Question

Please tell urgent question 2 maths

Answer 1

in 1st case

a and b both are 2

in 2nd case a is 5 and b is 6

Answer 2

Answer:

1. The value of a = 2 and b = -1
2. The value of a = 2 and b = -⅚.

Step-by-step explanation:

Question:-

If a and b are rational numbers, find the value of a and b in the following:

$\tt{(i) \: \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1} } = a + b \sqrt{3}$

$\tt{(ii) \: \frac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } } = a - b \sqrt{6}$

Solution:-

Let's solve the problem

We have,

$\tt{(i) \: \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1} } = a + b \sqrt{3}$

The denominator is √3+1. Multiplying the numerator and denominator by √3-1, we get

$⟹ \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1} \times \frac{ \sqrt{3} - 1}{ \sqrt{3} - 1}$

$⟹ \frac{( \sqrt{3} - 1)( \sqrt{3} - 1)}{( \sqrt{3} + 1)( \sqrt{3} - 1) }$

⬤ Applying Algebraic Identity

1. (a-b)(a-b) = (a-b)² = a²-b²+2ab to the numerator and
2. (a+b)(a-b) = a² - b² to the denominator.

We get,

$⟹ \frac{( \sqrt{3} - 1 {)}^{2} }{( \sqrt{3} {)}^{2} - (1 {)}^{2} }$

$⟹ \frac{( \sqrt{3} {)}^{2} + (1 {)}^{2} - 2 \times \sqrt{3} \times 1 }{3 - 1}$

$⟹ \frac{3 + 1 - 2 \times \sqrt{3} \times 1}{2}$

$⟹ \frac{ {4}- 2 \sqrt{3} }{ {2}}$

$⟹2 - \sqrt{3}$

$∴ \: 2 - \sqrt{3} = a - b \sqrt{3}$

On, Comparing the value of

$⟹a = 2 \: and \: b \: = - 1 \: \: Ans.$

________________________

$\tt{(ii) \: \frac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } } = a - b \sqrt{6}$

The denominator is 3√2-2√3. Multiplying the numerator and denomination by 3√2+2√3, we get

$⟹ \frac{ \sqrt{2} + \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} } \times \frac{3 \sqrt{2} - 2 \sqrt{3} }{3 \sqrt{2} - 2 \sqrt{3} }$

$⟹ \frac{( \sqrt{2} + \sqrt{3})(3 \sqrt{2} + 2 \sqrt{3} ) }{(3 \sqrt{2} - 2 \sqrt{3} )(3 \sqrt{2} + 2 \sqrt{3} }$

⬤ Applying Algebraic Identity

(a-b)(a+b) = a² - b² to the denominator

We get,

$⟹ \frac{3 \sqrt{2 \times 2} + 2 \sqrt{3 \times 2} + 3 \sqrt{2 \times 3} + 2 \sqrt{3 \times 3} }{(3 \sqrt{2} {)}^{2} - (2 \sqrt{3} {)}^{2} }$

$⟹ \frac{3 \times 2 + 2 \sqrt{6} + 3 \sqrt{6} + 2 \times 3}{18 - 12}$

$⟹ \frac{12 + 5 \sqrt{6} }{6}$

$⟹2 + \frac{5}{6} \sqrt{6}$

$∴ \: \: a - b \sqrt{6} = 2 + \frac{5}{6} \sqrt{6}$

On comparing the value of

$a = 2 \: \: and \: \: b \: = - \frac{5}{6} \: \: \: \: Ans.$

:)