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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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1. A
   (A) - (ii), (B) - (iv), (C) - (iii), (D) - (i)
2. B
   (A) - (ii), (B) - (iii), (C) - (iv), (D) - (i)
3. C
   (A) - (i), (B) - (iii), (C) - (ii), (D) - (iv)
4. D
   (A) - (iii), (B) - (ii), (C) - (i), (D) - (iv)

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1. A
   The series limit of Lyman series, third member of balmer series and second member of paschen series
2. B
   The first member of the Lyman series, third member of Balmer series and second member of paschen series
3. C
   The ionization potential of hydrogen, second member of Balmer series and third member of Paschen series
4. D
   The series limit of Lyman series, second memebr of Balmer series and second member of Paschen series.

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1. A
   $f = r$
2. B
   $f = - \frac{1}{2}r$
3. C
   $f = + \frac{1}{2}r$
4. D
   $f = - r$

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1. A
   $I_{1} = I_{2} = I_{3} < I_{4}$
2. B
   $I_{1} + I_{2} = I_{3} + \frac{5}{2}I_{4}$
3. C
   $I_{1} + I_{3} < I_{2} + I_{4}$
4. D
   $I_{1} = I_{2} = I_{3} > I_{4}$

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1. A
   $\left\lbrack M^{0}{LT}^{0} \right\rbrack$
2. B
   $\left\lbrack M^{2}{LT}^{2} \right\rbrack$
3. C
   $\left\lbrack {MLT}^{- 2} \right\rbrack$
4. D
   $\left\lbrack {ML}^{2}{\text{ }T}^{- 2} \right\rbrack$

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1. A
   7
2. B
   0.875
3. C
   0.5
4. D
   3.5

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

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