a) A committee is to be chosen from 'a' boys and 6 girls and is to consist 2 boys and 3 girls. If 120 committees are formed, what is the number of boys represented by 'a' ? [2] b) The square roots…

a) Number of boys represented by 'a'

Given:

• Number of girls = 6
• Committee composition = 2 boys and 3 girls
• Total number of possible committees = 120

Step 1: Calculate the number of ways to choose 3 girls from 6 girls. The number of ways to choose 3 girls from 6 is given by the combination formula: $${}^6{C}_3=\frac{6!}{3!(6-3)!}=\frac{6\times 5\times 4}{3\times 2\times 1}=20$$

Step 2: Let the number of boys be 'a'. Calculate the number of ways to choose 2 boys from 'a' boys. The number of ways to choose 2 boys from 'a' boys is given by: $${}^a{C}_2=\frac{a!}{2!(a-2)!}=\frac{a(a-1)}{2}$$

Step 3: Total number of committees is the product of the number of ways to choose boys and girls. $${}^a{C}_2\times {}^6{C}_3=120$$ $$\frac{a(a-1)}{2}\times 20=120$$

Step 4: Solve for 'a'. $$\frac{a(a-1)}{2}\times 20=120$$ $$a(a-1)\times 10=120$$ $$a(a-1)=12$$ $$a^2-a-12=0$$

Step 5: Solve the quadratic equation. $$a=\frac{1\pm \sqrt{1+48}}{2}=\frac{1\pm \sqrt{49}}{2}=\frac{1\pm 7}{2}$$ $$a=\frac{1+7}{2}=4\text{(since the number of boys cannot be negative)}$$

Final Answer: The number of boys represented by 'a' is 4.

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b) Complex number in polar form

Given:

• Square roots of a complex number are $(\sqrt{3}+i)$ and $(-\sqrt{3}-i)$.

Step 1: Let the complex number be $z=x+iy$. The square roots of $z$ are given by: $$\sqrt{z}=\pm (\sqrt{3}+i)$$

Step 2: Square the square root to find $z$. $$z=(\sqrt{3}+i{)}^2=(\sqrt{3}{)}^2+2\cdot \sqrt{3}\cdot i+(i{)}^2=3+2\sqrt{3}i-1=2+2\sqrt{3}i$$

Step 3: Express $z$ in polar form. The polar form of a complex number $z=x+iy$ is given by: $$z=r(\cos \theta +i\sin \theta )$$ where $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$.

Step 4: Calculate the modulus $r$. $$r=\sqrt{2^2+(2\sqrt{3}{)}^2}=\sqrt{4+12}=\sqrt{16}=4$$

Step 5: Calculate the argument $\theta$. $$\theta ={\tan }^{-1}\left(\frac{2\sqrt{3}}{2}\right)={\tan }^{-1}(\sqrt{3})=60^{\circ }=\frac{\pi }{3}\text{ radians}$$

Step 6: Write the polar form. $$z=4\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$$

Final Answer: The complex number in polar form is $4\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$.