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Find the set $K$ of all real $x$ where $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable.
Check $x=0$: Write $h(x)=\sin|x|-|x|$. For $x>0$: $h(x)=\sin x-x$, $h'(x)=\cos x-1$. For $x<0$: $h(x)=-\sin x+x$, $h'(x)=-\cos x+1$. At $x=0$: left limit $=0$, right limit $=0$ β differentiable.
The term $2(x-\pi)\cos|x|$ is clearly differentiable everywhere (product of smooth functions, since $\cos|x|$ is even and smooth).
At $x=\pi$: $\cos|\pi|=\cos\pi=-1$, and all parts are smooth for $x>0$.
Therefore $f$ is differentiable everywhere and $K=\boldsymbol{\emptyset}$.
Formula for number of diagonals in an \(n\-sided polygon: \(\dfrac{n(n-3)}{2}\
For decagon (\(n = 10\): \(\dfrac{10 \times 7}{2} = \dfrac{70}{2} = \mathbf{35}\
This formula counts all connections between non-adjacent vertices. Total connections = \(\binom{10}{2} = 45\, minus 10 sides = 35 diagonals.
Matching logic:
- Lysine is an amino acid β detected by ninhydrin (Q), which gives a purple colour with primary amines/amino acids.
- Furfural is an aldehyde from furan β detected by 1-naphthol (P) via the Tollens-type or colour reaction specific to furfural.
- Benzyl alcohol (PhCH$_2$OH) is a primary alcohol β detected by ceric ammonium nitrate (S), which gives a red colour with alcohols.
- Styrene (PhβCH=CH$_2$) has a C=C bond β oxidized by KMnO$_4$ (R), which gets decolourized.
Correct match: (A)β(Q), (B)β(P), (C)β(S), (D)β(R)
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