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Match the following

List- I

List – II
A) The number of non-zero complex numbers   $Z$ such that $| Z | + Z - 3 \bar{Z} = 0$ is P) 4
B) Let $f (x) = \int_{1}^{x} \frac{\tan^{- 1} t}{t} d t, (x > 0)$ then the integral part of $f (e^{2}) - f (\frac{1}{e^{2}})$ Q) 3
C) Suppose   $f (x)$ is a differentiable function such that $f (x) + f' (x) \leq 1$ for all $x$ and $f (0) = 0$ then the maximum value of $f (1)$ is $K$ then the integral part of $K$ R) 0
D) Let $α = \int_{0}^{1} \tan^{- 1} (\frac{\tan x}{2}) d x,$ $β = \int_{0}^{1} \tan^{- 1} (\frac{\tan x - 2 \cot x}{3}) d x$ then the integral part of   $α - β$ equals S) 1
T) 2

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$A - P, B - Q, C - R, D - T$

$A - S, B - T, C - R, D - P$

$A - S, B - T, C - P, D - R$

$A - R, B - Q, C - R, D - S$

answer is D.

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Detailed Solution

A) $z = x + i y \Rightarrow y = 0 (o r) \sqrt{x^{2} + y^{2}} = 2 x \Rightarrow y = 0, x = 0 \Rightarrow z = 0 is only solution$

B) $G . E = \int_{1}^{e^{2}} \frac{\tan^{- 1} t}{t} - \int_{1}^{1 / e^{2}} \frac{\tan^{- 1} t}{t}$

$= \int_{1}^{e^{2}} \frac{\tan^{- 1} t}{t} + \int_{1}^{e^{2}} \frac{\tan^{- 1} (\frac{1}{z})}{z} d z = \int_{1}^{e^{2}} \frac{π}{2} . \frac{1}{t} d t = \frac{π}{2} \times 2 = π \Rightarrow [π] = 3$

$\begin{array}{l} C) e^{x} (f (x) + f' (x)) \leq e^{x} \\ \Rightarrow d (e^{x} f (x)) \leq d (e^{x}) \Rightarrow \int_{0}^{1} d (e^{x} f (x)) \leq \int_{0}^{1} d (e^{x}) \Rightarrow e f (1) \leq e - 1 \\ \Rightarrow f (1) \leq \frac{e - 1}{e} \end{array}$

$∴ M a x . v a l u e o f f (1) = \frac{e - 1}{e} = K \Rightarrow K = 0$

$\begin{array}{l} D) β = \int_{0}^{1} \tan^{- 1} \frac{(\frac{\tan x}{2} - \cot x)}{1 + \frac{1}{2}} d x = \int_{0}^{1} \tan^{- 1} (\frac{\tan x}{2}) d x - \int_{0}^{1} \tan^{- 1} (\cot x) d x \\ \Rightarrow β = α - (\frac{π}{2} - \frac{1}{2}) \Rightarrow α - β = \frac{π - 1}{2} [α - β] = 1 \end{array}$

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