1. A
   It is discontinuous at $x=-2$ but continuous at every other point.
2. B
   It is continuous only in the interval $(-3,-2)$.
3. C
   It is discontinuous at $x=0$ but continuous at every other point.
4. D
   It is discontinuous at every point.

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1. A
   $[1,\infty)$
2. B
   $(0,\infty)$
3. C
   $\left(\dfrac{1}{2},\infty\right)$
4. D
   None of the above

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

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1. A
   1 and 2 only
2. B
   2 and 3 only
3. C
   1 and 3 only
4. D
   1, 2 and 3

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

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1. A
   $f(x)$ is continuous at $x = 0$ but not differentiable at $x = 0$
2. B
   $f(x)$ is continuous as well as differentiable at $x = 0$
3. C
   $f(x)$ is discontinuous at $x = 0$
4. D
   None of the above

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1. A
   $f'(x) = 2x$ for $0 < x \leq 1$
2. B
   $f'(x) = -2x$ for $0 < x \leq 1$
3. C
   $f'(x) = -2x$ for $0 < x < 1$
4. D
   $f'(x) = 0$ for $0 < x < \infty$

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1. A
   odd
2. B
   even
3. C
   both even and odd
4. D
   neither even nor odd

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1. A
   1 square unit
2. B
   $2\sqrt{2}$ square units
3. C
   2 square units
4. D
   $2\sqrt{3}$ square units

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1. A
   $(-1)^k(k-1)\pi$
2. B
   $(-1)^{k-1}(k-1)\pi$
3. C
   $(-1)^k k\pi$
4. D
   $(-1)^{k-1}k\pi$

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1. A
   $(0, \infty)$
2. B
   $(-\infty, \infty)$
3. C
   $(-\infty, 0) \cup (0, \infty)$
4. D
   $(-1, \infty)$

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1. A
   Graph (a): horizontal line at y=1
2. B
   Graph (b): line at +1 for x>0, line at -1 for x<0 (step)
3. C
   Graph (c): line at +1 for x>0, touching origin
4. D
   None of the above

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1. A
   251
2. B
   250
3. C
   1
4. D
   0

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1. A
   all integers
2. B
   all integers except 0 and 1
3. C
   all integers except 0
4. D
   all integers except 1

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1. A
   Zero (No solution)
2. B
   One
3. C
   Two
4. D
   Three

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1. A
   $f(x^2)=[f(x)]^2$
2. B
   $f(|x|)=|f(x)|$
3. C
   $f(x+y)=f(x)+f(y)$
4. D
   None of the above

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1. A
   $f(x)$ is continuous at $x=0$ and $x=1$
2. B
   $f(x)$ is continuous at $x=0$ but not at $x=1$
3. C
   $f(x)$ is continuous at $x=1$ but not at $x=0$
4. D
   $f(x)$ is neither continuous at $x=0$ nor at $x=1$

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

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1. A
   $\sqrt{2}-1$
2. B
   $1-\sqrt{2}$
3. C
   $2(\sqrt{2}-1)$
4. D
   $\sqrt{3}-1$

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1. A
   $(-\infty,0)$
2. B
   $(0,\infty)$
3. C
   $(-\infty,\infty)$
4. D
   $(-\infty,0]$

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1. A
   The function is not continuous at x = -3
2. B
   The function is continuous at x = 3
3. C
   The function is differentiable at x = 0
4. D
   The function is differentiable at x = -3

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1. A
   $b-a$
2. B
   $a-b$
3. C
   $0$
4. D
   $2(b-a)$

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1. A
   2
2. B
   3
3. C
   4
4. D
   9

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1. A
   $\frac{2^{-\frac{\pi}{6}}(6\ln 2-\sqrt{3}\pi)}{6}$
2. B
   $\frac{2^{\frac{\pi}{6}}(6\ln 2+\sqrt{3}\pi)}{6}$
3. C
   $\frac{2^{-\frac{\pi}{6}}(6\ln 2+\sqrt{3}\pi)}{6}$
4. D
   $\frac{2^{\frac{\pi}{6}}(6\ln 2-\sqrt{3}\pi)}{6}$

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1. A
   $\cos x+\sin x$
2. B
   $-(\cos x+\sin x)$
3. C
   $\pm(\cos x+\sin x)$
4. D
   None of the above

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