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1. A
   $\frac{1}{1+25x^2}$
2. B
   $\frac{5}{1+25x^2}$
3. C
   $\frac{1}{1+5x^2}$
4. D
   $\frac{5}{1+5x^2}$

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1. A
   $ad - bc > 0$
2. B
   $ad - bc < 0$
3. C
   $ab - cd > 0$
4. D
   $ab - cd < 0$

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1. A
   $\frac{19}{25}$
2. B
   $\frac{18}{25}$
3. C
   $\frac{16}{25}$
4. D
   $\frac{7}{25}$

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1. A
   $\frac{1}{\sqrt{6}}(-\hat{i} + 2\hat{j} - \hat{k})$
2. B
   $\frac{1}{\sqrt{6}}(\hat{i} + 2\hat{j} + \hat{k})$
3. C
   $\frac{1}{\sqrt{6}}(2\hat{i} - \hat{j} - \hat{k})$
4. D
   $\frac{1}{\sqrt{6}}(-\hat{i} - 2\hat{j} + \hat{k})$

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1. A
   $\frac{87}{90^4}$
2. B
   $\frac{348}{90^4}$
3. C
   $\frac{874}{90^4}$
4. D
   $\frac{4}{10^4}$

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

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

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1. A
   $\frac{244}{3}$
2. B
   $\frac{316}{3}$
3. C
   $\frac{200}{3}$
4. D
   $\frac{400}{3}$

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

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1. A
   $-\frac{4}{7}\tan^{-4/7}x + c$, where c is a constant of integration.
2. B
   $\frac{4}{7}\tan^{4/7}x + c$, where c is a constant of integration.
3. C
   $-\frac{7}{4}\tan^{-4/7}x + c$, where c is a constant of integration.
4. D
   $\frac{7}{4}\tan^{4/7}x + c$, where c is a constant of integration.

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1. A
   27
2. B
   28
3. C
   4
4. D
   3

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1. A
   (2, 3)
2. B
   [2, 3)
3. C
   [2, 3]
4. D
   (2, 3]

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1. A
   6
2. B
   -6
3. C
   8
4. D
   -8

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1. A
   $20\sqrt{2}$
2. B
   $\frac{10\sqrt{2}}{3}$
3. C
   $\frac{20\sqrt{2}}{3}$
4. D
   $10\sqrt{2}$

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1. A
   $\cos^{-1}\left(\frac{2}{3}\right)$
2. B
   $\cos^{-1}\left(\frac{1}{3}\right)$
3. C
   $\sin^{-1}\left(\frac{1}{3}\right)$
4. D
   $\sin^{-1}\left(\frac{2}{3}\right)$

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1. A
   16
2. B
   8
3. C
   2
4. D
   4

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1. A
   $8x + y - 58 = 0$
2. B
   $8x - 2y - 52 = 0$
3. C
   $8x - y - 54 = 0$
4. D
   $8x + 2y - 60 = 0$

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

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1. A
   $x + y = 3e^{-2}$
2. B
   $x - y = 3e^{-2}$
3. C
   $x - y = 3e^{2}$
4. D
   $x + y = 3e^{2}$

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1. A
   $\tan^{-1}x \cdot e^{\tan^{-1}x} + c$
2. B
   $\left(\tan^{-1}x\right)^2 e^{\tan^{-1}x} + c$
3. C
   $2\tan^{-1}x \cdot e^{\tan^{-1}x} + c$
4. D
   $2\left(\tan^{-1}x\right)^2 e^{\tan^{-1}x} + c$

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1. A
   $8x + 14y + 13z - 37 = 0$
2. B
   $8x - 14y - 13z - 34 = 0$
3. C
   $8x - 14y - 13z + 37 = 0$
4. D
   $8x + 14y + 13z + 37 = 0$

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1. A
   (-1, 2)
2. B
   (1, -2)
3. C
   (-1, -2)
4. D
   (1, 2)

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1. A
   $-\frac{\pi}{4}$
2. B
   $\frac{4\pi}{3}$
3. C
   $\frac{\pi}{12}$
4. D
   $\frac{7\pi}{12}$

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1. A
   $\tan x = (c + \sec x)y$, where c is constant of integration.
2. B
   $\sec y = (c + \tan y)x$, where c is constant of integration.
3. C
   $\sec x = (c + \tan x)y$, where c is constant of integration.
4. D
   $\cos y = (c + \tan y)$, where c is constant of integration.

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

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