$\frac{x}{x+4}$ is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its
(i) third permitted energy level to the second level and
(ii) the highest permitted energy level to the second permitted level. The value of $x$ will be

A Zener of breakdown voltage $V_{Z} = 8$ V and maximum Zener current, $I_{\mathrm{ZM}} = 10$ mA is subjected to an input voltage $V_{i} = 10$ V with series resistance $R = 100\ \Omega$. In the given circuit $R_{\mathrm{L}}$ represents the variable load resistance. The ratio of maximum and minimum value of $R_{L}$ is \_\_\_\_.

A potential barrier of 0.4 V exists across a p-n junction. An electron enters the junction from the $n$-side with a speed of $6.0 \times 10^5 \mathrm{~m~s}^{-1}$. The speed with which electron enters the $p$ side will be $\frac{x}{3} \times 10^5 \mathrm{~m~s}^{-1}$, then the value of $x$ is $\_\_\_\_$. (Given mass of electron $= 9 \times 10^{-31}$ kg, charge on electron $= 1.6 \times 10^{-19}$ C.)

A telegraph line of length 100 km has a capacity of $0.01\,\mu\mathrm{F\,km^{-1}}$ and it carries an alternating current at 0.5 kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is $\_\_\_\_$ mH. (If $\pi = \sqrt{10}$)

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

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) $\Leftrightarrow$

The statement $( p \wedge q ) \Rightarrow ( p \wedge r )$ is equivalent to
(1) $q \Rightarrow ( p \wedge r )$
(2) $p \Rightarrow ( p \wedge r )$
(3) $( p \wedge r ) \Rightarrow ( p \wedge q )$
(4) $( p \wedge q ) \Rightarrow r$

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) $q \Rightarrow \sim p \vee q$

The number of choices for $\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \}$, such that $( p \Delta q ) \Rightarrow ( ( p \Delta \sim q ) \vee ( ( \sim p ) \Delta q ) )$ is a tautology, is
(1) 1
(2) 2
(3) 3
(4) 4

Let the operations $*, \odot \in \{\wedge, \vee\}$. If $p * q \odot p \odot {\sim}q$ is a tautology, then the ordered pair $(*, \odot)$ is
(1) $(\vee, \wedge)$
(2) $(\vee, \vee)$
(3) $(\wedge, \wedge)$
(4) $(\wedge, \vee)$

The statement $(p \Rightarrow (q \vee p)) \Rightarrow r$ is NOT equivalent to:
(1) $p \wedge \sim r \Rightarrow q$
(2) $\sim q \Rightarrow (\sim r \vee p)$
(3) $p \Rightarrow (q \vee r)$
(4) $p \wedge \sim q \Rightarrow r$

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) $\sim ( R \vee Q ) \rightarrow \sim P$