MENU

RD Chapter 4 Algebraic Identities Ex 4.3 Solutions

Question - 11 : -
Evaluate each of the following:
(i)  (103)3
(ii) (98)3
(iii) (9.9)3
(iv) (10.4)3
(v) (598)3
(vi) (99)3

Answer - 11 : -

We know that (a + bf = a3 + b3 +3ab(a + b) and (a – b)3= a3 – b3 – 3ab(a – b)
Therefore,
(i)  (103)3 = (100 + 3)3
= (100)3 + (3)3 + 3 x 100 x 3(100 +3)    {∵ (a +b)3 = a3 + b3 + 3ab(a + b)}
= 1000000 + 27 + 900 x 103
= 1000000 + 27 + 92700
= 1092727
(ii) (98)3 = (100 – 2)3
= (100)3 – (2)3 – 3 x 100 x 2(100 – 2)
= 1000000 – 8 – 600 x 98
= 1000000 – 8 – 58800
= 1000000-58808
= 941192
(iii) (9.9)3 = (10 – 0.1)3
= (10)3 – (0.1)3 – 3 X 10 X 0.1(10 – 0.1)
= 1000 – 0.001 – 3 x 9.9
= 1000 – 0.001 – 29.7
= 1000 – 29.701
= 970.299
(iv) (10.4)3 = (10 + 0.4)3
= (10)3 + (0.4)3 + 3 x 10 x 0.4(10 + 0.4)
= 1000 + 0.064 + 12(10.4)
= 1000 + 0.064 + 124.8 = 1124.864
(v) (598)3 = (600 – 2)3
= (600)3 – (2)3 – 3 x 600 x 2 x (600 – 2)
= 216000000 – 8 – 3600 x 598
= 216000000 – 8 – 2152800
= 216000000 – 2152808
= 213847192
(vi) (99)3 = (100 – 1)3
= (100)3 – (1)3 – 3 x 100 x 1 x (100 – 1)
= 1000000 – 1 – 300 x 99
= 1000000 – 1 – 29700
= 1000000 – 29701
= 970299

Question - 12 : -
Evaluate each of the following:
(i)  1113 – 893
(ii) 463 + 343
(iii) 1043 + 963
(iv) 933 – 1073

Answer - 12 : -

We know that a3 + b3 = (a + bf –3ab(a + b) and a3 – b3 = (a – bf + 3 ab(a – b)
(i) 1113 – 893
= (111 – 89)3 + 3 x ill x 89(111 – 89)
= (22)3 + 3 x 111 x 89 x 22
= 10648 + 652014 = 662662
(OR)
(a + b)3 – (a – b)3 = 2(b3 + 3a2b)
= 1113 – 893 = (100 + 11)3 –(100 – 11)3
= 2(113 + 3 x 1002 x 11]
= 2(1331 + 330000]
= 331331 x 2 = 662662
(a + b)3 + (a- b)3 = 2(b3 + 3ab2)
(ii) 463 + 343 = (40 + 6)3 +(40 – 6)3
= 2[(40)3 + 3 x 40 x 62]
= 2[64000 + 3 x 40 x 36]
= 2[64000 + 4320]
= 2 x 68320 = 136640
(iii) 1043 + 963 = (100 + 4)3 +(100 – 96)3
= 2 [a3 + 3 ab2]
= 2[(100)3 + 3 x 100 x (4)2]
= 2[ 1000000 + 300 x 16]
= 2[ 1000000 + 4800]
= 1004800 x 2 = 2009600
(iv) 933 – 1073 = -[(107)3 –(93)3]
= -[(100 + If – (100 – 7)3]
= -2[b3 + 3a2b)]
= -2[(7)3 + 3(100)2 x 7]
= -2(343 + 3 x 10000 x 7]
= -2[343 + 210000]
= -2[210343] = -420686

Question - 13 : -

Answer - 13 : -


Question - 14 : -
Find the value of 27X3 + 8y3 if
(i) 3x + 2y = 14 and xy = 8
(ii) 3x + 2y = 20 and xy = 149

Answer - 14 : -


Question - 15 : - Find the value of 64x3 – 125z3, if 4x – 5z = 16 and xz = 12.

Answer - 15 : -

4x – 5z = 16, xz = 12
Cubing both sides,
(4x – 5z)3 = (16)3
⇒ (4x)3 –(5y)3 – 3 x 4x x 5z(4x – 5z) = 4096
⇒ 64x3 –125z3 – 3 x 4 x 5 x xz(4x – 5z) = 4096
⇒ 64x3 – 125z3 – 60 x 12 x 16 = 4096
⇒ 64x3 –125z3 – 11520 = 4096
⇒ 64x3 – 125z3 = 4096 + 11520 = 15616

Question - 16 : -

Answer - 16 : -


Question - 17 : - Simplify each of the following:

Answer - 17 : -


Question - 18 : -

Answer - 18 : -


Question - 19 : -

Answer - 19 : -


Free - Previous Years Question Papers
Any questions? Ask us!
NodeJS Interview Questions Answers
ASP.Net Interview Questions Answers
Interview Questions Answers
×