Let there be N members, other than father .
Father's share = 1/4, other's share = 3/4.
Each of other's share = 3/4N

Correct Option: C

Let there be N members, other than father .
Father's share = 1/4, other's share = 3/4.
Each of other's share = 3/4N
∵ 3 x 3/4N = 1/4
∴ N = 9
Hence, the total number of members = N + 1 = 10

Suppose Ravi earns Rs. N in each of the 11 months.
Then earning in January = Rs. 2 x N.
∴ Total annual income = (11 x N + 2 x N) = Rs. 13 x N

Correct Option: A

Suppose Ravi earns Rs. N in each of the 11 months.
Then earning in January = Rs. 2 x N.
∴ Total annual income = (11 x N + 2 x N) = Rs. 13 x N
Part of total earning in January = (2 x N) / (13 x N) = 2/13

(?)² = √13² + 28 / 4 - (3)³ + 107
= √169 + 7 - 27 + 107

Correct Option: D

(?)² = √13² + 28 / 4 - (3)³ + 107
= √169 + 7 - 27 + 107
= √256
= √16
= 4

We know that, (a + b)² = a² + b² + 2ab
= 234 + 2 x 108 = 450
(a - b)² = a² + b² - 2ab
= 234 - 2 x 108 = 18
∴ (a + b)²/(a - b)² = 450/18 = 25

Correct Option: C

We know that, (a + b)² = a² + b² + 2ab
= 234 + 2 x 108 = 450
(a - b)² = a² + b² - 2ab
= 234 - 2 x 108 = 18
∴ (a + b)²/(a - b)² = 450/18 = 25
⇒ [(a + b)/(a - b)]² = 25
∴ (a + b)/(a - b) = √25 = 5

a² + 1 = a
⇒ a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³

Correct Option: B

a² + 1 = a
⇒ a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³
⇒ a⁶ + 1/a⁶ + 3a² x 1/a² (a² + 1/a²) = -1
⇒ a⁶ + 1/a⁶ + 3 x (-1) = -1
Now, a⁶ + 1/a⁶ + 1 = 3
As, a¹² + a⁶ + 1 can be written as a⁶ + 1/a⁶ + 1
∴ a¹² + a⁶ + 1 = 3