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

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1. A
   0.8275
2. B
   1.1225
3. C
   1.8275
4. D
   2.0725

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1. A
   1
2. B
   0
3. C
   2
4. D
   infinitely many.

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1. A
   $\frac{49}{169}$
2. B
   $\frac{24}{169}$
3. C
   $\frac{52}{169}$
4. D
   $\frac{25}{169}$

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1. A
   $\frac{7}{8}$
2. B
   $\frac{7}{4}$
3. C
   $\frac{7}{2}$
4. D
   $\frac{7}{12}$

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1. A
   $\frac{-5\hat{i}+7\hat{j}+2\hat{k}}{\sqrt{78}}$
2. B
   $\frac{5\hat{i}-7\hat{j}+\hat{k}}{\sqrt{53}}$
3. C
   $\frac{5\hat{i}-7\hat{j}-\hat{k}}{\sqrt{53}}$
4. D
   $\frac{5\hat{i}+7\hat{j}-\hat{k}}{\sqrt{53}}$

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

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

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

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1. A
   $\sin\sqrt{x}-2\sqrt{x}\cos\sqrt{x}+c$
2. B
   $2\cos\sqrt{x}-2\sqrt{x}\sin\sqrt{x}+c$
3. C
   $\cos\sqrt{x}-2\sqrt{x}\sin\sqrt{x}+c$
4. D
   $2\sin\sqrt{x}-2\sqrt{x}\cos\sqrt{x}+c$

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1. A
   $\sqrt{45}$ units
2. B
   $\sqrt{18}$ units
3. C
   $\sqrt{72}$ units
4. D
   $\sqrt{33}$ units

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1. A
   f(x) is strictly increasing function
2. B
   f(x) is bounded
3. C
   f(x) has a local maxima
4. D
   f(x) is a strictly decreasing function

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1. A
   $5x-4\log(x-1)+c$
2. B
   $x-4\log(x-1)+c$
3. C
   $5x+4\log(x-1)+c$
4. D
   $5x+\log(x-1)+c$

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1. A
   $\frac{1-2\cos^{2}\alpha}{2}$
2. B
   $\frac{1+2\cos^{2}\alpha}{2}$
3. C
   $\frac{1-4\cos^{2}\alpha}{2}$
4. D
   $\frac{1+4\cos^{2}\alpha}{2}$

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1. A
   $14x+10y+9z=13$
2. B
   $14x+10y+9z=33$
3. C
   $14x+10y+9z=-15$
4. D
   $14x+10y+9z=-33$

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

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

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1. A
   $m+n=68$
2. B
   $m=n=78$
3. C
   $m=n=68$
4. D
   $n=m-8$

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

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

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

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1. A
   occurs only at unique point
2. B
   occurs only at two distinct points
3. C
   occurs at infinitely many points
4. D
   does not exist

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1. A
   $x+2y-2z=0$
2. B
   $3x+2y-2z=0$
3. C
   $x-2y+z=0$
4. D
   $5x+2y-4z=0$

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1. A
   Circuit (a)
2. B
   Circuit (b)
3. C
   Circuit (c)
4. D
   Circuit (d)

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1. A
   -16
2. B
   -7
3. C
   16
4. D
   7

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