An atom absorbs a photon of wavelength 500 nm and emits another photon of wavelength 600 nm. The net energy absorbed by the atom in this process is $n \times 10 ^ { - 4 } \mathrm { eV }$. The value of $n$ is [Assume the atom to be stationary during the absorption and emission process] (Take $h = 6.6 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm {~s}$ and $c = 3 \times 10 ^ { 8 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$).

In an amplitude modulation, a modulating signal having amplitude of $X \mathrm {~V}$ is superimposed with a carrier signal of amplitude $Y \mathrm {~V}$ in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude $2Y \mathrm {~V}$. The ratio of modulation index in the two case respectively will be :
(1) $1 : 2$
(2) $1 : 1$
(3) $2 : 1$
(4) $4 : 1$

The radius of fifth orbit of $\mathrm{Li}^{++}$ is $\_\_\_\_$ $\times 10^{-12}$ m. Take: radius of hydrogen atom $= 0.51\AA$

A monochromatic light is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of light and subsequently emit radiation of six different wavelengths. The frequency of incident light is $x \times 10^{15}$ Hz. The value of $x$ is $\_\_\_\_$ (Given $h = 4.25 \times 10^{-15}$ eVs)

A solution of sugar is obtained by mixing 200 g of its $25\%$ solution and 500 g of its $40\%$ solution (both by mass). The mass percentage of the resulting sugar solution is $\_\_\_\_$ (Nearest integer)

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is
(1) symmetric but neither reflexive nor transitive
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) transitive but neither reflexive nor symmetric

The compound statement $( \sim ( P \wedge Q ) ) \vee ( ( \sim P ) \wedge Q ) \Rightarrow ( ( \sim P ) \wedge ( \sim Q ) )$ is equivalent to
(1) $( ( \sim P ) \vee Q ) \wedge ( ( \sim Q ) \vee P )$
(2) $( \sim Q ) \vee P$
(3) $( ( \sim P ) \vee Q ) \wedge ( \sim Q )$
(4) $( \sim P ) \vee Q$

Consider: S1: $p \Rightarrow q \vee (p \wedge \sim q)$ is a tautology. S2: $\sim p \Rightarrow (\sim q \wedge \sim p) \vee q$ is a contradiction. Then
(1) only S2 is correct
(2) both S1 and S2 are correct
(3) both S1 and S2 are wrong
(4) only S1 is correct

The relation $R = a , b : \operatorname { gcd} a , b = 1 , \quad 2 a \neq b , \quad a , \quad b \in \mathbb { Z }$ is:
(1) transitive but not reflexive
(2) symmetric but not transitive
(3) reflexive but not symmetric
(4) neither symmetric nor transitive

The negation of the expression $q \vee ((\sim q) \wedge p)$ is equivalent to
(1) $(\sim p) \wedge (\sim q)$
(2) $p \wedge (\sim q)$
(3) $(\sim p) \vee (\sim q)$
(4) $(\sim p) \vee q$

$50^{\text{th}}$ root of a number $x$ is 12 and $50^{\text{th}}$ root of another number $y$ is 18. Then the remainder obtained on dividing $(x + y)$ by 25 is $\_\_\_\_$.

Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)\, R\, (c,d)$ if and only if $ad(b-c) = bc(a-d)$. Then $R$ is
(1) symmetric but neither reflexive nor transitive
(2) transitive but neither reflexive nor symmetric
(3) reflexive and symmetric but not transitive
(4) symmetric and transitive but not reflexive

The negation of the statement $(p \vee q) \wedge (q \vee \sim r)$ is
(1) $p \vee r \wedge \sim q$
(2) $(\sim p) \vee r \wedge \sim q$
(3) $\sim p \vee \sim q \vee \sim r$
(4) $\sim p \vee \sim q \wedge \sim r$

Negation of $p \wedge ( q \wedge \sim ( p \wedge q ) )$ is
(1) $( \sim ( p \wedge q ) ) \vee p$
(2) $p \vee q$
(3) $\sim ( p \vee q )$
(4) $( \sim ( p \wedge q ) ) \wedge q$

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

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

Let $R$ be a relation on $\mathbb{R}$, given by $R = \{(a, b) : 3a - 3b + \sqrt{7}$ is an irrational number$\}$. Then $R$ is
(1) Reflexive but neither symmetric nor transitive
(2) Reflexive and transitive but not symmetric
(3) Reflexive and symmetric but not transitive
(4) An equivalence relation

Let $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$. Then the number of elements in the relation $R = \{ ( ( a _ { 1 } , b _ { 1 } ) , ( a _ { 2 } , b _ { 2 } ) ) \in ( A \times B ) \times ( A \times B ) : a _ { 1 }$ divides $b _ { 2 }$ and $a _ { 2 }$ divides $b _ { 1 } \}$ is
(1) 36
(2) 24
(3) 18
(4) 12

Let $f$, $g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x) = \left\{ \begin{array}{cc} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{array} \right.$, $\quad g(x) = \left\{ \begin{array}{cc} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{array} \right.$ and $h(x) = 2[x] - f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is
(1) 1
(2) $\sin(1)$
(3) $-1$
(4) 0

Let $p$ and $q$ be two statements. Then $\sim ( p \wedge ( p \rightarrow \sim q ) )$ is equivalent to
(1) $p \vee ( p \wedge ( \sim q ) )$
(2) $p \vee ( ( \sim p ) \wedge q )$
(3) $( \sim p ) \vee q$
(4) $p \wedge q$

The number of values of $r \in \{p, q, \sim p, \sim q\}$ for which $((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$ is a tautology, is:
(1) 1
(2) 2
(3) 4
(4) 3

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

Consider the following statements: $P$: I have fever $Q$: I will not take medicine $R$: I will take rest The statement ``If I have fever, then I will take medicine and I will take rest'' is equivalent to:
(1) $((\sim P) \vee \sim Q) \wedge ((\sim P) \vee R)$
(2) $((\sim P) \vee \sim Q) \wedge ((\sim P) \vee \sim R)$
(3) $(P \vee Q) \wedge ((\sim P) \vee R)$
(4) $(P \vee \sim Q) \wedge (P \vee \sim R)$

Among the two statements $\left( S _ { 1 } \right) : ( p \Rightarrow q ) \wedge ( p \wedge ( \sim q ) )$ is a contradiction and $\left( S _ { 2 } \right) : ( p \wedge q ) \vee ( ( \sim p ) \wedge q ) \vee ( p \wedge ( \sim q ) ) \vee ( ( \sim p ) \wedge ( \sim q ) )$ is a tautology
(1) only ( $S _ { 2 }$ ) is true
(2) only ( $S _ { 1 }$ ) is true
(3) both are false
(4) both are true

Among the statements $(S1): (p \Rightarrow q) \vee ((\sim p) \wedge q)$ is a tautology $(S2): (q \Rightarrow p) \Rightarrow ((\sim p) \wedge q)$ is a contradiction
(1) Neither $(S1)$ and $(S2)$ is True
(2) Both $(S1)$ and $(S2)$ are True
(3) Only $(S2)$ is True
(4) Only $(S1)$ is True