Given expression
= (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x (1 - 1/5) ...(1 - 1/m - 1) x (1 - 1/m)
= ((2 - 1)/2) x ((3 - 1)/3) ((4 - 1)/4) ((5 - 1)/5) ...((m-1 - 1)/m - 1) x ((m - 1)/m)
= 1/2 x 2/3 x 3/4 x 4/5 x ... x (m -2)/(m-1) x ((m - 1)/m

Correct Option: A

Given expression
= (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x (1 - 1/5) ...(1 - 1/m - 1) x (1 - 1/m)
= ((2 - 1)/2) x ((3 - 1)/3) ((4 - 1)/4) ((5 - 1)/5) ...((m-1 - 1)/m - 1) x ((m - 1)/m)
= (1/2) x (2/3) x (3/4) x (4/5) x ... x (m -2)/(m-1) x ((m - 1)/m
use the multiplication rule of Algebra,
= 1 x 1/m
= 1/m

We know that algebraic formula,
(x + y)³ = x³ + y³ + 3xy (x + y)
Put the value of x + y is above equation and solve.

Correct Option: B

We know that algebraic formula,
(x + y)³ = x³ + y³ + 3xy (x + y)
put the value of x + y in given equation. [ given, x + y = 1]
1 = x³ + y³ + 3xy X 1
⇒ x³ + y³ + 3xy = 1

p/q + q/p = (p² + q²)/pq
Apply the formula of algebra
a ²+ b² = (a + b)² - 2ab

p/q + q/p = ( (p + q)² - 2pq ) / pq
By substituting the pq and p + q values in given equation.

Correct Option: B

p/q + q/p = (p² + q²)/pq
Apply the formula of algebra
a ²+ b² = (a + b)² - 2ab

p/q + q/p = ( (p + q)² - 2pq ) / pq
By substituting the pq and p + q values in given equation.
p/q + q/p = ( (p + q)² - 2pq ) / pq
p/q + q/p = ((10)² - 2 x 5 ) / 5
p/q + q/p = (100 - 10 )/ 5 = 90/5 = 18
p/q + q/p = 90/5 = 18
p/q + q/p = 18

Let, (m - n) = a,
(n -r) = b
(r - m) = c,
Now a + b + c = (m - n) + (n -r) + (r - m)
⇒ a + b + c = m - n + n - r + r - m
⇒ a + b + c = 0;

Correct Option: A

Let, (m - n) = a,
(n -r) = b
(r - m) = c,
Now a + b + c = (m - n) + (n -r) + (r - m)
⇒ a + b + c = m - n + n - r + r - m
⇒ a + b + c = 0............................. (i)
As we know the Algebra formula,
a³ + b³ + c³ ? 3abc = (a+b+c) X 1/2[(a?b)²+(b?c)²+(a?c)²]
Put the value of a + b + c from equation (i).
⇒ a³ + b³ + c³ ? 3abc = 0 X 1/2[(a?b)²+(b?c)²+(a?c)²]
⇒ a³ + b³ + c³ ? 3abc = 0
∴ a³ + b³ + c³ = 3abc
∴ Given expression in question is
[ (m - n)³ + (n - r)³ + (r - m)³ ]/ 6(m - n) (n - r) (r - m)
= ( a³ + b³ + c³ ) / 6abc
= 3abc/6abc
= 1/2

Given in question
x + y = 18
By using algebraic formula ,
∵ x² + y² = (x + y)² - 2xy

Correct Option: C

Given in question,
x + y = 18
By using algebraic formula
∵ x² + y² = (x + y)² - 2xy
Put the value of x + y and xy as per given question,
⇒ x² + y² = (18)² - 2 x 72
⇒ x² + y² = 324 - 144
⇒ x² + y² = 180