1. A
   $\left(-\dfrac{1}{3}, \dfrac{7}{3}\right)$
2. B
   $\left(-1, \dfrac{7}{3}\right)$
3. C
   $\left(\dfrac{1}{3}, \dfrac{7}{3}\right)$
4. D
   $\left(1, \dfrac{7}{3}\right)$

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1. A
   $\left(1, \dfrac{\sqrt{3}}{2}\right)$
2. B
   $\left(\dfrac{2}{3}, \dfrac{1}{\sqrt{3}}\right)$
3. C
   $\left(\dfrac{2}{3}, \dfrac{\sqrt{3}}{2}\right)$
4. D
   $\left(1, \dfrac{1}{\sqrt{3}}\right)$

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

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1. A
   $2<r<8$
2. B
   $r=2$ or $r=8$
3. C
   $r<2$
4. D
   $r>2$

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1. A
   $x^2+y^2+3x+2=0$
2. B
   $x^2+y^2+3x+12y+2=0$
3. C
   $x^2+y^2+2x=0$
4. D
   $x^2+y^2=5$

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

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1. A
   $\dfrac{x^2}{64}+\dfrac{y^2}{60}=1$
2. B
   $\dfrac{x^2}{60}+\dfrac{y^2}{64}=1$
3. C
   $\dfrac{x^2}{20}+\dfrac{y^2}{24}=1$
4. D
   $\dfrac{x^2}{24}+\dfrac{y^2}{20}=1$

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1. A
   $2x+3y-4=0$
2. B
   $2x+3y-5=0$
3. C
   $x+y-1=0$
4. D
   $3x-2y+7=0$

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1. A
   $\tan^{-1}\!\left(\dfrac{1}{2\sqrt{6}}\right)$
2. B
   $\tan^{-1}\!\left(\dfrac{1}{\sqrt{2}}\right)$
3. C
   $\tan^{-1}(3)$
4. D
   $\tan^{-1}\!\left(\dfrac{1}{\sqrt{3}}\right)$

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1. A
   5, 2
2. B
   2, 5
3. C
   1, 0
4. D
   0, 0

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1. A
   parallel to x-axis
2. B
   parallel to y-axis
3. C
   parallel to z-axis
4. D
   equally inclined to all the axes

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1. A
   $\left(\dfrac{a+b+c}{3}, \dfrac{a+b+c}{3}, \dfrac{a+b+c}{3}\right)$
2. B
   $(a, b, c)$
3. C
   $\left(\dfrac{a}{2}, \dfrac{b}{2}, \dfrac{c}{2}\right)$
4. D
   $\left(\dfrac{a}{3}, \dfrac{b}{3}, \dfrac{c}{3}\right)$

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1. A
   vertices of a rhombus which is not a square
2. B
   non-coplanar
3. C
   collinear
4. D
   coplanar but not collinear

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1. A
   $\left(0, -\dfrac{7}{2}, \dfrac{5}{2}\right)$
2. B
   $\left(0, \dfrac{7}{2}, \dfrac{1}{2}\right)$
3. C
   $\left(0, -\dfrac{7}{2}, -\dfrac{5}{2}\right)$
4. D
   $\left(0, \dfrac{7}{2}, -\dfrac{5}{2}\right)$

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1. A
   $ae+cg-1=0$
2. B
   $ae+bf-1=0$
3. C
   $ae+cg+1=0$
4. D
   $ag+ce+1=0$

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1. A
   $m=2$ and $n=\pm1$
2. B
   $m=\pm2$ and $n=-1$
3. C
   $m=2$ and $n=-1$
4. D
   $m=\pm2$ and $n=1$

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1. A
   $2\overrightarrow{OP}$
2. B
   $4\overrightarrow{OP}$
3. C
   $6\overrightarrow{OP}$
4. D
   $8\overrightarrow{OP}$

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1. A
   $\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{AC}+\overrightarrow{DB}$
2. B
   $\overrightarrow{BA}+\overrightarrow{CD}-\overrightarrow{BD}+\overrightarrow{CA}$
3. C
   $\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{AC}+\overrightarrow{BD}$
4. D
   $\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{BC}+\overrightarrow{AD}$

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1. A
   $\vec{a}, \vec{b}, \vec{c}$ are orthogonal in pairs and $|\vec{a}|=|\vec{c}|$ and $|\vec{b}|=1$
2. B
   $\vec{a}, \vec{b}, \vec{c}$ are non-orthogonal to each other
3. C
   $\vec{a}, \vec{b}, \vec{c}$ are orthogonal in pairs but $|\vec{a}|\neq|\vec{c}|$
4. D
   $\vec{a}, \vec{b}, \vec{c}$ are orthogonal in pairs but $|\vec{b}|\neq1$

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

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1. A
   $0$
2. B
   $\dfrac{1}{2}$
3. C
   $1$
4. D
   $2$

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1. A
   $\dfrac{1}{2}$
2. B
   $1$
3. C
   $\sqrt{3}$
4. D
   None of the above

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1. A
   $\dfrac{1}{2}\ln\left|\dfrac{x^7-1}{x^7+1}\right|+c$
2. B
   $\dfrac{1}{7}\ln\left|\dfrac{x^7+1}{x^7}\right|+c$
3. C
   $\ln\left|\dfrac{x^7-1}{7x}\right|+c$
4. D
   $\dfrac{1}{7}\ln\left|\dfrac{x^7}{x^7+1}\right|+c$

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1. A
   $[0, \pi]$ and $[-1, 1]$
2. B
   $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ and $[-1, 1]$
3. C
   $[0, \pi]$ and $(-1, 1)$
4. D
   $[0, \pi]$ and $[0, 1]$

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1. A
   $f\!\left(-\dfrac{a}{a+1}\right)$
2. B
   $f(a^2)$
3. C
   $f\!\left(\dfrac{1}{a}\right)$
4. D
   $f(-a)$

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