1. A
   $\overrightarrow{r} \cdot (\widehat{i} - 7\widehat{j} + 3\widehat{k}) = \frac{7}{3}$
2. B
   $\overrightarrow{r} \cdot (\widehat{i} + 7\widehat{j} + 3\widehat{k}) = 7$
3. C
   $\overrightarrow{r} \cdot (3\widehat{i} + 7\widehat{j} + 3\widehat{k}) = 7$
4. D
   $\overrightarrow{r} \cdot (\widehat{i} + 7\widehat{j} + 3\widehat{k}) = \frac{7}{3}$

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1. A
   $( - 2,8)$
2. B
   $(1,5)$
3. C
   $(3,13)$
4. D
   $(2,8)$

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1. A
   $\frac{71}{256}$
2. B
   $- \frac{69}{256}$
3. C
   $\frac{69}{256}$
4. D
   $- \frac{71}{256}$

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1. A
   -4
2. B
   -5
3. C
   $- \sqrt{2} - \sqrt{3} - 1$
4. D
   $- \sqrt{2} - \sqrt{3} + 1$

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1. A
   . $a = 1,b = 1,c = 0$
2. B
   $a = - 1,b = 1,c = 1$
3. C
   $a = 1,b = 0,c = 1$
4. D
   $a = \frac{1}{2},b = \frac{1}{2},c = 1$

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1. A
   $\sim p \land q$
2. B
   $p \land \sim q$
3. C
   $\sim p \vee q$
4. D
   $p \vee \sim q$

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1. A
   (B) and (E) only
2. B
   (C) and (D) only
3. C
   (A) and (D) only
4. D
   (A) and (E) only

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1. A
   $x^{2} + 9y^{2} = 9$
2. B
   $2x^{2} - 18y^{2} = 9$
3. C
   $y^{2} = \frac{1}{6\sqrt{3}}x$
4. D
   $x^{2} + y^{2} = 7$

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1. A
   $1200\sqrt{3m}$
2. B
   $1800\sqrt{3}\text{ }m$
3. C
   $3600\sqrt{3}\text{ }m$
4. D
   $2400\sqrt{3m}$

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1. A
   (A) is a tautology but not (B)
2. B
   (A) and (B) both are not tautologies.
3. C
   (A) and (B) both are tautologies.
4. D
   (B) is a tautology but not (A).

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1. A
   a unique solution
2. B
   exactly two solutions
3. C
   infinitely many solutions
4. D
   no solution

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1. A
   $\frac{n(n + 1)^{2}(n + 2)}{12}$
2. B
   $\frac{n(n - 1)(2n + 1)}{6}$
3. C
   $\frac{n(n + 1)(2n + 1)}{6}$
4. D
   $\frac{n(2n + 1)(3n + 1)}{6}$

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1. A
   5
2. B
   $\frac{62}{5}$
3. C
   $\frac{31}{5}$
4. D
   10

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1. A
   $9\sqrt{3}$ square units
2. B
   $12\sqrt{3}$ square units
3. C
   $11\sqrt{3}$ square units
4. D
   $6\sqrt{3}$ square units

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

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

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

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1. A
   0
2. B
   nRT $\left( \ln2 + \frac{1}{2} \right)$
3. C
   nRTIn2
4. D
   nRT $\left( \ln2 - \frac{1}{2} \right)$

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

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1. A
   $3.6\text{ }V$
2. B
   $10.0\text{ }V$
3. C
   $5.6\text{ }V$
4. D
   $2.0\text{ }V$

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1. A
   $K = \frac{Y\eta}{9\eta - 3Y}N/m^{2}$
2. B
   $\eta = \frac{3YK}{9\text{ }K + Y}N/m^{2}$
3. C
   $Y = \frac{9K\eta}{3K - \eta}N/m^{2}$
4. D
   $Y = \frac{9K\eta}{2\eta + 3K}N/m^{2}$

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1. A
   $2\pi\sqrt{\frac{d^{3}}{3Gm}}$
2. B
   $\frac{1}{2\pi}\sqrt{\frac{3Gm}{d^{3}}}$
3. C
   $\frac{1}{2\pi}\sqrt{\frac{d^{3}}{3Gm}}$
4. D
   $2\pi\sqrt{\frac{3Gm}{d^{3}}}$

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

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1. A
   Statemnet-I is false but Statement-II is true
2. B
   Both Statement-I and Statement-II are true
3. C
   Both Statement-I and Statement-II are false
4. D
   Statement-I is true but Statement-II is false

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1. A
   $2100C$
2. B
   $260C$
3. C
   $2250C$
4. D
   $11250C$

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