Let the two numbers be P and Q.
According to given question,
Sum of two numbers = 24
∴ P + Q = 24
Product of two numbers = 143
and, PQ = 143
As we know the formula,
∴ P ² + Q² = (P + Q)² – 2PQ

Correct Option: C

Let the two numbers be P and Q.
According to given question,
Sum of two numbers = 24
∴ P + Q = 24 ...................... (1)
Product of two numbers = 143
and, PQ = 143 .......................... (2)
As we know the formula,
∴ P ² + Q² = (P + Q)² – 2PQ
Put the value from the equation (1) and (2), We will get
∴ P ² + Q² = (24)² – 2 × 143
∴ P ² + Q² = 576 – 286 = 290

Let two alternate odd integers odd integers be (2x+1) and (2x+5).
Then according to the question,
(2x + 1) (2x + 5) = 3(2x + 1) + 12
⇒ (2x + 1)(2x + 5 - 3) = 12
⇒ 2x² + 3x - 5 = 0

Correct Option: B

Let two alternate odd integers odd integers be (2x+1) and (2x+5).
Then according to the question,
(2x + 1) (2x + 5) = 3(2x + 1) + 12
⇒ (2x + 1) (2x + 5 - 3) = 12
⇒ 2x² + 3x - 5 = 0
On solving this quadratic equation,we get
x = 1 and x = -5/2
x = -5/2 is not a integer ∴ x = 1
Then, larger integer = 2x + 5 = 2 x 1 + 5 = 7

x - y = 0.9 ...(i)
and 11(x + y)^-1=2
⇒ 11/ (x + y) = 2
⇒ 2(x + y) =11

Correct Option: A

x - y = 0.9 ...(i)
and 11(x + y)^-1=2
⇒ 11/ (x + y) = 2
⇒ 2(x + y) =11
⇒ x + y = 11/2 ...(ii)
On solving Eqs.(i) and (ii),we get
x = 3.2
and y = 2.3

x = 1 + √2
∴ x⁴ - 4x³ + 4x² = x²(x² - 4x + 4)
= x²(x - 2)²
= (1 + √2)²(1 + √2 - 2)²

Correct Option: C

x = 1 + √2
∴ x⁴ - 4x³ + 4x² = x²(x² - 4x + 4)
= x²(x - 2)²
= (1 + √2)²(1 + √2 - 2)²
=(√2 + 1)² (√2 - 1)²
=[(√2)² - (1)²]²
=(2 - 1)² =1