A transistor is used in an amplifier circuit in common emitter mode. If the base current changes by $100 \mu \mathrm {~A}$, it brings a change of 10 mA in collector current. If the load resistance is $2 \mathrm { k } \Omega$ and input resistance is $1 \mathrm { k } \Omega$, the value of power gain is $x \times 10 ^ { 4 }$. The value of $x$ is $\_\_\_\_$ .

A student in the laboratory measures thickness of a wire using screw gauge. The readings are $1.22$ mm, $1.23$ mm, $1.19$ mm and $1.20$ mm. The percentage error is $\frac{x}{121}\%$. The value of $x$ is \_\_\_\_.

The Vernier constant of Vernier callipers is 0.1 mm and it has zero error of $(-0.05$ cm$)$. While measuring diameter of a sphere, the main scale reading is 1.7 cm and coinciding vernier division is 5. The corrected diameter will be $\_\_\_\_$ $\times 10^{-2}$ cm.

Using the rules for significant figures, the correct answer for the expression $\frac{0.02858 \times 0.112}{0.5702}$ will be:
(1) 0.005613
(2) 0.00561
(3) 0.0056
(4) 0.006

The remainder when $( 2021 ) ^ { 2023 }$ is divided by 7 is
(1) 2
(2) 3
(3) 4
(4) 5

The remainder when $( 2021 ) ^ { 2022 } + ( 2022 ) ^ { 2021 }$ is divided by 7 is
(1) 0
(2) 1
(3) 2
(4) 6

The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is
(1) 0
(2) 2
(3) 3
(4) 4

$\sum_{r=1}^{20} (r^2 + 1) \cdot r!$ is equal to
(1) $22! - 21!$
(2) $22! - 2 \cdot 21!$
(3) $21! - 2 \cdot 20!$
(4) $21! - 20!$

The number of elements in the set $S = \left\{x \in \mathbb{R} : 2\cos\left(\frac{x^2 + x}{6}\right) = 4^x + 4^{-x}\right\}$ is
(1) 1
(2) 3
(3) 0
(4) infinite

Let $\boldsymbol { p }$ : Ramesh listens to music. $\boldsymbol { q }$ : Ramesh is out of his village $\boldsymbol { r }$ : It is Sunday $\boldsymbol { s }$ : It is Saturday Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as
(1) $( ( \sim q ) \wedge ( r \vee s ) ) \Rightarrow p$
(2) $( q \wedge ( r \vee s ) ) \Rightarrow p$
(3) $p \Rightarrow ( q \wedge ( r \vee s ) )$
(4) $p \Rightarrow ( ( \sim q ) \wedge ( r \vee s ) )$

Let $p, q, r$ be three logical statements. Consider the compound statements $S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$ and $S_2 : p \rightarrow (q \vee r)$ Then, which of the following is NOT true?
(1) If $S_2$ is True, then $S_1$ is True
(2) If $S_2$ is False, then $S_1$ is False
(3) If $S_2$ is False, then $S_1$ is True
(4) If $S_1$ is False, then $S_2$ is False

If $\lim _ { x \rightarrow 0 } \frac { \alpha e ^ { x } + \beta e ^ { - x } + \gamma \sin x } { x \sin ^ { 2 } x } = \frac { 2 } { 3 }$, where $\alpha , \beta , \gamma \in R$, then which of the following is NOT correct?
(1) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6$
(2) $\alpha \beta + \beta \gamma + \gamma \alpha + 1 = 0$
(3) $\alpha \beta ^ { 2 } + \beta \gamma ^ { 2 } + \gamma \alpha ^ { 2 } + 3 = 0$
(4) $\alpha ^ { 2 } - \beta ^ { 2 } + \gamma ^ { 2 } = 4$

The statement $( \sim ( p \Leftrightarrow \sim q ) ) \wedge q$ is:
(1) a tautology
(2) a contradiction
(3) equivalent to $( p \Rightarrow q ) \wedge q$
(4) equivalent to $( p \Rightarrow q ) \wedge p$

Consider the following statements: $A$ : Rishi is a judge. $B$ : Rishi is honest. $C$ : Rishi is not arrogant. The negation of the statement ``if Rishi is a judge and he is not arrogant, then he is honest'' is
(1) $B \rightarrow ( A \vee C )$
(2) $( \sim B ) \wedge ( A \wedge C )$
(3) $B \rightarrow ( ( \sim A ) \vee ( \sim C ) )$
(4) $B \rightarrow ( A \wedge C )$

Consider the following two propositions : $P _ { 1 } : \sim p \rightarrow \sim q$ $P _ { 2 } : p \wedge \sim q \wedge ( \sim p \vee q )$ If the proposition $p \rightarrow ( \sim p \vee q )$ is evaluated as FALSE, then
(1) $P _ { 1 }$ is TRUE and $P _ { 2 }$ is FALSE
(2) $P _ { 1 }$ is FALSE and $P _ { 2 }$ is TRUE
(3) Both $P _ { 1 }$ and $P _ { 2 }$ are FALSE
(4) Both $P _ { 1 }$ and $P _ { 2 }$ are TRUE

Let $r \in ( P , q , \sim p , \sim q )$ be such that the logical statement $r \vee ( \sim p ) \Rightarrow ( p \wedge q ) \vee r$ is a tautology. Then $r$ is equal to
(1) $p$
(2) $q$
(3) $\sim p$
(4) $\sim q$

Let $\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \}$ be such that $( p \wedge q ) \Delta ( ( p \vee q ) \Rightarrow q )$ is a tautology. Then $\Delta$ is equal to
(1) $\wedge$
(2) $\vee$
(3) $\Rightarrow$
(4) $\Leftrightarrow$

Let $\Delta , \nabla \in \{ \wedge , \vee \}$ be such that $p \nabla q \rightarrow ( ( p \Delta q ) \nabla r )$ is a tautology. Then $( p \nabla q ) \Delta r$ is logically equivalent to
(1) $( p \Delta r ) \vee q$
(2) $( p \Delta r ) \wedge q$
(3) $( p \wedge r ) \Delta q$
(4) $( p \nabla r ) \wedge q$

Which of the following statements is a tautology?
(1) $\sim p \vee q \Rightarrow p$
(2) $p \Rightarrow \sim p \vee q$
(3) $\sim p \vee q \Rightarrow q$
(4) $q \Rightarrow \sim p \vee q$

Which of the following statement is a tautology?
(1) $( ( \sim q ) \wedge p ) \wedge q$
(2) $( ( \sim q ) \wedge p ) \wedge ( p \wedge ( \sim p ) )$
(3) $( ( \sim q ) \wedge p ) \vee ( p \vee ( \sim p ) )$
(4) $( p \wedge q ) \wedge ( \sim ( p \wedge q ) )$

$( p \wedge r ) \Leftrightarrow ( p \wedge ( \sim q ) )$ is equivalent to $( \sim p )$ when $r$ is
(1) $p$
(2) $\sim p$
(3) $q$
(4) $\sim q$

If the truth value of the statement $( P \wedge ( \sim R ) ) \rightarrow ( ( \sim R ) \wedge Q )$ is $F$, then the truth value of which of the following is $F$ ?
(1) $P \vee Q \rightarrow \sim R$
(2) $R \vee Q \rightarrow \sim P$
(3) $\sim ( P \vee Q ) \rightarrow \sim R$
(4) $\sim ( R \vee Q ) \rightarrow \sim P$

Let $R _ { 1 }$ and $R _ { 2 }$ be two relations defined on $\mathbb { R }$ by $a \mathrm { R } _ { 1 } b \Leftrightarrow a b \geq 0$ and $a R _ { 2 } b \Leftrightarrow a \geq b$, then
(1) $R _ { 1 }$ is an equivalence relation but not $R _ { 2 }$
(2) $R _ { 2 }$ is an equivalence relation but not $R _ { 1 }$
(3) both $R _ { 1 }$ and $R _ { 2 }$ are equivalence relations
(4) neither $R _ { 1 }$ nor $R _ { 2 }$ is an equivalence relation

Let $f , g : \mathbb { N } - \{ 1 \} \rightarrow \mathbb { N }$ be functions defined by $f ( \mathrm { a } ) = \alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p ^ { \alpha }$ divides $a$, and $g ( a ) = a + 1$, for all $a \in \mathbb { N } - \{ 1 \}$. Then, the function $f + g$ is
(1) one-one but not onto
(2) onto but not one-one
(3) both one-one and onto
(4) neither one-one nor onto

Let $A = \sum _ { i = 1 } ^ { 10 } \sum _ { j = 1 } ^ { 10 } \min \{ i , j \}$ and $B = \sum _ { i = 1 } ^ { 10 } \sum _ { j = 1 } ^ { 10 } \max \{ i , j \}$. Then $A + B$ is equal to $\_\_\_\_$.