If x = 2 + 3 , then find the value of x + 1 x 2 .

Given, x = 2 + 3 We need to find the value of x + 1 x 2 . Now, 1 x = 1 2 + 3 = 1 2 + 3 × 2 - 3 2 - 3 Applying the identity, a + b a - b = a 2 - b 2 ⇒ 1 x = 2 - 3 4 - 3 = 2 - 3 1 ⇒ 1 x = 2 - 3 So, x + 1 x 2 = 2 + 3 + 2 - 3 2 ⇒ x + 1 x 2 = 4 2 = 16 Hence, the required answer is 16 .

If x = 2 + 3 , then find the value of x + 1 x 2 .

Given, x = 2 + 3 We need to find the value of x + 1 x 2 . Now, 1 x = 1 2 + 3 = 1 2 + 3 × 2 - 3 2 - 3 Applying the identity, a + b a - b = a 2 - b 2 ⇒ 1 x = 2 - 3 4 - 3 = 2 - 3 1 ⇒ 1 x = 2 - 3 So, x + 1 x 2 = 2 + 3 + 2 - 3 2 ⇒ x + 1 x 2 = 4 2 = 16 Hence, the required answer is 16 .

Rationalise the denominator of 2 3 - 5 .

If the denominator of an expression contains a term with a square root (or a number under a radical sign), then the process of converting it to an equivalent expression whose denominator is a rational number, is called rationalising the denominator. In 2 3 - 5 , the denominator contains square root. Thus it needs to be rationalised. 2 3 - 5 = 2 3 - 5 × 3 + 5 3 + 5 = 2 3 + 5 3 2 - 5 2 = 2 3 + 5 3 - 5 = 2 3 + 5 - 2 = - 3 + 5 ∴ 2 3 - 5 = - 3 + 5

If x - a is a factor of 4 x 2 - m x - n a , prove that a = m + n 4 .

Given, x - a is the factor of the polynomial 4 x 2 - m x - n a We know that, According to the factor theorem, if f ( x ) is a polynomial of degree n ≥ 1 and ' a ' is any real number, then, x - a is a factor of f ( x ) , if f ( a ) = 0 . Substituting x = a , ⇒ f a = 0 ⇒ 4 a 2 - m a - n a = 0 ⇒ 4 a 2 - m a - n a = 0 ⇒ 4 a 2 = a m + n ⇒ a = m + n 4 Hence, proved.

Find the value of x , if 6 5 x 5 6 2 x = 125 216 .

Given: 6 5 x 5 6 2 x = 125 216 ⇒ 5 6 - x 5 6 2 x = 5 3 6 3 Using a b m = b a - m ⇒ 5 6 - x + 2 x = 5 6 3 using a m × a n = a m + n and a x b x = a b x ⇒ 5 6 x = 5 6 3 On comparing, we get x = 3 Hence, the required value of x is 3 .