1. A
   should be 0
2. B
   should be 1
3. C
   should be 2
4. D
   cannot be determined

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1. A
   $y=\tan x+c$
2. B
   $y=\tan(x+c)$
3. C
   $\tan^{-1}(y+c)=x$
4. D
   $\tan^{-1}(y+c)=2x$

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

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1. A
   $[-1,1]$
2. B
   $[1,3]$
3. C
   $[0,5]$
4. D
   $[-2,1]$

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

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

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1. A
   $\frac{my}{nx}$
2. B
   $-\frac{my}{nx}$
3. C
   $\frac{mx}{ny}$
4. D
   $-\frac{ny}{mx}$

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1. A
   $\frac{1}{n}\ln\left(\frac{x^n}{x^n+1}\right)+c$
2. B
   $\ln\left(\frac{x^n+1}{x^n}\right)+c$
3. C
   $\ln\left(\frac{x^n}{x^n+1}\right)+c$
4. D
   $\frac{1}{n}\ln\left(\frac{x^n+1}{x^n}\right)+c$

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

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

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1. A
   1 only
2. B
   2 only
3. C
   Both 1 and 2
4. D
   Neither 1 nor 2

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

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1. A
   0
2. B
   $-1$
3. C
   1
4. D
   Limit does not exist

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

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

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

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1. A
   1 only
2. B
   2 only
3. C
   Both 1 and 2
4. D
   Neither 1 nor 2

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1. A
   The domain is $\{x\in\mathbf{R}\mid x\neq1\}$ and the range is the set of reals.
2. B
   The domain is $\{x\in\mathbf{R}\mid x\neq1\}$, the range is $\{y\in\mathbf{R}\mid y\neq0\}$ and the graph intersects y-axis at $(0,-1)$.
3. C
   The domain is the set of reals and the range is the singleton set {0}.
4. D
   The domain is $\{x\in\mathbf{R}\mid x\neq1\}$ and the range is the set of points on the y-axis.

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

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

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

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1. A
   $\frac{5}{6}$
2. B
   $\frac{5}{9}$
3. C
   $\frac{1}{2}$
4. D
   $\frac{5}{12}$

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1. A
   $P(B)=\frac{2}{3}$
2. B
   $P(A\cap B)=P(A)P(B)$
3. C
   $P(A\cup B)>P(A)+P(B)$
4. D
   $P(\text{not A and not B})=P(\text{not A})P(\text{not B})$

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1. A
   50
2. B
   60
3. C
   80
4. D
   100

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

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