Day 151: MHT-CET Maths

1. A.
   $4 : 3$ internally, $(0, 2, 33)$
2. B.
   $3 : 4$ internally, $(0, 2, 33)$
3. C.
   $3 : 4$ externally, $(0, 2, 33)$
4. D.
   $4 : 3$ externally, $(2, 0, 33)$

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1. A.
   $\dfrac{1}{2}\log\left|\csc\left(x - \dfrac{\pi}{3}\right) + \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
2. B.
   $\dfrac{1}{2}\log\left|\csc\left(x - \dfrac{\pi}{3}\right) - \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
3. C.
   $\dfrac{1}{2}\log\left|\csc\left(x + \dfrac{\pi}{3}\right) - \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
4. D.
   $\dfrac{1}{2}\log\left|\csc\left(x + \dfrac{\pi}{3}\right) - \cot\left(x + \dfrac{\pi}{3}\right)\right| + c$

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1. A.
   $0$
2. B.
   $1$
3. C.
   $2$
4. D.
   $-1$

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1. A.
   $n\pi + (-1)^n\left(\dfrac{\pi}{3}\right), n \in \mathbb{Z}$
2. B.
   $2n\pi + (-1)^n\left(\dfrac{\pi}{2}\right), n \in \mathbb{Z}$
3. C.
   $n\pi + (-1)^{n+1}\left(\dfrac{\pi}{2}\right), n \in \mathbb{Z}$
4. D.
   $2n\pi + (-1)^n + \left(\dfrac{\pi}{4}\right), n \in \mathbb{Z}$

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1. A.
   $\dfrac{1}{2}$
2. B.
   $\dfrac{\sqrt{3}}{2}$
3. C.
   $\dfrac{1}{\sqrt{2}}$
4. D.
   $\dfrac{3}{4}$

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