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

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

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1. A
   $\frac{2}{3}$
2. B
   $\frac{4}{3}$
3. C
   $\frac{9}{2}$
4. D
   $\frac{16}{3}$

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

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

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1. A
   $-\frac{7}{4}$
2. B
   $-\frac{15}{4}$
3. C
   $-\frac{17}{4}$
4. D
   $-\frac{5}{4}$

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

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

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1. A
   $\frac{\pi}{16}\log 2$
2. B
   $\frac{\pi}{2}\log 2$
3. C
   $\frac{\pi}{8}\log 2$
4. D
   $\frac{\pi}{4}\log 2$

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1. A
   $x=\frac{a}{2}+\frac{\sqrt{a^2+b^2}}{2}\cos\theta,\ y=\frac{b}{2}+\frac{\sqrt{a^2+b^2}}{2}\sin\theta$
2. B
   $x=-\frac{a}{2}+\frac{\sqrt{a^2+b^2}}{4}\sin\theta,\ y=-\frac{b}{2}+\frac{\sqrt{a^2+b^2}}{4}\cos\theta$
3. C
   $x=\frac{a}{2}+\frac{\sqrt{a^2+b^2}}{2}\sin\theta,\ y=\frac{b}{2}+\frac{\sqrt{a^2+b^2}}{2}\cos\theta$
4. D
   $x=\frac{a}{2}+\frac{\sqrt{a^2+b^2}}{4}\cos\theta,\ y=\frac{b}{2}+\frac{\sqrt{a^2+b^2}}{4}\sin\theta$

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

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1. A
   $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}$
2. B
   $\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1$
3. C
   $\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1$
4. D
   $x+y+z=6$

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

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1. A
   $\lambda\vec{c}$
2. B
   $\lambda\vec{b}$
3. C
   $\lambda\vec{a}$
4. D
   $\vec{0}$

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

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1. A
   ₹ 36
2. B
   ₹ 31
3. C
   ₹ 30
4. D
   ₹ 40

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1. A
   $3\sqrt{10}$
2. B
   $10\sqrt{3}$
3. C
   $\frac{10}{\sqrt{3}}$
4. D
   $\frac{20}{\sqrt{3}}$

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

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1. A
   $x=n\pi,\ n\in\mathbb{Z}$
2. B
   $x=2n\pi+\frac{\pi}{2}\ \text{or}\ 2n\pi,\ n\in\mathbb{Z}$
3. C
   $x=n\pi+(-1)^n\frac{\pi}{4}-\frac{\pi}{4},\ n\in\mathbb{Z}$
4. D
   not existing

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1. A
   $a+b=c$
2. B
   $b+c=a$
3. C
   $a+c=b$
4. D
   $b=c$

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1. A
   (10,2,2)
2. B
   (2,–10,–2)
3. C
   (10,–2,–2)
4. D
   (–2,10,2)

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

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

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

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1. A
   8
2. B
   4
3. C
   12
4. D
   2

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