Let p(N) = 8 (N⁵ - N³ + N)
= 4 x 2 x N (N⁴ - N² + 1)
and q(N) = 28 (N⁶ + 1)
= 7 x 4 [( N²)³ + (1)³]
= 4 x 7 (N² + 1) ( N⁴ - N² + 1)

Correct Option: A

Let p(N) = 8 (N⁵ - N³ + N)
= 4 x 2 x N (N⁴ - N² + 1)
and q(N) = 28 (N⁶ + 1)
= 7 x 4 [( N²)³ + (1)³]
= 4 x 7 (N² + 1) ( N⁴ - N² + 1)
∴ HCF of p(N) and q (N) = 4 (N⁴ - N² + 1)

Let f (x) = (x⁴ - y⁴)
= (x² - y²) (x² + y²)
= (x - y) (x + y) (x² + y²)
and g(x) = (x⁶ - y⁶)
= (x³)² - (y³)²
= (x³ + y³) (x³ - y³)
= (x + y) (x² - xy + y²) (x - y)(x² + xy + y²)
= (x - y) (x + y) (x² - xy + y²)
(x² + xy + y²)
∴ HCF of [f(x), g(x)] = (x - y) (x + y)
= x² - y²

Correct Option: A

Let f (x) = (x⁴ - y⁴)
= (x² - y²) (x² + y²)
= (x - y) (x + y) (x² + y²)
and g(x) = (x⁶ - y⁶)
= (x³)² - (y³)²
= (x³ + y³) (x³ - y³)
= (x + y) (x² - xy + y²) (x - y)(x² + xy + y²)
= (x - y) (x + y) (x² - xy + y²)
(x² + xy + y²)
∴ HCF of [f(x), g(x)] = (x - y) (x + y)
= x² - y²

Let f(x) = x³ - y² - 2x = x( x² - x- 2)
= x( x² - 2x + x - 2)
= x {x (x - 2) + 1 (x -2)}
= x (x +1) (x - 2)
and g(x) = x³ + x²
= x² (x +1) = x. x (x +1)
∴ LCM of [ f(x), g(x)] = x (x +1) . x . (x - 2)

Correct Option: C

Let f(x) = x³ - y² - 2x = x( x² - x- 2)
= x( x² - 2x + x - 2)
= x {x (x - 2) + 1 (x -2)}
= x (x +1) (x - 2)
and g(x) = x³ + x²
= x² (x +1) = x. x (x +1)
∴ LCM of [ f(x), g(x)] = x (x +1) . x . (x - 2)
= x²(x + 1)(x - 2)
= x²(x² - x - 2)
= x⁴ - x³ - 2x²

a² b⁴ + 2 a² b² = a² b² ( b² +2).......................(i)
and (ab)⁷ - 4 a² b⁹ = a⁷ b⁷ - 4 a² b⁹ = a² b²(a⁵ b⁵ - 4b⁷)................(ii)

from Eqs. (i) and (ii)
HCF = a² b²

Correct Option: C

a² b⁴ + 2 a² b² = a² b² ( b² +2).......................(i)
and (ab)⁷ - 4 a² b⁹ = a⁷ b⁷ - 4 a² b⁹ = a² b²(a⁵ b⁵ - 4b⁷)................(ii)

from Eqs. (i) and (ii)
HCF = a² b²

Given that
x = 1305, y = 4665, z = 6905

Then ,
|x - y| = |1305 - 4665| = 3360
|y - z] = |4665 - 6905| = 2240
|z - x| =|6905 -1305| = 5600

∴ Required number = HCF of 3360, 2240 and 5600 = 1120

On dividing 1305 by 1120, remainder is 185.
On dividing 4665 by 1120, remainder is 185.
On dividing 6905 by 1120, remainder is 185.
∴ Common remainder = 185

Correct Option: C

Given that
x = 1305, y = 4665, z = 6905

Then ,
|x - y| = |1305 - 4665| = 3360
|y - z] = |4665 - 6905| = 2240
|z - x| =|6905 -1305| = 5600

∴ Required number = HCF of 3360, 2240 and 5600 = 1120

On dividing 1305 by 1120, remainder is 185.
On dividing 4665 by 1120, remainder is 185.
On dividing 6905 by 1120, remainder is 185.
∴ Common remainder = 185