Please answer the questions
Answers
Answer :-
(1)
a²+b² = 241 ;
a - b = 11 ..(1)
(a-b) ² = 121
2a - 3b = ?
(a+b)²+(a-b)² = 2(a²+b²)
(a+b)² + 121 = 2(241)
(a+b)² = 482 - 121
(a+b)² = 361
a+b = 19 ..(2)
By elimination,
Frm (1) & (2) ,
2a = 30
a = 15 & b = 4
To find:-
2a - 3b = 2(15) - 3(4)
= 30 - 12
= 18 •●✿●•
(2)
Given :-
x³ - y³ = 784 ; x - y = 4 ..(1)
(x - y)³ = 64
x³ - y³- 3xy(x-y) = 64
784 - 3xy(4) = 64
12xy = 720
xy = 60
x = 60/y in (1)
60/y - y = 4
y²+4y-60 = 0
y²-6y+10y+60 = 0
(y-6)(y+10) = 0
y = 6 (or) -10
y = 6; x = 4-y = 4-6 = -2
y = -10; x = 4-y = 4+10 = 14
y = 6; x = -2 ,
2x - 3y = 2(-2) - 3(6)
= -4-18
= -22
y = -10 ; x = 14 ,
2x - 3y = 2(14) - 3(-10)
= 28+30
= 58 •●✿●•
(3)
x³ + y³ = 126 ; x²-xy+y² = 21
x³ + y³ = 126
(x+y)(x²-xy+y²) = 126
(x+y) (21) = 126
x+y = 126/21
x+y = 6 ..(1)
(x+y)² = 36
x²+y²+2xy = 36
x²+y²-xy+3xy = 36 (add and subtract 3xy)
21+3xy = 36
3xy = 15
xy = 5
(x-y) ² = x²+y²-2xy = (x²+y²-xy) -xy
= 21-5
(x-y) ² = 16
x-y = 4 ..(2)
By elimination, 2x = 10
x = 5 & y = 1
2x-3y = 2(5) -3(1) = 10-3 = 7
2x-3y = 7 •●✿●•
(4) x^4+x²y²+y^4 = 7371 ; x²-xy+y² = 63
(x²+y²)² = (63+xy)²
x^4+y^4+2x²y² = 3969+x²y²+126xy
7371+x²y² = 3969+x²y²+126xy
126xy = 7371-3969
xy = 3402 /126
xy = 27
(x+y)² = x²+y²+2xy
(add and subtract 3xy)
= (x²+y²-xy)+3xy
= 63 + 3(27)
= 63+81
(x+y)² = 144
x+y = 12 ..(1)
(x-y) ² = (x²+y²-xy)-xy
= 63-27
(x-y) ² = 36
x-y = 6.. (2)
By elimination, 2x = 18
x = 9 & y = 3
2x-3y = 2(9)-3(3)
= 18 - 9
= 9 •●✿●•
(5) x - y = 6 ; x²+y²+5xy = 316
x - y = 6 ..(1)
(x-y) ² = 36
x²+y²-2xy = 36
x²+y²+5xy = 316
(x²+y²-2xy)+7xy = 316 (add and sub 7xy)
36+7xy = 316
7xy = 280
xy = 40
x²+y²+5xy = 316
x²+y²+5(40) = 316
x²+y² = 316 - 200
x²+y² = 116
(x+y)² = x²+y²+2xy = 116+2(40)
(x+y)² = 196
x+y = 14 ..(2)
By elimination, 2x = 20
x = 10 & y = 4
2x-3y = 2(10)-3(4) = 20 - 12 = 8
2x-3y = 8 •●✿●•