1. A
   The relation is an equivalent relation
2. B
   The relation is not reflexive but it is symmetric and transitive
3. C
   The relation is not symmetric but it is reflexive and transitive
4. D
   The relation is not transitive but it is reflexive and symmetric

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1. A
   45
2. B
   120
3. C
   165
4. D
   330

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1. A
   0
2. B
   1
3. C
   $i$
4. D
   $-i$

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1. A
   $(-3, -2] \cup [2, 3)$
2. B
   $(-3, 3)$
3. C
   $[-3, -2] \cup [2, 3]$
4. D
   None of the above

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1. A
   $p^2 m = l^2 q$
2. B
   $m^2 p = l^2 q$
3. C
   $m^2 p - q^2 l$
4. D
   $m^2 p^2 = l^2 q$

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1. A
   $1$
2. B
   $-1$
3. C
   $i$
4. D
   $-i$

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1. A
   1233
2. B
   1322
3. C
   1323
4. D
   1332

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1. A
   $\dfrac{1}{3}\left[n - \dfrac{1}{9}\left(1 - \dfrac{1}{10^n}\right)\right]$
2. B
   $\dfrac{1}{3}\left[n - \dfrac{2}{9}\left(1 - \dfrac{1}{10^n}\right)\right]$
3. C
   $\dfrac{1}{3}\left[n - \dfrac{1}{3}\left(1 - \dfrac{1}{10^n}\right)\right]$
4. D
   $\dfrac{1}{3}\left[n - \dfrac{1}{9}\left(1 + \dfrac{1}{10^n}\right)\right]$

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

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1. A
   $mn$
2. B
   $m+n$
3. C
   $2(m+n)$
4. D
   $-(m+n)$

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

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1. A
   Both the roots are real
2. B
   One root is real and the other is complex
3. C
   Both the roots are complex
4. D
   Cannot say

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1. A
   0
2. B
   4
3. C
   6
4. D
   10

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

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1. A
   $\dfrac{c-1}{b}$
2. B
   $\dfrac{1-c}{b}$
3. C
   $\dfrac{b}{c-1}$
4. D
   $\dfrac{b}{1-c}$

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1. A
   AP
2. B
   GP
3. C
   HP
4. D
   None of the above

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1. A
   AP
2. B
   GP
3. C
   HP
4. D
   None of the above

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1. A
   254
2. B
   255
3. C
   256
4. D
   257

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1. A
   5200
2. B
   4320
3. C
   3000
4. D
   2160

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1. A
   5
2. B
   7
3. C
   8
4. D
   15

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1. A
   2475
2. B
   2530
3. C
   4905
4. D
   5049

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1. A
   $2^n - n - 1$
2. B
   $1 - 2^{-n}$
3. C
   $2^{-n}+n-1$
4. D
   $2^n-1$

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

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

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1. A
   $\{-1\}$
2. B
   $\{0\}$
3. C
   $\{1\}$
4. D
   an empty set

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