Q.1 $If 𝑡𝑎𝑛 (𝐴 + 𝐵) = \surd 3 and 𝑡𝑎𝑛 (𝐴 - 𝐵) = \frac{1}{\sqrt{3}}; 0° < (𝐴 + 𝐵) \le 90° ; 𝐴 > 𝐵, then 𝑓𝑖𝑛𝑑 𝐴 𝑎𝑛𝑑 𝐵.$

 

(OR)

 

Q.2 Find the number of plates 1.5 cm in diameter and 0.2 cm thick that can be fitted completely inside a right circular cylinder of height 10 cm and diameter 4.5 cm.

Given that, 

$𝑡𝑎𝑛 (𝐴 + 𝐵) = \surd 3$

$𝑡𝑎𝑛 (𝐴 + 𝐵) = 𝑡𝑎𝑛 60⁰$

$\Rightarrow 𝐴 + 𝐵 = 60⁰ \dots (𝑖)$

$𝑡𝑎𝑛 (𝐴 - 𝐵) = \frac{1 }{\sqrt{3}}$

$𝑡𝑎𝑛 (𝐴 - 𝐵) = 𝑡𝑎𝑛 30⁰$

$\Rightarrow 𝐴 - 𝐵 = 30⁰ \dots (𝑖𝑖)$

Adding both equations we get,  

$$A+B+A-B=60⁰+30⁰$$                      

$$2A=90⁰$$

$$A=45⁰$$

Substitute $A = 45 ⁰$in $ii$we get, 

$$45⁰-B=30⁰$$

$$B=45⁰-30⁰$$

$B=15⁰$

The values of A and B are 45⁰ and 15⁰ respectively. 

(OR)

Given that,  

Plates are fitted completely inside a right circular cylinder 

Diameter of plate$= 1.5 cm$

Thickness of plate$= 0.2 cm$

Height of right circular cylinder $= 10 cm$

Diameter of right circular cylinder$= 4.5 cm$

Radius of right circular cylinder$=2.25 cm$

Volume of right circular cylinder $= \pi r²h$

$=\pi \times 2.25\times 2.25\times 10$

$= \pi \times 0. 5625 \times 0. 2$

$=\frac{405\pi }{8} cm³$

Volume of plate $= \pi r²h$

$=\pi \times 0.75\times 0.75\times 0.2$

$=\pi \times 0.5625\times 0.2$

$=\frac{ 9\pi }{80} 𝑐𝑚³$

$Number of plates =\frac{Volume of right circular cylinder }{Volume of plate }$

$= \frac{{\displaystyle \frac{405\pi }{8 𝑐𝑚³}} }{{\displaystyle \frac{9\pi }{80 𝑐𝑚³}}}$

$= \frac{ 405 \pi }{8} \times \frac{ 80 }{ 9 \pi }$

$= 45 \times 10$

$= 450 plates$

Hence, the number of plates are 450.