( 2 3 / 3 2 ) 2 / (-3/2) - 3 Y = ( 2 3 / 3 2 ) 2 x (-3/2) 3 = (2 6 /2 3 ) x (-3 3 / 3 4 ) = 2 3 / (-3) Y = -8/3 Ans 9. Sol: ( 5/4) -2 Y = (1/2) -3 ( 4/5) 2 Y = (2) 3 ( 4/5) 2 = (2) 3 x Y Y = (4/5) 2 (2) 3 Y = 2 4 / 5 2 x (1/2) 3 Y = 2/5 2 = 2/25 Ans. 10. Simplify: [ (-2/5) -9 x (-2/5) 9 ] (-2/5) 2 = (-2/5) -7+9 (-2/5) 2 = (-2/5) 2 x (-5/2) 2 = 1 Ans. 12 Scientific Notation. (i) 0.00002 = 2/10000 = 2/10 5 = 2 x10-5 Ans. (ii) 0,00000542 = 5.42/1000000 = 5.42 / 10 6 = 5.42 x 10-6 Ans. (iii) 9-3 x 10-8 Ans (iv) 3.142 x 10-3 Ans 13. Standard form (Answers) (i) 0.00006 (ii) 0.000532 (iii) 0.000000096 (MCQs) 15 Ans (c) 1 16 16. Ans (d) 1/64 17 7 Ans 18 Ans (c) 4 x 10-6 m 19 Ans (c) 6.049 x 10-2.

4. SQUARES AND SQUARE ROOTS Exercise 3 (A) 1. Prime factorization method: (i) 400 2 400 2 200 2 100 2 50 5 25 5 5 1 400 = (2 x 2) x(2 x 2) x (5 x 5), 400 = 2 x 2 x 5 = 20 Yes 400 is a Perfect Square Number (ii) 768 2 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 3 3 1 768 = (2 x 2) x( 2 x 2) x( 2 x 2) x (2 x 2) x 3 We find that 3 is left, so 768 is not a perfect Square Number. 2. Smallest number must be multiplied (i) 512 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 512 = (2 x 2) x (2 x 2) x (2 x 2) x (2 x 2) x 2 Ans : If we multiply 512 by 2, the Product is a Perfect Square number. (ii) 700

2 700 2 350 5 175 5 35 7 7 1 700 = (2x2) x (5x5) x 7 Ans: If we multiply 700 by 7, the product is a perfect Square Number. 3. Smallest number Should be divided (i) 180 2 180 2 90 5 45 3 9 3 3 1 180 = (2x2) x 5 x (3x3) Ans: If We divide 180 by 5 the Quotient is a perfect Square number. (ii) 1575 5 1575 5 315 3 63 3 21 7 7 1 1575 = (5x5) x (3x3) x 7 Ans: If we divide 1575 by 7 the Quotient is a perfect square number. 4. True or False (i) False (ii) False (iii) True (iv) False (v) False 5. Reason: (i) 2367 ends in 7 (ii) 57000 ends in odd number of zero(s) (iii) 35943 ends in 3 (iv) 4368 ends in 8 (v) 333222 ends in 2 6. Answers: (i) 1 (ii) 1 (iii) 9 (iv) 6 (v) 9 (vi) 5 (vii) 4 (viii) 0 7. Fill in the blanks: (i) 37 (ii) 59 (iii) 173 8. Match the following: (i) 25 (ii) 64 (iii) 9

(iv) 36 (v) 100 9. Pythagorean triplets: (i) (10, 24, 26) (2m, m 2 1, m 2 + 1) are in Pythagorean triplet. Here 2m = 10 => m = 5 m 2 1 = 5 2-1 = 25 1 = 24 m 2 + 1 = 5 2 + 1 = 25 + 1 = 26 Ans: Yes (5, 24,26) are Pythagorean triplet. (MCQs) 10. Ans (c) 288 11. Ans (b) 5 12, Ans (d) (n+1) + n Exercise 3 (B) 1. Square root using Prime factorization metho: (i) 2916 2 2916 2 1458 3 729 3 243 3 81 3 27 3 9 3 3 1 2916 = (2x2) x (3x3) x (3x3) x (3x3), 2916 = 2 x 3 x 3 x 3 = 54 ans. (ii) 2704 2 2704 2 1352 2 676 2 338 13 169 13 13 1 2704 = (2x2) x (2X2) X (13x13), 2704 = 2 x 2 x 13 = 52 Ans. 2. Square Root for Fractions: (i) 25/49 25/49 = 5/7 Ans (ii) 196/484 196/484 = 14/22 Ans

(iv). 0.0009 = 9/10000 9/10000 = 3/100 = 0.03 Ans 3. Simplify ; i 5 2 4 2 = 25 16 = 9 = 3 Ans (iii). (1/2) 2 + 0.25 = 1/4 + 0.5 = 0.25 + 0.5 = 0.75 = 3/4 ans (iv). [- 4/9] X [- 81/100] = (- 2/3) X (-9/10) = 18/30 = 3/5 Ans 4. Square Root By Long Division Method: (i) 6724 82 8 67,24 64 162 3 24 3 24 0 6724 = 82 Ans (ii) 7921 89 8 7921 64 169 1521 1521 0 7921 = 89 Ans 5. Find Square Root: (i) 4556.25 67.5 6 4556.25 36

127 0956 889 1345 6725 6725 0 6. Evaluate: 4556.25 = 67.5 Ans (i). 2809 = 2809 / 4096 4096 53 5 2809 25 103 309 309 0 64 6 4096 36 124 496 496 0 2809 = 53 4096 = 64 7. Find 9216 from this Calulate 92.16 + 9.216 2809 4096 = 53/64 Ans 96 9 9216 81 186 1116 1116 0 9216 = 96 92.16 = 9.6 + 9.216 = 18.816 Ans. 14. Number of Rows = 3600 3600 = 36 X 100 15.Area = 6889m 2 Side Length = 6889 = 6 X 10 = 60 Ans 83 8 6889 64

163 489 489 0 Side Length of a Square = 83 m Perimeter = 4 X side length = 4 X 83 = 332m Ans. 19. Least Number must be added to 594 to make the sum a perfect Square Number 25 2 594 4 45 194 225-31 31 must be added to 594 then 594 + 31 = 625 becomes a perfect square number 625 = 25 ans 21. Least Six digit number is 1,00,000 1,00,489 = 317 ans 317 3 100000 9 61 100 61 1345 3900 4389-489 489 is added to 1,00,000 then 1,00,000+489 = 1,00,489 becomes a perfect square number So,1,00,489 is a least six digit square number Exercise 3(C) 1. Find Square root (correct to two places of decimal) (i) 3 1.732 1 3.00 00 00 1 27 2 00 1 89 343 11 00 10 29 3462 7100 6924 Ans 3=1.732 correct to 2 decimal places. 176 2. Find Square root (i) 175.01 = 175.01 00 000

13.229 1 1.75 01 00 00 175.01 00 000 =13.23 correct to 2 decimal places 1 23 75 69 262 6 01 5 24 2642 77 00 52 84 26449 24 16 00 23 80 41 35 51 3. Find Square root (i) 4/7 4.00 00 00 00/7 = 0.571428 4 7 = 0.571428 0.755 0 0.57 14 28 0 7 57 49 145 819 725 1505 8928 7525 1403 4 7 = 0.76 Correct to 2 decimal places (ii) 13/11 = 1 (2/11) = 1+ 2.00 00 00/11 = 1 + 0.18 18 18 = 1.18 18 18 1.087 3 1.18 18 18 1 208 0 18 18 16 64 2167 154 18 151 69 2 49 13 11 = 1.09 correct to 2 decimal places 4. CUBES AND CUBE ROOTS Exercise 4 1. Find cube: (i) 13 13 3 = 13 x 13 x 13 = 2197 Ans (ii) 400 400 3 = 400 x 400 x 400 = 64000000 Ans (iii) (-4/9) 3 = -64/729 Ans

(iv) 2 (5/7) = 19/7 = (19/7) 3 = 6859/343 = 19 (342/343) ans (v) (0.3) 3 = 0,027 Ans 2. Perfect Cube or not (i) 64 2 64 2 32 2 16 2 8 2 4 2 2 1 64 = 2x2x2 x (2x2x2) 64 = 2 x 2 = 4 64 = 2 x 2 = 4 (Yes 64 is a perfect cube) Ans. (ii) 125 5 125 5 25 5 5 1 125 = 5 x 5 x 5 3 125 = 5 (Yes 125 is a perfect cube) Ans (iii). 243 3 243 3 81 3 27 3 9 3 3 1 243 = (3x3x3) x 3 x 3 Ans: 243 is not a perfect cube. 3. 675 5 675 5 135 3 27 3 9 3 3 1 675 = 5 x 5 x (3x3x3) Resolving into prime factors and forming triplets of equal prime factors we see that the triplet 5 is incomplete. Ans: So, 675 should be multiplied by 5 to make it a perfect cube. 5, 6 3 = 216 = 27 x 8 Resolve 216, 729, 1728, 3375 and 5832 into factors. 9 3 = 729 = 27 x 27 Hence we can say that the cube of a natural number 12 3 = 1728 = 27 x 64 which is a multiple of 3 is a multiple of 27 also. 15 3 = 3375 = 27 x 125 18 3 = 5832 = 27 x 216

6 Find Cube root (i) 125 = 5x5x5 Ans 5 (ii) 343 = 7x7x7 Ans 7 (iii) 2744 2 2744 2 1372 2 686 7 343 7 49 7 7 1 2744 = (2x2x2) x (7x7x7) 2744 = 2 x 7 = 14 ans (iv) -729 = (-9) x (-9) x (-9) 3 729 7 Evaluae (i) = -9 ans 0,216 = 0.6 Ans. (ii) 4.096 2 4096 2 2048 2 1024 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 4096 = (2x2x2) x (2x2x2) x (2x2x2) x (2x2x2) 4096 = 2 x 2 x 2 x 2 =16 4.096 = 1.6 Ans 3 8. 27 x 125 = 27 x 125 3 = 3x3x3 x (5x5x5) = 3 x 5 = 15 ans 3 9. Find: 392 3 x 448 = 392 x 448 2 392 2 196 2 98

7 49 7 7 1 2 448 2 224 2 112 2 56 2 28 2 14 7 7 1 392 x 448 = (2x2x2) x (2x2x2) x (2x2x2) x (7x7x7) 3 392 x 448 = 2 x 2 x 2 x 7 = 56 Ans 10. 17496 2 17496 2 8748 2 4374 3 2187 3 729 3 243 3 81 3 27 3 9 3 3 1 17496 = (2x2x2) X (3x3x3) x (3x3x3) x 3 So, 17496 must be divided by 3 17896 3 = 5832 becomes a perfect cube number 3 5932 = 18 Ans UNIT: 2 ALGEBRA 5. Algebraic Expressions and Special Products Exercise 5 (A) 1. Add: (i) 5xy, -2xy, -11xy, 8xy = 5xy + 8xy - 2xy -11xy = 13xy 13xy = 0 Ans (ii) 3a 2 b, -5a 2 b, 10a 2 b, -7a 2 b, 3a 2 b + 10a 2 b - 5a 2 b - 7a 2 b = 13a 2 b 12a 2 b = a 2 b Ans. (iii) a + b, 3a + 2b = a+b+3a+2b =(a+3a)+(b+2b) = 4a + 3b Ans 2. Subtract: (i) 3x 5 from 8x 17

8x -17 (3x 5) = 8x -17-3x + 5 = 8x -3x -17 + 5 = 5x -12 Ans (ii) -8xy from 7xy 7xy (-8xy) = 7xy + 8xy = 15xy Ans. (iii) 2x 2 5x +10 from 5x 2 11x + 19 5x 2 11x + 19 (2x 2 5x +10) = 5x 2 11x + 19-2x 2 + 5x 10 =( 5x 2 2x 2 ) - ( 11x + 5x) +(19 10) = 3x 2 6x +9 Ans. 3. Adjacent sides of a Rectangle are 3x 2-5y 2 and 7x 2 xy Perimeter = 2 (3x 2-5y 2 + 7x 2 xy) = 2 (3x 2 + 7x 2 5y 2 xy) = 2(10x 2-5y 2 xy) = 20x 2 10y 2-2xy Ans. 4. Perimeter of a Triangle = 7p 2 8p + 9 Two of its sides are 2p 2 p + 1 and 11p 2 3p + 5 Let the third side be X => 2p 2 p + 1 + 11p 2 3p + 5 + X = 7p 2 8p + 9 X = 7p 2 8p + 9 (2p 2 p + 1 + 11p 2 3p + 5) = 7p 2 8p + 9-2p 2 + p - 1-11p 2 + 3p - 5 = -6p 2 4p +3 Length of the Third side is -6p 2 4p +3 Ans MCQs 5. Ans (c) Trinomial : 5 6. Ans ( c) (3t 2-5t) (-5t 2 4t) 1. Multiply (i) -3x 2 y by -5xy 2 = (-3x 2 y) X (-5xy 2 ) = 15 x 3 y3 ans. Exercise 5 (B) (ii) 3 8 x4 y z X (-16a 2 y 4 b 3 ) = 3 8 X (-16) x4 y y 4 z a 2 b 3 = - 6 x 4 y 5 z a 2 b 3 Ans. 2. Find the Products: (- 2xy) (-3x 3 y 2 ) ( 1 6 x2 y 7 )

= -2 x (-3) x ( 1 6 ) x (x. x3. x 2 ) X (y. y 2. Y 7 ) = 1 x 6 y 10 = x 6 y 10 Ans 4 Find the Value. (i) (-10m 2 n 6 ) X (-20mn 15 ) for m =1.2 and n = 1 =[-10 x (-20)] (m 2. m ) (n 6. n 15 ) = 200 m 3 n 21 = 200 (1.2) 3 (1) 15 = 200 X 1.728 X 1 = 345. 6 Ans 5. -4 (5y + 7) = (-4 x 5y) + (-4 x 7) = - 20y -28 Ans. 7. -2 ( 2x 2 + 3xy 4y 2 ) = (-2 x 2x 2 ) + (-2 x 3xy) + (-2 x -4y 2 ) = - 4x 2 6xy + 8y 2 Ans. 8. 2y (- y 2 + 3y 4) = [2y x (-y 2 )] + (2y x 3y) - ( 2y x 4) = - 2y 3 + 6y 2 8y Ans. 12. 100x (0.01x - 0.19yz) = (100x X 0.01x) (100x X 0.19yz) = x 2 19xyz Ans. 13. Simplify: x(y z) + y (z x) + z (x y) = xy xz + yz yx + zx zy = 0 Ans. MCQ s. 15. (a) 11 16. -12x 5 y 4 Exercise 5 (C) 1. Multiply: (x + 2) (x + 5) = (x + 2) x + (x + 2) 5 = x 2 + 2x + 5x + 10 = x 2 + 7x + 10 Ans. 4 (x 2 + 5) (x 2 6) = (x 2 + 5) x 2 (x 2 + 5) 6 = x 4 + 5x 2 6x 2 + 30

= x 4 x 2 + 30 Ans. 6. (5x 2 + 2y 2 ) (3x 2-7y 2 ) = (5x 2 + 2y 2 ) X 3x 2 (5x 2 + 2y 2 ) X 7y 2 = 15x 4 + 6x 2 y 2-35 x 2 y 2 14 y 4 = 15x 4 29 x 2 y 2 14y 4 Ans 7. (3x + 5y 7 ) (2x - 3y 7 ) = (3x + 5y 7 ) X 2x - (3x + 5y 7 ) X 3y 7 = 6x 2 + 10xy 7 9xy 7 15y2 49 = 6x 2 + xy 7 -- 15y2 49 Ans 10 (4x 2 + 3y) (3x 2-4y) = (4x 2 + 3y) X 3x 2 - (4x 2 + 3y) X 4y = 12x 4 + 9x 2 y - 16 x 2 y 12y 2 = 12x 4-7 x 2 y 12y 2 When x = 1 and y= 2 [(4 x 1 2 ) + (3x 2) ] [ (3 x 1 2 ) (4x2) ] = (12 x 1 4 ) (7x1x2) (12 x2 2 ) = (4 + 6) (3 8) = 12 14 48 = 10 x (-5) = (-50) = (-50) = (-50) verified 11 Simplify: (i) 2 7 (14x2-21y 2 ) (-5x 2 3y 2 ) = 2 7 (14x2-21y 2 ) X (-5x 2 ) - (14x 2-21y 2 ) X 3y 2 = 2 7 (-70x2 + 105 x 2 y 2-42 x 2 y 2 + 63y 4 ) = 2 7 (-70x2 + 63 x 2 y 2 + 63y 4 ) = 2 (-10x 2 + 9 x 2 y 2 + 9y 4 ) = (-20x 2 + 18 x 2 y 2 + 18y 4 ) Ans 19. Simplify: (3x 4) (2x 2-5x + 1) - (2x 1) (3x 2 +7x 5)

(MCQs) (3x 4) (2x 2-5x + 1) = 3x (2x 2-5x + 1) 4 (2x 2-5x + 1) = 6x 3-15x 2 + 3x 8x 2 + 20x -4 = 6x 3-15x 2 8x 2 + 3x + 20x -4 = 6x 3-23x 2 + 23x - 4. (1) (2x 1) (3x 2 + 7x 5) = 2x (3x 2 + 7x 5) - 1 (3x 2 + 7x 5) = 6x 3 + 14x 2 10x -3x 2-7x +5 = 6x 3 + 14x 2-3x 2-7x - 10 x + 5 = 6x 3 + 11x 2 17x + 5 (2) (1) (2) we get 6x 3-23x 2 + 23x - 4-6x 3-11x 2 + 17x 5 = -34 x 2 + 40 x 9 Ans 20. Ans (a) (x +1) (x 6) 21. (c) x 3 y 3 1. Divide (i) x 5 x 3 = x 5-3 = x 2 Ans Exercise 5 (D) (v). a 3 b 4 a 2 b = a 3 2 b 4 1 = a b 3 Ans (vii) -54x 4 y 5 6x 2 y 3 = -54/6 x 4 2 y 5-3 = -9x 2 y 2 Ans 2. Divide: (i) 8x + 4 by 2 = 8x/2 + 4/2 = 4x +2 Ans (MCQs) (vi) x 3 x 2 by x 3. Ans -90/m 3 4. 4. Ans 0 = x 3 /x - x 2 /x = x 2 - x Ans 1. Divide: a 2 + 6a + 5 by a + 1 a + 5 a + 1 a 2 + 6a + 5 (-) (-) a 2 + a 5a + 5 (-) (-) 5a + 5 0 Exercise 5 (E)

Ans : a 2 + 6a + 5 = (a + 1) (a + 5) 4 14-17x + 5x 2 by 2 x 7 5x 2 - x 14-17x + 5x 2 (-) (+) 14 7x -10x + 5x 2 (+) (-) -10x + 5x 2 0 Ans (7 5x) 7. x 4 + 4x 2x 2 + x 3-10 by x 2 = x 4 + x 3 2x 2 + 4x 10 by x-2 x 3 + 3x 2 + 4x + 12 x 2 x 4 + x 3 2x 2 + 4x 10 (-) (+) x 4 2x 3 +3x 3 2x 2 (-) (+) 3x 3 6x 2 +4x 2 + 4x (-) (+) 4x 2-8x +12x 10 (-) (+) 12x - 24 14 Ans. Q = x 3 + 3x 2 + 4x + 12 R = 14 8 8x 2 + 21x + x 3 + 18 by 6 + 5x + x 2 x + 3 x 2 +5x + 6 x 3 + 8x 2 + 21x + 18 (-) (-) (-) x 3 + 5x 2 + 6x 3x 2 + 15x + 18 (-) (-) (-) 3x 2 + 15x + 18 0 Ans : Q = x + 3 R = 0 9. Find whether or not the first polynomial is a factor of the second x 7; x 2 3x - 28 x + 4

x - 7 x 2 3x - 28 (-) (+) x 2 7x 4x - 28 (-) (+) 4x - 28 0 Yes x 7 is a factor of x 2 3x - 28 Q = x + 4, R = 0 13 6p 3 + 5p 2 + 4 by 2p + 1 3p 2 + p 2p + 1 6p 3 + 5p 2 + 0 + 4 (-) (-) 6p 3 + 3p 2 2p 2 + 0 + 4 (-) (-) 2p 2 + p -p + 4 Check = Divisor X Quotient + Remainder 6p 3 + 5p 2 + 4 = (2p + 1) ( 3p 2 + p) -p + 4 14 Sol: = 6p 3 + 3p 2 + 2p 2 + p p + 4 = 6p 3 + 5p 2 + 4 Verified 3x 3 + 2x 2 9x - 1/2 2x 2-3 6x 5 + 4x 4-27x 3 7x 2-27x - 6 (-) (+) 6x 5-9x 3 4x 4-18x 3 7x 2 (-) (+) 4x 4-6x 2-18x 3 - x 2 27x (+) (-) -18x 3 + 27x - x 3 54x - x 3 54x - 6-54x 15/2 Ans: -54x 15/2 should be added to 6x 5 + 4x 4-27x 3 7x 2-27x - 6,then it exactly divisible by 2x 2-3 Exercise: 5 (F) 1. Write down the squares of (i) p + 5 (p + 5) 2 = p 2 + 10p + 25 (Here we use the identity (a + b) 2 = a 2 + 2ab + b 2 )

(iii). (5c + 2d) 2 = 25c 2 + 20cd + 4d 2 (v). (3x + 1/3x) 2 = 9x 2 + 2 + 1/9x 2 (ix). (t 3) 2 = t 2-2t + 9 2. Simplify: (i) (a 3 b xy 3 ) 2 = a 6 b 2 2a 3 bxy 3 + x 2 y 6 3. Evaluate: (i) 102 2 = (100 + 2) 2 (ii) 709 2 = (700 + 9) 2 (iii) 59 2 = (60 1) 2 = 100 2 + 2 x 100 x 2 + 2 2 = 10000 + 400 + 4 = 10404 Ans = 700 2 + 2 x 700 x 9 + 9 2 = 490000 + 12600 + 81 = 502681 Ans = 60 2 2 x 60 x 1 + 1 2 = 3600 120 + 1 = 3481 Ans 4. 25x 2 + 70x + 49 if x = (-1) = 25 X (-1) 2 + 70 X (-1) + 49 =25 70 + 49 =74 70 = 4 Ans 5. x +1/x = 12 find x 2 + 1/x 2 (x + 1/x) 2 = x 2 + 1/x 2 + 2 12 2 = x 2 + 1/x 2 + 2 144 = x 2 + 1/x 2 + 2 => x 2 +1/x 2 = 144 2 = 142 Ans. 6. x + 1/x = 5 find x 2 + 1/x 2 and x 4 + 1/x 4 (x + 1/x) 2 = x 2 + 1/x 2 + 2 5 2 = x 2 + 1/x 2 + 2 25 = x 2 + 1/x 2 + 2 => x 2 +1/x 2 = 25 2 = 23 (x 2 + 1/x 2 ) 4 = x 4 + 1/x 4 + 2 23 2 = x 4 + 1/x 4 + 2 529 = x 4 + 1/x 4 + 2 => x 4 +1/x 4 = 529 2 = 527 7. x - 1/x = 7 find x 2 + 1/x 2 (x - 1/x) 2 = x 2 + 1/x 2-2 7 2 = x 2 + 1/x 2-2 49 = x 2 + 1/x 2-2 => x 2 +1/x 2 = 49 + 2 = 51 Ans.

8. x - 1/x = 8 find x 2 + 1/x 2 and x 4 + 1/x 4 (x - 1/x) 2 = x 2 + 1/x 2-2 8 2 = x 2 + 1/x 2-2 64 = x 2 + 1/x 2-2 => x 2 +1/x 2 = 64 + 2 = 66 (x 2 + 1/x 2 ) 4 = x 4 + 1/x 4 + 2 66 2 = x 4 + 1/x 4 + 2 4356 = x 4 + 1/x 4 + 2 => x 4 +1/x 4 = 4356 2 = 4354 9. (x + y) 2 = x 2 + 2xy + y 2 => 9 2 = x 2 + 2 X 16 + y 2 81 = x 2 + 32 + y 2 81 32 = x 2 + y 2 x 2 + y 2 = 49 Ans. Exercise: 5 (G) 1. Find the following Products: (i) (x+2) (x 2) = x 2 2 2 = x 2 4 Ans (iv). (x 2 y 2 ) (x 2 + y 2 ) = ( x 2 ) 2 - (y 2 ) 2 = x 4 y 4 Ans (v). ( 2 5 ab c ) ( 2 5 ab + c ) = ( 2 5 ab)2 - c 2 = 4 25 a2 b 2 c 2 Ans 2. (i) (a +1 ) (a 1) (a 2 + 1) => (a 2-1) (a 2 + 1) = (a 4-1) Ans (iii) (x 1/x) (x + 1/x) (x 2 + 1/x 2 ) => (x 2-1/x 2 ) (x 2 + 1/x 2 ) = (x 4-1/x 4 ) Ans 3 Find the Value of : (i) 18 X 22 = (20 2) (20 + 2) = 20 2 2 2 = 400 4 = 396 Ans (ii) (84) 2 - (76) 2 = (84 + 76) (84 76) = 160 X 8 = 1280 Ans. 5. Find the value of x: (i) 13 x = (58) 2 - (45) 2 = (58 + 45) (58 45) 13x = 103 X 13 => x + 103 Ans. (iii) Given exp. = (297) 2 - (203) 2 / 94 = (297 + 203) (297 203)/94 = 500 X 94 /94 = 500 Ans. (6) FACTORIZATION OF ALGEBRAIC EXPRESSIONS Exercise 6 (A) Factorize: 1. 3x 12 = 3 (x 4) Ans

2. 80 + 16y = 16 (5 + y) Ans 3. 8x 8y = 8 ( x y) Ans 4. 7x + 14 = 7 ( x + 2) Ans 5. a 2 + 7a = a ( a + 7) Ans. 6. 5y 7y 2 = y(5 7y) Ans 7. 8c 2-24c = 8c(c 3) Ans 8. 5m 2-6m = m ( 5m 6) Ans 9. 12 u 20 u v = 4u ( 3 5v) Ans 10. 10pq 12p 2 q = 2pq (5 6p) Ans Exercise 6 (B) Factorize: 1. z (z -1) + 2 (z 1) = (z + 2) (z 1) Ans 3 2y (y + 5) 3 (y + 5) = (2y 3) (y + 5) Ans 5 5(2 + b) 6b(2 + b) = (5 6b) (2 + b) Ans 9 (x + 2) 2 + 5 (x + 2) = (x + 2) (x + 2) + 5 (x + 2) = (x + 2 + 5) (x + 2) = (x + 7) (x + 2) Ans 11 14 (3y 5z) 3 + 7 (3y 5z) 2 = 14 (3y 5z) (3y 5z) 2 + 7 (3y 5z) 2 = 7 (3y 5z) 2 [ 2(3y 5z) + 1] = 7(3y 5z) 2 (6y 10z +1) Ans Factorize: 1. ax 2 + by 2 + bx 2 + ay 2 = ax 2 + bx 2 + ay 2 + by 2 = x 2 (a + b) + y 2 (a + b) = (x 2 + y 2 ) (a + b) Ans 3 px + qx + py +qy = x(p + q) + y (p + q) = (x + y) (p + q) Ans Exercise 6 (C) 5) x 2 ax bx + ab = x (x a) b (x a) = (x a) (x b) Ans 7) 8pr + 4qr + 6ps + 3qs = 4r (2p + q) + 3s (2p + q) = (4r + 3s) (2p + q) Ans 9) 3mn + 2pn + 3mq + 2pq = n (3m + 2p) + q (3m + 2p) = (n + q) (3m + p) Ans 11) ab 2 bc 2 ab + c 2 = b (ab c 2 ) 1(ab c 2 ) = (b 1) (ab c 2 ) Ans

13.) 6pm + 9mq + 8pn + 12qn = 3m (2p + 3q) + 4n (2p + 3q) = (3m +4n) (2p + 3q) Ans. 15) axy 2 + 3x + 2a 2 y 2 + 6a = axy 2 + 2a 2 y 2 + 3x + 6a = ay 2 ( x +2a) + 3 (x + 2a) = (ay 2 + 3) (x + 2a) Ans. MCQs 16. Ans (c) (yz + x) (zx + y) 17. Ans (d) (a + b) (a 1) Exercise 6 (D) A. Which are perfect Trinomial Squares? Write Yes or No. 1. a 2 + 10 a + 16 = (a + 8) (a + 2) Ans : No 2. b 2 + 14b - 49 Ans: No 3. No 4. Yes 5. No 6. Yes 7. No 8. Yes 9. Yes 10. No 11. yes B. Fill in the Blanks: 12) 14n 13) 10x 14) 6b 15) x 2 16) 25q 2. 17) 4x 2 C. Factorize: 18. a 2 + 2a + 1 = a 2 + (2) (a) (1) + 1 = (a + 1) 2 Ans 19. b 2 + 4b + 4 =. b 2 + (2) (2) (b) + 2 2 = (b + 2) 2 Ans 20. y 2 8y + 16 = y 2 (2) (4) (y) + 4 2 = (y - 4) 2 Ans 21. 36 12r + r 2 = 6 2 (2) (6) (r) + r 2 = (r 6) 2 Ans 24. 1 8ax + 16a 2 x 2 = 1 (2) (4ax) (1) + (4ax) 2 = (4ax 1) 2 Ans 26. x 2 + 5x + 25/4 = x 2 + (2) (5/2) (1) + (5/2) 2 = (x + 5/2) 2 Ans

29. x 2 /4y 2 2/3 + 4y 2 /9x 2 = (x/2y) 2 2 (x/2y) (2y/3x) + (2y/3x) 2 = (.x/2y - 2y/3x) 2 Ans D. Find the Square root: 31. 64 16y + y 2 = (8) 2 - (2) (8) (y) + y 2 = ( 8 y) 2 64 16y + y2 = (8 y) Ans. 35. a 2 /b 2 + 2 + b 2 /a 2 = (a/b) 2 + 2 (a/b) (b/a) + (b/a) 2 = (a/b + b/a) 2 a 2 /b 2 + 2 + b 2 /a 2 = a/b + b/a Ans. MCQs 36) Ans (c) (ax ay+ bx +by) 2 37) Ans (c) (2x + 5) Exercise 6 (E) Factorize: 1. a 2 4 = (a + 4) (a 4) 2. 25 x 2 = (5 + x) (5 x) 3. 9b 2 16 = (3b + 4) (3b 4) 4. 49p 2-16 = (7p + 4) (7p 4) 5. a 2 - b 2 = (a + b) (a b) 32. x 2 y 2 +2x + 1 = (x 2 + 2x +1) y 2 = (x + 1) 2 - y 2 = (x + 1 + y) (x = 1 y) Ans MCQ s. 34. Ans (b) (11x 7y) (-x + 17y) 35. Ans (b) (x + y z) Exercise 6 (F) Factorise: 1. a 2 + 5a +6 Sum of the roots = 5 (2+ 3), Product of the roots = 6 (2 x 3) So, a 2 + 5a +6 = a 2 + 3a + 2a + 6 = (a 2 + 3a) + (2a + 6) = a(a + 3) + 2 (a + 3) = (a + 2) (a + 3) Ans 2. x 2 + 7x +12 Sum of the roots = 7 (4+ 3), Product of the roots = 12 (4 x 3) So, x 2 + 7x +12 = x 2 + 4x + 3x + 12

= (x 2 + 4x) + (3x + 12) = x(x + 4) + 3 (x + 4) = (x + 3) (x + 4) Ans 3. m 2 + 13m +42 Sum of the roots = 13 (6+ 7), Product of the roots = 42 (6 x 7) So, m 2 + 13m +42 = m 2 + 6m + 7m + 42 = (m 2 + 6m) + (7m +42) = m (m + 6) + 7 (m + 6) = (m + 7) (m + 6) Ans 4. a 2 + 25a - 54 Sum of the roots = (-25) (27-2), Product of the roots = 54 [27 x (-2)] So, a 2 +25a - 54 = a 2 + 27a - 2a - 54 = (a 2 + 27a) - (2a + 54) = a(a + 27) - 2 (a + 27) = (a - 2) (a + 27) Ans MCQs 24. Ans (d) (x +2) (x 12) 25) Ans (c) (x y 9) Exercise 6 (G) Factorise: 1. 4x 2 + 5x + 1 Sum of the roots = 5 (1 + 4), Product of the roots = 4 x 1 = 4 (4 x 1) So, 4x 2 + 5x + 1 = 4x 2 + 4x + x + 1 = ( 4x 2 + 4x) + ( x + 1) = 4x (x + 1) + 1 (x + 1) = (4x + 1) (x + 1) Ans. 2. 2x 2 + 11x + 14 Sum of the roots = 11 (7 + 4), Product of the roots = 2 x 14 = 28 (7 x 4) So, 2x 2 + 11x + 14 = 2x 2 + 7x + 4x + 14 = (2x 2 + 7x) + (4 x + 14) = x (2x + 7) + 2(2x + 7) = (x + 2) (2x + 7) Ans. 3. 2x 2 + 11x + 12 Sum of the roots = 11 (8 + 3), Product of the roots = 2 x 12 = 24 (8 x 3) So, 2x 2 + 11x + 12 = 2x 2 + 8x + 3x + 12 = (2x 2 + 8x) + (3x + 12) = 2x (x + 4) + 3( x + 4) = (2x + 3) (x + 4) Ans. 4. 3x 2 + 13x + 4 Sum of the roots = 13 (12 + 1), Product of the roots = 3 x 4 = 12 (12 x 1) So, 3x 2 + 13x + 4 = 3x 2 + 12x + x + 4 = (3x 2 + 12x) + ( x + 4) = 3x (x + 4) + 1(x + 4)

= (3x + 1) (x + 4) Ans. 5. 2x 2 5x - 12 Sum of the roots = -5 (-8 + 3), Product of the roots = 2 x (-12) = (-24) (-8 x 3) So, 2x 2-5x 12 = 2x 2-8x + 3x + 12 = (2x 2-8x) + (3x + 12) = 2x (x + 4) + 3(x + 4) = (2x + 3) (x + 4) Ans. MCQs 18. Ans (c) (5x + 2) 19. Ans (d) b( 1-2b) (1 + 2b) (1 + 3b 2 ) Solve and Check your Answer: 1 (i) 10p - (3p 4) = 4 (p +1 ) + 9 10p 3p + 4 = 4p + 4 + 9 7p + 4 = 4p + 13 7p 4p = 13 4 3p = 9 => p =3 Check: 10 X 3 (3X3 4) = 4 ( 3 + 1) + 9 30 5 = 16 + 9 25 = 25 6. LINEAR EQUATIONS Exercise 7 (A) (ii) 7 + 2 (a + 1) 3a = 5a 7 + 2a + 2 3a = 5a 9 a = 5a 9 = 6a => a = 9/6 = 3/2 Check: 7 + 2(3/2 + 1) -3X3/2 = 5 X 3/2 7 + 2(5/2) 9/2 = 15/2 7 + 5 9/2 = 15/2 15/ 2 = 15/2 2. (i) x 3 5 = 8 => x 15 = 8 3 => x 15 = 24 => x = 39 Ans 3. (i) 4y + y 20y + y = 21 => = 21 5 5 => 21y= 105 => y = 105/21 = y = 5 Ans 4. 2 3 (x + 3 5 ) = 7 2 => 2 3 ( 5x+3 5 ) = 7 2 ( 5x+3 ) = 7/2 X 3/2 5

21 5x + 3 = 5 X 4 5x + 3 = 105/4 => 5x = ( 105/4) - 3 5x = (105-12) /4 x= 93 / 20 x = 4 (13/20) Ans 5. (x + 3) / 7 (2x 5) / 3 = (3x 5) /5-25 Multiplying both the sides of the equation by the LCM of 7, 3 and 5 = 105 105 X (x + 3) / 7 105 (2x 5)/ 3 = 105 (3x 5) / 5 105 X 21 15 (x + 3) - 35 (2x 5) = 21 ( 3x 5) 2625 15x + 45-70x + 175 = 63x -105 2625 15x 70x 63x = (-105 2625 45 175) (-118 x) = (- 2950) x = 25 Ans 6. (3x 2) /10 (x + 3) / 7 + (4x 7) / 3 = (x 1) [LCM of 10, 7 and 3 = 210) 210 X (3x 2) / 10 210 X(x + 3) / 7 + 210 (4x 7) / 3 = 210(x 1) 21(3x -2) - 30 (x + 3) + 70 (4x 7) = 210x - 210 63x 42 30x - 90 + 280x 490 = 210x 210 63x 30x + 280x 210x = (- 210 + 42 + 90 + 490) 103 x = 412 x = 412 / 103 = > x = 4 Ans 10 3/ (x + 1) = 5/2x 3 X 2x = 5 (x + 1) 6x = 5x + 5 6x 5x = 5 x = 5 Ans. 11 6 / (3m + 1) = 9/ ( 5m 3) 6 (5m 3) = 9 (3m + 1) 30m 18 = 27m + 9 30m 27m = 9 + 18 3m = 27 => m = 9 Ans 15 (x 2) / (x 4) = (x + 4) / (x 2) (x -2) 2 = (x + 4) (x 4) x 2 4x + 4 = x 2 16 x 2 - x 2-4x = (-16 4) -4x = (-20) x = 5 Ans 18 [(x/4) (3/5) ] / (4/3) 7x = (-3) / 20 20 [ (x/4) (3/5)] = (-3) [ (4/3) -7x]

5x 12 = (-4) + 21x 5x 21x = (-4) + 12 (-16x) = 8 x = (- 1/2) Ans MCQs 23. Ans x =5 24. X = (-1/8) 25. X = 3 1. Let the Smaller number = x Then, Larger number = x + 12 Given, x + x + 12 = 10 => 2x + 12 = 10 Exercise 7 (B) x = (10-12) /2 => x = -1 Ans : Smaller number = (-1), Larger Number = (-1) +12 = 11 2. Let the smaller Number = x, Larger number = 5x Given, 5x 18 = 3x => 5x 3x = 18 => 2x = 18 => x = 9 Ans: The numbers are 9 and 45 (5 X 9) 3. (i) Let the Three consecutive numbers be x, x + 1, x + 2 Then, x + x + 1 + x + 2 = 108 3x + 3 = 108 3x = 108-3 => 3x = 105 => x= 35 Ans : The numbers are 35, 36 and 37 (ii). Let the Three consecutive odd numbers be 2x + 1, 2x + 3 and 2x + 5 Then, 2x + 1 + 2x + 3 + 2x + 5 = 93 6x + 9 = 93 6x = 93 9 => 6x = 84 => x= 14 Ans : The numbers are 2 X 14 +1, 2 X 14 + 3, 2 X 14 + 5 i.e., 29, 31, 33 (iii) Let the three consecutive even numbers be 2x, 2x + 2, 2x + 4 Then, 2x + 2x + 2 + 2x + 4 = 246 6x + 6 = 246 6x = 240 => x = 40 Ans : The numbers are 2 X 40, 2 X 40 + 2, 2 X 40 + 4 i.e. 80, 82, 84 (iv) Given 7x + 7x + 7 + 7x + 14 = 777 21x + 21 = 777 21 x = 777 21 => 21x = 756 => x = 36 Ans : The numbers are 252, 259, 266

4. Let the first part x. Then, Second part = 2x 32, Third part = x + 18 Given, x + 2x 32 + x + 18 = 534 4x 14 = 534 4x = 548 => x = 137 Ans: The Three parts are 137, 2 X 137 32, 137 + 18 i.e., 137, 242, 155 5. Let the three consecutive odd numbers be 2x + 1, 2x + 3, 2x + 5 Then, 3(2x + 1) 7 = 2 (2x + 5) 6x + 3 7 = 4x + 10 6x 4x = 10 + 4 2x = 14 => x = 7 Ans : Three consecutive odd numbers are 2 X 7 + 1, 2 X 7 + 3, 2 X 7 + 5 i.e., 15, 17, 19 6. Let the Smaller number = x, Then the larger number = x + 7 Given (x + 7) 2 = x 2 + 77 x 2 + 14x = 49 = x 2 + 77 14x = 77-49 => x = 28/14 => x = 2 Ans : The numbers are 2 and 9. 7 Let the two Consecutive even numbers be 2x and 2x + 2 Given (2x + 2) 2 (2x) 2 = 36 => 4x 2 + 8x + 4 4x 2 = 36 8x = 36 4 => 8x = 32 => x = 4 Ans The two numbers are 2 X 4, 2 X 4 + 2 i.e., 8 and 10 8) Let the numerator of the fraction be x Then, the denominator = x + 3 Given (x + 5) / (x + 3 + 5) = 4/5 (x + 5) / (x + 8) = 4/5 5 ( x + 5) = 4 (x + 8) 5x + 25 = 4x + 32 x = 7 Ans : The fraction is (7 / 10) 9) Let the two integers be x and 50 + x Given, x / ( 50+ x) = 1/3 3x = 50 + x 3x x = 50 2x = 50 => x = 25 Ans: Two Integers are 25 and 75 10) Let the units digit be x. Then the ten s digit = 7 x The number formed by these digits = 10 (7 x) + x = 70 9x The numbers formed by interchanging the digits = 10x - (7 x) = 9x + 7 Given, (9x + 7) - (70 9x) = 27 9x + 7 70 + 9x = 27 18x - 63 = 27

18x = 27 + 63 18x = 90 => x = 5 Ans : The number is 25 11) Let Vijay s age = x years. Then Sushma s age = (x + 15) years. In 3 more years, Sushma s age = (x + 5 + 3) = x + 18 years Three years ago Vijay s age = (x 3) years Given, x + 18 = 8 ( x 3) x + 18 = 8x 24 18 + 24 = 8x x 42 = 7x => x= 6 Years Ans : Vijay s age = 6 years, Sushma s age = 21 years. 12) Let the brother s age = x years. Then Sanjay s age = (1/2) x years Their ages after 6 years from now = (x + 6) years and [(1/2) x + 6] years Given [(1/2) x + 6] = 3/5 (x + 6) 5 [(1/2) x + 6] = 3x + 18 5x + 60 = 6x + 36 60 36 = 6x 5x x = 24 Years Ans; Sanjay s age = 12 years, His brother s age = 24 years. 13) (a) Area = Length X Breadth => 8 X (x 5) = 3x 8x 40 = 3x 8x 3x = 40 5x = 40 => x = 8 (b) Perimeter = 2 (l + b) => 2 (x 5 + 8) = 40 => 2 (x + 3) = 40 => 2x + 6 = 40 => 2x = 34 => x = 17 14) From the Given figure, Breadth = Area / Length = 225 / 25 = 9m Perimeter of the Rectangular enclosure = 2 (l + b) 2 ( 25 = 9) = 68m Given, 4(x 2) = 68 => x 2 = 68 / 4 x - 2 = 17 => x = 19 Ans 15). Let each side of the square garden be y metre Area = (y X y) m 2 = y 2 m 2 Length of the garden = (x + 3) m, Width of the garden = (x + 1)m Area of the enlarged garden = (x + 3) (x + 1) = x 2 + 4x + 3 Given, New area Original area = 19 => x 2 + 4x + 3 x 2 =19 4x = 16 => x = 4, Length of a side now = (4 + 3)m = 7m

MCQs 24) Ans (b) 54 25 Ans (c) 3 : 5 26) Ans (d) 900km UNIT: 4 - GEOMETRY 1. Fourth Angle = 360 0 (65 0 + 55 0 + 130 0 ) = 360 0-250 0 = 110 0 12. QUADRILATERALS Exercise 12 2. Let the two equal angles be x 0 each, then x + x + 75 0 + 125 0 = 360 0 =>.2x + 200 0 = 360 0 2x = 360 0 200 0 => 2x = 160 0 => x = 80 0 The two angles are 80 0 each 3. Let B = c = D = x 0. Then A + B + c + D = 360 0 150 0 + x + x + x = 360 0 => 150 0 + 3x = 360 0 3x = 360 0 150 0 => x = 210 0 / 3 => x = 70 0 Ans B = c = D = 70 0 4. OMPN is a Quadrilateral, in which O = 40 0, N = 90 0, M = 90 0 O + M + N + P = 360 0 40 0 + 90 0 + 90 0 + A = 360 0 => 220 0 + A = 360 0 A = 140 0 Ans 5. Measure of each angle = 360 0 / 4 = 90 0 Ans 6. Fourth Angle = 360 0 (55 0 + 55 0 + 140 0 ) = 360 0-250 0 = 110 0 7. Let the angles be 3x 0,, 5x 0, 7x 0, 9x 0, then 3x + 5x + 7x + 9x = 360 0 => 24x = 360 0 => 360 0 / 24 => x = 15 0 Ans: The angles are 3 X 15, 5 X 15, 7 X 15, 9 X 15, i.e., 45 0, 75 0, 105 0 and 135 0 8. In the Quadrilateral ABCD A + B + C + D = 360 0, A + B + 70 0 + 50 0 = 360 0 A + B = 360 0 120 0 => A + B = 240 0 AO bisects A and BO bisects B, OAB = 1/2 A, OBA = 1/2 B OAB + OBA = 1/2 ( A + B ) In OAB, OAB + OBA + AOB = 180 0 1/2 ( A + A ) + AOB = 180 0 => 1/2 X 240 0 + AOB = 180 0

120 0 + AOB = 180 0 AOB = 60 0 MCQs 9. Ans (c) Obtuse 10. Ans (b) 91 0 11. Ans (d) 2 0 13. PARALLELOGRAMS Exercise 13(A) 1. Adjacent angles of a Parallelogram are supplementary ans opposite angles are equal Given, A = 80 0 => A + D = 180 0 => D = 180 0-80 0 = 100 0 Ans : A = c = 80 0 and B = D = 100 0 2. A + B = 180 0 ( adjacent angles of a Parallelogram are Supplementary) Given, A = B => A = B = 180 0 / 2 = 90 0 Also c = A = 90 0 and B = D = 90 0 3. Perimeter of the Parallelogram = 2 (l + b) = 2( 5cm + 6cm) = 2 X 11cm = 22 cm 4. Let the two consecutive ( adjacent ) angles be x 0 and 3x 0 Let x + 3x = 180 0 => 4x = 180 0 => x = 45 0 The Two angles are 45 0 and 135 0, The other two angles are also 45 0 and 135 0 5. Let one side of the Parallelogram = x cm. Then, other side = (x + 30) cm Perimeter of the Parallelogram = 2(l + b) Given, 2(l + b) = 180 => 2 (x + x + 30) = 180 4x + 60 = 180 => 4x = 180 60 4x = 120 => x = 30 Ans: Length of the Adjacent Sides are 30cm and 60cm 6. Let one angle = x 0, Then, the adjacent angle = 2x 0-20 Sum of adjacent sides of a llgm = 180 0 X + 2x 20 = 180 0 => 3x = 180 0 + 20 0 => x = 200 0 / 3 = 66 (2/3) 0 Other angle = 2 X (200/3) 0 20 0 = (400 0 60 0 ) / 3 = 113 ( 1/ 3) 0 Ans : The angles of a Parallelogram are 66(2/3) 0 113(1/3) 0 66(2/3) 0 113(1/3) 0 7. Y + 56 0 = 125 0 (opp angles of a llgm are equal) Y = 69 0 (z = y ) = 180 0 ( AE ll DC) co interior angles are supplementary) z = 180 69 => 111 0 Now, 125 0 + x + y + z = 360 0 125 0 + x + 69 0 + 111 0 = 360 0 x = 360 0 305 0 x = 55 o Ans

8. In BCD, 2x + 3x + 5x = 180 0 10x = 180 0 => x = 18 0 BDC = 2 X 18 0 = 36 0 BCD = 5 X 18 0 = 90 0 BDC = 3 X 18 0 = 54 0 A = BCD = 90 0 ( (opp angles of llgm are equal) ADC + BCD = 180 0 => ADC= 90 0 (adjacent angles are equal) ABC = ADC = 90 0 (opp, angles are equal) Exercise 13(B) 1. True or False: (a) True (b) True (c) False (d) True (e) False (f) True 2. (a) Rhombus (b) Rectangle (c) Rectangle (d) Rhombus 3. Complete the following: (i) EG (ii) HF at right angles (iii) 50 0 (iv) 4 (v) 4 (vi) 66 0 (vii) 12 (viii) 90 0 4. Given ABCD is a Rectangle, BOC = 44 0 To find : OAD Sol: BOC = AOD = 44 0 (vertically opposite angles are equal) OAD = ODA => 44 0 + 2 OAD = 180 0 (Angle sum of a Triangle) 2 OAD = 180 0-44 0 = 136 0 OAD = 136 / 2 = 68 0 Ans 5. Given ABCD is a Rhombus, ABC = 56 0 = OBA To find : CAD Sol: BOA = AOC = COD = DOB=90 0 (Diagonals of a Rhombus bisect at Right angle) In BOA, BOA + OAB + OBA = 180 0 90 0 + 56 0 + OAB = 180 0 ( Angle sum of a Triangle) OAB = 180 0 146 0 = 34 0 OAB = CAD = 34 0 (Diagonals of a Rhombus bisect interior angles) 6. Given ABCD is a Rhombus, DAC = 50 0 To find : (a) ACD, (b) CAB (c) ABC (a) DAC = ACD = 50 0 (b) CAB = 50 0 (c) ABC = 80 0 7. A + B + C + D = 360 0 40 0 + 40 0 + 2 C = 360 0 80 0 + 2 C = 360 0 2 C = 280 0 = C = 140 0 Ans: C + D = 140 0 ( C = D)

8. (i) Rectangle Ans : x = 69 0 9. Given, BCD = 40 0 To find: (a) BDC CBD = BDC (Angles Opposite to equal sides are equal) 2 BDC + 40 0 = 180 0 (angle sum of a riangle) 2 BDC = 140 0 => BDC = 70 0 (b) ABC = 130 0 13. Diagonals of a Rhombus are 12cm and 16cm In Rhombus ABCD, AC = 12cm and BD = 16 cm AO = 6cm and BO = 8cm AOB = 90 0 Using Pythagorous Theorem AB 2 = AO 2 + BO 2 AB 2 = 6 2 + 8 2 AB 2 = 36 + 64 = 100 AB = 100 = 10 In Rhombus all its sides are equal length AB = BC = CD = AD = 10cm *** Refer Class work *** 1 (i) Triangular Prism (ii) Faces 5, Edges 9, Vertices 6 3. (i) Tetrahedron (ii) Cylinder (iii) cube, cuboid (iv) Square pyramid and Rectangular pyramid (v) regular octahedron (vi) Hexagonal prism 4. (i) square (ii) Triangle and Rectangle (iii). Triangle (iv) Circle and Rectangle 5. (i) 8 Faces, 8 Edges, 12 vertices 14. CONSTRUCTION OF QUADRILATERALS 15. VISUALISING SOLID SHAPES Exercise 15 8. (i) Triangular Pyramid (ii) Square Pyramid (iii). Hexagonal Pyramid. MCQs 9. Ans (d) 10. Ans (c) 20