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1. A
   $\frac{\log x}{x^3} + C$
2. B
   $\frac{\log x^3}{3} + C$
3. C
   $\frac{(\log x)^3}{2} + C$
4. D
   $\frac{(\log x)^3}{3} + C$

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1. A
   $-\frac{5}{2}[\tan x]^5 + C$
2. B
   $[\tan x]^5 + C$
3. C
   $\frac{2}{5}[\tan x]^5 + C$
4. D
   $\frac{5}{2}[\tan x]^5 + C$

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1. A
   $\sqrt{5}\ \text{cm}$
2. B
   $\sqrt{3}\ \text{cm}$
3. C
   $\sqrt{3}\ \text{cm}$
4. D
   $\sqrt{5}\ \text{cm}$

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1. A
   1, 2, 3
2. B
   0, 1, 2
3. C
   2, 3
4. D
   None of these

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1. A
   $\frac{3}{2}$
2. B
   0
3. C
   1
4. D
   $-\frac{1}{2}$

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

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1. A
   unbounded in first quadrant
2. B
   unbounded in first and second quadrants
3. C
   bounded in first quadrant
4. D
   None of the above

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1. A
   bounded feasible space
2. B
   unbounded feasible space
3. C
   both bounded and unbounded feasible space
4. D
   None of the above

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1. A
   8
2. B
   2
3. C
   5
4. D
   4

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

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1. A
   $\frac{9}{10}$
2. B
   $\left(\frac{1}{10}\right)^5$
3. C
   $\left(\frac{9}{10}\right)^5$
4. D
   $\left(\frac{1}{2}\right)^5$

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1. A
   $\frac{16}{3}$
2. B
   $\frac{4}{3}$
3. C
   $\frac{5}{3}$
4. D
   $\frac{10}{3}$

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1. A
   $\frac{16a^2}{3}$
2. B
   $\frac{14a^2}{3}$
3. C
   $\frac{13a^2}{3}$
4. D
   $16a^2$

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1. A
   $\frac{\pi}{2}$
2. B
   $\frac{\pi}{4}$
3. C
   $\frac{\pi}{3}$
4. D
   $\pi$

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1. A
   $\tan^{-1}2$
2. B
   $\tan^{-1}\frac{1}{\sqrt{3}}$
3. C
   $\tan^{-1}1$
4. D
   $\frac{\pi}{4} - \tan^{-1}2$

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1. A
   $4400 \times 10^{-7}$
2. B
   $4200 \times 10^{-7}$
3. C
   $4800 \times 10^{-7}$
4. D
   $4600 \times 10^{-7}$

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1. A
   $-\frac{\Delta\phi}{\Delta t + R}$
2. B
   $\frac{\Delta\phi}{\Delta t \times R}$
3. C
   $\frac{\Delta\phi}{R}$
4. D
   $\frac{\Delta\phi}{\Delta t}$

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1. A
   6400 km
2. B
   800 km
3. C
   1600 km
4. D
   3200 km

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1. A
   $P_x$ is maximum
2. B
   $P_z$ is maximum
3. C
   $P_y$ is maximum
4. D
   $P_x = P_y = P_z$

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1. A
   1 cm
2. B
   8 cm
3. C
   4 cm
4. D
   2 cm

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1. A
   $v^2 T = \text{constant}$
2. B
   $\frac{v^2}{T} = \text{constant}$
3. C
   $v T^2 = \text{constant}$
4. D
   $\frac{v}{T^2} = \text{constant}$

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1. A
   $2\varepsilon_0 E A d$
2. B
   $\frac{1}{2}\varepsilon_0 E^2 A d$
3. C
   $\frac{\varepsilon_0 E^2}{2A d}$
4. D
   $\frac{E^2 A d}{2\varepsilon_0}$

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1. A
   $f\!\left(\frac{1}{K_1} + \frac{1}{K_2}\right)$
2. B
   $f\!\left(\frac{1}{K_1} - \frac{1}{K_2}\right)$
3. C
   $f(K_1 + K_2)$
4. D
   $f(K_1 - K_2)$

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1. A
   $\frac{T}{12}$ s
2. B
   $\frac{T}{2}$ s
3. C
   $\frac{T}{4}$ s
4. D
   $\frac{T}{6}$ s

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1. A
   $\frac{4\varepsilon_0 n h^3}{m e^2}$
2. B
   $\frac{4\varepsilon_0 n^2 h^2}{m e^2}$
3. C
   $\frac{4\varepsilon_0^2 n^3 h^3}{m e^4}$
4. D
   $\frac{4\varepsilon_0^2 n^2 h^3}{m e^3}$

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