Day 281: NDA Maths

5 questions (8 parts). Try each one first, then open its answer.

Question 1 (case study, 2 parts)
Differentiation. Asked once: NDA 1 2025
Let $x = \sec\theta - \cos\theta$ and $y = \sec^4\theta - \cos^4\theta$ .
71 $\left(\frac{dy}{dx}\right)^2$
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72 $\left(\dfrac{x^2+4}{y^2+4}\right)\dfrac{dy}{dx}\left[(x^2+4)\dfrac{d^2y}{dx^2}-16y\right]$
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Question 2 (case study, 2 parts)
Functions. Asked once: NDA 1 2025
A function $f$ is such that $f(xy) = f(x+y)$ for all real values of $x$ and $y$ , and $f(5) = 10$ .
97 $f(0)$
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98 $f(20) + f(-20)$
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Question 3
Indefinite Integration. Asked once: NDA 2 2021
81 $f(x)=1+x^{2}+x^{4}$
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Question 4
Inverse Trigonometry. Asked once: NDA 2 2021
28 $\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\frac{3\pi}{2}$
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Question 5 (case study, 2 parts)
Limits & Continuity. Asked once: NDA 2 2025
For the following two (02) items: Let $f(x)=\begin{cases}\dfrac{1-\cos2x}{x^2}, & x<0\\ 9, & x=0\\ \dfrac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}, & x>0\end{cases}$
71 $\displaystyle\lim_{x\to 0^-}f(x)$
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72 $\displaystyle\lim_{x\to 0^+}f(x)$
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Question 1 (case study, 2 parts)