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1. A
   $\alpha = 1, \beta = 1$
2. B
   $\alpha = 2, \beta = 2$
3. C
   $\alpha = 1, \beta = 2$
4. D
   $\alpha = 2, \beta = 1$

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1. A
   $\frac{\pi}{2}$
2. B
   $\frac{5\pi}{4}$
3. C
   $\frac{7\pi}{4}$
4. D
   $\frac{3\pi}{2}$

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1. A
   $\begin{pmatrix} 0 & 8 & -4 \\ -1 & 3 & 2 \\ 1 & -7 & 2 \end{pmatrix}$
2. B
   $\begin{pmatrix} 0 & -8 & 4 \\ -1 & 3 & -2 \\ 1 & -7 & 2 \end{pmatrix}$
3. C
   $\begin{pmatrix} 0 & 8 & -4 \\ 1 & -3 & 2 \\ -1 & 7 & -2 \end{pmatrix}$
4. D
   $\begin{pmatrix} 0 & -8 & 4 \\ 1 & 3 & 2 \\ -1 & -7 & 2 \end{pmatrix}$

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

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1. A
   $\pm\frac{1}{\sqrt{2}}(-\hat{j}-\hat{k})$
2. B
   $\frac{\hat{j}-\hat{k}}{\sqrt{2}}$
3. C
   $\frac{-\hat{j}+2\hat{k}}{\sqrt{5}}$
4. D
   $\pm\frac{1}{\sqrt{26}}(\hat{j}+5\hat{k})$

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1. A
   Horses have wings but crows do not have tails. or crows have tails but horses do not have wings.
2. B
   Horses do not have wings if and only if crows do not have tails.
3. C
   Horses do not have wings and crows have tails. or Horses have wings and crows have tails.
4. D
   Horses do not have wings and crows do not have tails.

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1. A
   $\frac{2}{\sqrt{5}}$ units
2. B
   $\frac{2\sqrt{2}}{\sqrt{5}}$ units
3. C
   $\sqrt{\frac{2}{5}}$ units
4. D
   2 units

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

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1. A
   $\sqrt{e^x-1} + \tan^{-1}\sqrt{e^x-1} + c$
2. B
   $2\sqrt{e^x-1} + \tan^{-1}\sqrt{e^x-1} + c$
3. C
   $2\sqrt{e^x-1} - 2\tan^{-1}\sqrt{e^x-1} + c$
4. D
   $2\sqrt{e^x-1} - \tan^{-1}\sqrt{e^x-1} + c$

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1. A
   35
2. B
   45
3. C
   60
4. D
   90

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1. A
   7
2. B
   -7
3. C
   no real value
4. D
   4

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

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1. A
   $\frac{2}{\sqrt{3}}$ m, $\frac{3}{\sqrt{3}}$ m
2. B
   $\frac{3}{\sqrt{3}}$ m, $\frac{2}{\sqrt{3}}$ m
3. C
   3 m, 6 m
4. D
   6 m, 3 m

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1. A
   7.096
2. B
   8.096
3. C
   7.934
4. D
   8.934

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1. A
   4
2. B
   8
3. C
   10
4. D
   2

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1. A
   $0 < x \leq 1$
2. B
   $0 \leq x \leq 1$
3. C
   $-\infty < x \leq 0$
4. D
   $-\infty < x < 1$

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

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1. A
   $-\frac{7}{9}$
2. B
   $\frac{7}{9}$
3. C
   $\frac{9}{7}$
4. D
   $-\frac{9}{7}$

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

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1. A
   $x + y = a\sqrt{2}$
2. B
   $x + y = \frac{a}{\sqrt{2}}$
3. C
   $x + y = \frac{a}{2\sqrt{2}}$
4. D
   $x + y = \frac{a}{8}$

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1. A
   $\frac{\pi^2}{16}$
2. B
   $\frac{3\pi^2}{16}$
3. C
   $-\frac{3\pi^2}{16}$
4. D
   $-\frac{\pi^2}{16}$

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1. A
   $\frac{1}{5}\left(\frac{\pi}{4} - \tan^{-1}\frac{1}{3\sqrt{3}}\right)$
2. B
   $\frac{1}{10}\left(\frac{\pi}{4} - \tan^{-1}\frac{1}{9\sqrt{3}}\right)$
3. C
   $\frac{1}{20}\tan^{-1}\frac{1}{9\sqrt{3}}$
4. D
   $\frac{\pi}{40}$

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1. A
   $x^2+y^2-6x-4y+5=0$
2. B
   $x^2+y^2-6x-4y+17=0$
3. C
   $x^2+y^2-6x-4y+9=0$
4. D
   $x^2+y^2-6x-4y+4=0$

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

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1. A
   $13x^2+12xy+3y^2=0$
2. B
   $13x^2-12xy+3y^2=0$
3. C
   $13x^2+12xy-3y^2=0$
4. D
   $13x^2-12xy-3y^2=0$

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