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1. A
   $\frac{\pi}{2}\log 2$
2. B
   $\frac{\pi}{4}\log 2$
3. C
   $\frac{\pi}{6}\log 2$
4. D
   $\frac{\pi}{8}\log 2$

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1. A
   $\frac{41}{50}$
2. B
   $\frac{41}{125}$
3. C
   $\frac{7}{25}$
4. D
   $\frac{7}{50}$

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

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1. A
   $e^y \log y = e^x \sin x + c$
2. B
   $e^y = e^x \sin x + c$
3. C
   $\log y = e^y \sin x + c$
4. D
   $y\log y = e^x \sin x + c$

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1. A
   5
2. B
   6
3. C
   7
4. D
   8

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

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1. A
   $\frac{4\sqrt{2}}{\sqrt{19}}$ units
2. B
   $\frac{3\sqrt{2}}{\sqrt{19}}$ units
3. C
   $\frac{5\sqrt{2}}{\sqrt{19}}$ units
4. D
   $\frac{2\sqrt{2}}{\sqrt{19}}$ units

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

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

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

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1. A
   $-\hat{i}+2\hat{j}-2\hat{k}$
2. B
   $-\hat{i}+\hat{j}-\hat{k}$
3. C
   $-\hat{i}-\hat{j}+\hat{k}$
4. D
   $\hat{i}+3\hat{j}-4\hat{k}$

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1. A
   $p\wedge q$
2. B
   $p\vee\sim q$
3. C
   $p\wedge\sim q$
4. D
   $(\sim p)\wedge(\sim q)$

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1. A
   $-2, 7$
2. B
   $2, -7$
3. C
   $7, -2$
4. D
   $-7, -2$

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

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1. A
   $a - 2a\log 2$
2. B
   $a + 2a\log 2$
3. C
   $a - a\log 2$
4. D
   $a + a\log 2$

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1. A
   $\sqrt{PQ\cdot RS}$
2. B
   $\frac{PQ+RS}{2}$
3. C
   $\frac{2\cdot PQ\cdot RS}{PQ+RS}$
4. D
   $\sqrt{\frac{PQ^2+RS^2}{2}}$

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1. A
   20
2. B
   15
3. C
   25
4. D
   10

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1. A
   $2\hat{i}-4\hat{j}+4\hat{k}$
2. B
   $\hat{i}-\hat{j}+2\hat{k}$
3. C
   $4\hat{i}+4\hat{j}-2\hat{k}$
4. D
   $2\hat{i}+4\hat{j}+4\hat{k}$

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

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

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1. A
   $\begin{pmatrix}0.8&0.6\\-0.6&0.8\end{pmatrix}$
2. B
   $\begin{pmatrix}-0.8&0.6\\-0.6&0.8\end{pmatrix}$
3. C
   $\begin{pmatrix}-0.8&-0.6\\0.6&0.8\end{pmatrix}$
4. D
   $\begin{pmatrix}8&-6\\6&8\end{pmatrix}$

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

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

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1. A
   $-\frac{\sqrt{6}}{5}$
2. B
   $\frac{2\sqrt{6}}{5}$
3. C
   $-\frac{2\sqrt{6}}{5}$
4. D
   $\frac{2\sqrt{5}}{6}$

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1. A
   $-\frac{3}{2}$
2. B
   $\frac{3}{2}$
3. C
   $-\frac{8}{3}$
4. D
   $\frac{8}{3}$

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