5. tan^-1x का x के सापेक्ष अवकलन है ................. [x, 11+x^2,1] 6. 6 वस्तुओं में से 3 चुनने के तरीके होंगे ................. [^6C_3,^6P_3,^6C_2] 7. सम्मिश्र संख्या 1+i का आयाम ................. होगा। [pi/4,pi/2,0] 8.

Step-by-step solution

1. 1

   Given Questions

   Four separate math questions on differentiation, combinations, complex numbers, and limits.
2. 2

   Key Formulas

   1. $\frac{d}{dx}(\\tan^{-1}x) = \frac{1}{1+x^2}$
   2. Choosing $r$ from $n$: $^{n}C_r = \frac{n!}{r!(n-r)!}$
   3. Argument of $z=x+iy$: $\arg(z) = \\tan^{-1}(\frac{y}{x})$
   4. $\lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-a} = na^{n-1}$
3. 3

   Q5: Differentiation

   The derivative of $\\tan^{-1}x$ with respect to $x$ is $\frac{1}{1+x^2}$.

   $$\frac{d}{dx}(\\tan^{-1}x) = \frac{1}{1+x^2}$$
4. 4

   Q6: Combinations

   Ways to choose 3 items from 6 is given by $^{6}C_3$.

   $$^{6}C_3$$
5. 5

   Q7: Complex Number Argument

   For $z=1+i$, $x=1, y=1$. $\arg(z) = \\tan^{-1}(\frac{1}{1}) = \\tan^{-1}(1) = \frac{\pi}{4}$.

   $$\arg(1+i) = \\tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}$$
6. 6

   Q8: Limit Evaluation

   The limit $\lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-a}$ is a standard formula, equal to $na^{n-1}$.

   $$\lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x-a} = na^{n-1}$$
7. 7

   Final Answers

   5. $\frac{1}{1+x^{2}}$
   6. $^{6}C_{3}$
   7. $\frac{\pi}{4}$
   8. $na^{n-1}$