Question

Find the value of (The following questions are from the chapter Expansions): i) mn if m+n= 8 and m-n= 2 ii) p^2 + q^2 if p-q = 6 and p+q = 14
iii) x- 1/x and x^2- 1/x^2 if x+1/x = √5

NOTE- I want the answers with step-by-step Explanation

Answer 1

Answer:

a) m+n= 8.....(i)

m-n = 2......(ii)

adding eq i & ii

2m=10

m=5

putting the value of m in eq i

5+n=8

n= 8-5

n =3

Now , mn= 5×3= 15 Ans

b) p+q=14.....(i)

p-q=6........(ii)

Adding eq i & ii

2p = 20

p =10

putting the value of p in eq i

10+q=14

q=4

Now, p²+q²= 10²+4²

= 100+16

=116 Ans..

Answer 2

Answer:

1. 15 2. 116 3. (X-1/X)=1 & (X²-1/X²)=✓5

Step-by-step explanation:

1. Given that m+n=8.........(1)

m-n=2..........(2)

eq. (1)+(2)

m+n+m-n=8+2

2m=10

m=10/2

m=5...........(3)

from eq.(3) put the value of m in eq.(1)

5+n=8

n=8-5

n=3

m.n=5*3

=15

2. Given that p+q=14.........(1)

p-q=6..........(2)

eq. (1)+(2)

p+q+p-q=14+6

2p=20

p=20/2

p=10...........(3)

from eq.(3) put the value of m in eq.(1)

10+q=14

q=14-10

q=4

p²+q²= 10²+4²

= 100+16

=116

3. Given that. X+1/X=✓5..........(1)

so. formula (a-b)²=(a+b)²-4ab

so. (X-1/X)=✓[(X+1/X)²-4X(1/X)]

put the value from eq (1)

so (X-1/X)=✓[(✓5)²-4*1]

(X-1/X)=✓(5-1)

(X-1/X)=✓1

(X-1/X)=1......................(2)......ANS

Square eq(1)

(X+1/X)²=(✓5)²

X²+1/X²+2*X*(1/X)=5

X²+1/X²+2=5

X²+1/X²=5-2

X²+1/X²=3.................(3)

so(X²-1/X²)=✓[(X²+1/X²)²-4X²(1/X²)]

(X²-1/X²)=✓[(3)²-4]

(X²-1/X²)=✓(9-4)

(X²-1/X²)=✓5.................ANS