1. A
   $\dfrac{x^2}{2}+c$
2. B
   $\ln(x+e)+c$
3. C
   $\ln(x^e+e^x)+c$
4. D
   $\dfrac{1}{e}\ln(x^e+e^x)+c$

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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
   $f(p)=g(m)$
2. B
   $f(q)=g(n)$
3. C
   $f(n)=g(q)$
4. D
   $f(m)=g(p)$

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

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1. A
   $-\left(\dfrac{d^2y}{dx^2}\right)^{-1}\!\!\left(\dfrac{dy}{dx}\right)^{-3}$
2. B
   $\left(\dfrac{d^2y}{dx^2}\right)^{-1}\!\!\left(\dfrac{dy}{dx}\right)^{-2}$
3. C
   $-\left(\dfrac{d^2y}{dx^2}\right)\!\!\left(\dfrac{dy}{dx}\right)^{-3}$
4. D
   $\left(\dfrac{d^2y}{dx^2}\right)^{-1}$

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1. A
   one-one and into
2. B
   neither one-one nor onto
3. C
   many-one and onto
4. D
   one-one and onto

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

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

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

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1. A
   Both the statements are true and Statement 2 is the correct explanation of Statement 1
2. B
   Both the statements are true but Statement 2 is not the correct explanation of Statement 1
3. C
   Statement 1 is true but Statement 2 is false
4. D
   Statement 1 is false but Statement 2 is true

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

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1. A
   $(x^2-2y^2)p^2-4pxy-x^2=0$
2. B
   $(x^2-2y^2)p^2+4pxy-x^2=0$
3. C
   $(x^2+2y^2)p^2-4pxy-x^2=0$
4. D
   $(x^2+2y^2)p^2-4pxy+x^2=0$

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1. A
   $x=y^2+cy$
2. B
   $x=2cy^2$
3. C
   $x=2y^2+cy$
4. D
   None of the above

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1. A
   $f(5)f'(0)$
2. B
   $f(5)-f'(0)$
3. C
   $f(5)f(0)$
4. D
   $f(5)+f'(0)$

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1. A
   $\int_0^a g(x)\,dx$
2. B
   $\int_0^a f(x)\,dx$
3. C
   $2\int_0^a f(x)\,dx$
4. D
   $0$

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1. A
   $y=xe^a+c$
2. B
   $x=ye^a+c$
3. C
   $y-\ln x+c$
4. D
   $x=\ln y+c$

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

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1. A
   $x^2-1$
2. B
   $x+\dfrac{3}{x}$
3. C
   $|x|$
4. D
   $x^2(x-3)$

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1. A
   $\dfrac{x\log_{10}e}{5x^2+3}$
2. B
   $\dfrac{2x\log_{10}e}{5x^2+3}$
3. C
   $\dfrac{10x\log_{10}e}{5x^2+3}$
4. D
   $\dfrac{10x\log_e 10}{5x^2+3}$

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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
   $\dfrac{3a^2}{4}$
2. B
   $\dfrac{a^2}{2}$
3. C
   $\dfrac{3\sqrt{3}a^2}{4}$
4. D
   $\dfrac{\sqrt{3}a^2}{4}$

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1. A
   $f(x)$ fluctuates in the interval
2. B
   $f(x)$ increases in the interval
3. C
   $f(x)$ decreases in the interval
4. D
   None of the above

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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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