Question

solve the following in the attachment given below

Answer 1

Q12.

Given :

• $\sf\:\frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}$

‎

To find :

• The value of a and b.

‎

Concept :

For calculating the value of a and b , we need to rationalize the ratio in the LHS of the given equation. Doing so and comparing LHS and RHS of the equation we would get our required value i.e., the value of a and b.

* Rationalisation is a process in which we remove the square root from the denominator of the given fraction. In order to do so we need to multiply the denominator with its conjugate.

* Conjugate - let's see by an example.

• Conjugate of 3-√5 is 3+√5

Also we need to keep in our mind some algebraic identities while proceeding with rationalisation. This identities are :

◛ $\sf\:(a+b) ^{2}=a^{2}+2ab+b^{2}$

◛ $\sf\:(a-b) ^{2}=a^{2}-2ab+b^{2}$

◛ $\sf\:a^{2}-b^{2}=(a+b)(a-b)$

‎

‎

Solution :

$\sf\:\frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}$

Rationalising the denominator

$\sf\to\:\frac{4+3\sqrt{5}}{4-3\sqrt{5}} \times \frac{4+3\sqrt{5}}{4+3\sqrt{5}}=a+b\sqrt{5}$

$\sf\to\:\frac{(4+3\sqrt{5})^{2}}{(4)^{2}-(3\sqrt{5})^{2}}=a+b\sqrt{5}$

Now using the identities

$\sf\to\:\frac{(4)^{2}+2 \times 4 \times 3\sqrt{5} + (3\sqrt{5})^{2}}{16-9 \times 5}=a+b\sqrt{5}$

$\sf\to\:\frac{16+8 \times 3\sqrt{5} + 9 \times 5}{16-45}=a+b\sqrt{5}$

$\sf\to\:\frac{16+24\sqrt{5} + 45}{16-45}=a+b\sqrt{5}$

$\sf\to\:\frac{61+24\sqrt{5}}{-29}=a+b\sqrt{5}$

$\sf\to\:\frac{61}{-29}+\frac{24\sqrt{5}}{-29}=a+b\sqrt{5}$

Comparing LHS and RHS we get,

⠀⠀⠀⠀⠀⠀❒ $\small{\mathfrak{a=\frac{61}{-29} ~oR~\frac{-61}{29}}}$

⠀⠀⠀⠀⠀⠀❒ $\small{\mathfrak{b=\frac{24}{-29} ~oR~\frac{-24}{29}}}$

════════════════════════

Q13.

Given :

• $\sf\:a~=~2-\sqrt{3}$

‎

To find :

The values of -

• $\sf\:(a-\frac{1}{a})$
• $\sf\:(a+\frac{1}{a})$
• $\sf\:(a^{2}-\frac{1}{a^{2}})$

‎

Solution :

We know - $\sf\:a~=~2-\sqrt{3}$

Henceforth,

• $\sf\:\frac{1}{a}~=~2+\sqrt{3}$

‎

Now let's calculate the values of given expressions one by one.

⁍ $\sf\:(a-\frac{1}{a})$

$\sf\rightarrow\:(2-\sqrt{3})-(2+\sqrt{3})$

$\sf\rightarrow\:2-\sqrt{3}-2-\sqrt{3}$

$\sf\rightarrow\:\cancel{2}-\cancel{2}-\sqrt{3}-\sqrt{3}$

$\small{\underline{\boxed{\mathrm{\rightarrow\:-2\sqrt{3}}}}}$

╌╌╌╌╌╌‎

⁍‎ $\sf\:(a+\frac{1}{a})$

$\sf\rightarrow\:(2-\sqrt{3})+(2+\sqrt{3})$

$\sf\rightarrow\:2-\sqrt{3}+2+\sqrt{3}$

$\sf\rightarrow\:2+2-\cancel{\sqrt{3}}+\cancel{\sqrt{3}}$

$\small{\underline{\boxed{\mathrm{\rightarrow\:4}}}}$

╌╌╌╌╌╌

⁍ $\sf\:a^{2}+\frac{1}{a^{2}}$

$\sf\rightarrow\:(2-\sqrt{3})^{2}+(2+\sqrt{3}^{2}$

$\sf\rightarrow\:\big[(2)^{2}-2 \times 2 \times \sqrt{3}+(\sqrt{3})^{2}\big]+\big[(2)^{2}+2 \times 2 \times \sqrt{3}+(\sqrt{3})^{2}\big]$

$\sf\rightarrow\:4-4\sqrt{3}+3+4+4\sqrt{3}+3$

$\sf\rightarrow\:4+3+4+3-\cancel{4\sqrt{3}}+\cancel{4\sqrt{3}}$

$\sf\rightarrow\:4+3+4+3$

$\small{\underline{\boxed{\mathrm{\rightarrow\:14}}}}$

════════════════════════

Scroll left to right to view your answer properly :D~‎

‎

‎