Four resistances of $15 \Omega , 12 \Omega , 4 \Omega$ and $10 \Omega$ respectively in cyclic order to form Wheatstone's network. The resistance that is to be connected in parallel with the resistance of $10 \Omega$ to balance the network is $\_\_\_\_$ $\Omega$.

A beam of electromagnetic radiation of intensity $6.4 \times 10 ^ { - 5 } \mathrm {~W} / \mathrm { cm } ^ { 2 }$ is comprised of wavelength, $\lambda = 310 \mathrm {~nm}$. It falls normally on a metal (work function $\varphi = 2 e V$) of surface area of $1 \mathrm {~cm} ^ { 2 }$. If one in $10 ^ { 3 }$ photons ejects an electron, total number of electrons ejected in 1 s is $10 ^ { x }$. ($h c = 1240 \mathrm { eVnm } , 1 \mathrm { eV } = 1.6 \times 10 ^ { - 19 } J$), then $x$ is $\_\_\_\_$

The balancing length for a cell is 560 cm in a potentiometer experiment. When an external resistance of $10 \Omega$ is connected in parallel to the cell, the balancing length changes by 60 cm. If the internal resistance of the cell is $\frac { \mathrm { n } } { 10 } \Omega$, where n is an integer then value of n is $\_\_\_\_$

Orange light of wavelength $6000 \times 10 ^ { - 10 }$ m illuminates a single slit of width $0.6 \times 10 ^ { - 4 }$ m. The maximum possible number of diffraction minima produced on both sides of the central maximum is $\_\_\_\_$

The surface of a metal is illuminated alternately with photons of energies $E_1 = 4\,\mathrm{eV}$ and $E_2 = 2.5\,\mathrm{eV}$ respectively. The ratio of maximum speeds of the photoelectrons emitted in the two cases is 2. The work function of the metal in (eV) is $\_\_\_\_$

A point object in air is in front of the curved surface of a plano-convex lens. The radius of curvature of the curved surface is 30 cm and the refractive index of the lens material is 1.5, then the focal length of the lens (in cm) is $\_\_\_\_$.

The shortest wavelength of $H$ atom in the Lyman series is $\lambda _ { 1 }$. The longest wavelength in the Balmer series of $\mathrm { He } ^ { + }$ is:
(1) $\frac { 36 \lambda _ { 1 } } { 5 }$
(2) $\frac { 5 \lambda _ { 1 } } { 9 }$
(3) $\frac { 9 \lambda _ { 1 } } { 5 }$
(4) $\frac { 27 \lambda _ { 1 } } { 5 }$

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

The contrapositive of the statement ``If I reach the station in time, then I will catch the train'' is
(1) If I do not reach the station in time, then I will catch the train.
(2) If I do not reach the station in time, then I will not catch the train.
(3) If I will catch the train, then I reach the station in time.
(4) If I will not catch the train, then I do not reach the station in time.

For two statements $p$ and $q$, the logical statement $(p \rightarrow q) \wedge (q \rightarrow \sim p)$ is equivalent to
(1) $p$
(2) $q$
(3) $\sim p$
(4) $\sim q$

Negation of the statement: $\sqrt { 5 }$ is an integer or 5 is irrational is:
(1) $\sqrt { 5 }$ is not an integer 5 is not irrational
(2) $\sqrt { 5 }$ is not an integer and 5 is not irrational
(3) $\sqrt { 5 }$ is irrational or 5 is an integer
(4) $\sqrt { 5 }$ is an integer and 5 irrational

Let $[ t ]$ denote the greatest integer $\leq t$. If $\lambda \varepsilon R - \{ 0,1 \} , \quad \lim _ { x \rightarrow 0 } \left| \frac { 1 - x + | x | } { \lambda - x + [ x ] } \right| = L$, then $L$ is equal to
(1) 1
(2) 2
(3) $\frac { 1 } { 2 }$
(4) 0

Contrapositive of the statement: 'If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is
(1) If a function $f$ is continuous at $a$, then it is not differentiable at $a$.
(2) If a function $f$ is not continuous at $a$, then it is not differentiable at $a$.
(3) If a function $f$ is not continuous at $a$, then it is differentiable at $a$.
(4) If a function $f$ is continuous at $a$, then it is differentiable at $a$.

If $p \rightarrow ( p \wedge \sim q )$ is false, then the truth values of $p$ and $q$ are respectively
(1) $F , F$
(2) $T , F$
(3) $T , T$
(4) $F , T$

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

Let $p , q , r$ be three statements such that the truth value of $( p \wedge q ) \rightarrow ( \sim q \vee r )$ is $F$. Then the truth values of $p , q , r$ are respectively :
(1) $T , T , F$
(2) $T , T , T$
(3) $T , F , T$
(4) $F , T , F$

Given the following two statements: $\left( \mathrm { S } _ { 1 } \right) : ( \mathrm { q } \vee \mathrm { p } ) \rightarrow ( \mathrm { p } \leftrightarrow \sim \mathrm { q } )$ is a tautology $\left( \mathrm { S } _ { 2 } \right) : \sim \mathrm { q } \wedge ( \sim \mathrm { p } \leftrightarrow \mathrm { q } )$ is a fallacy. Then :
(1) both ( $S _ { 1 }$ ) and ( $S _ { 2 }$ ) are not correct.
(2) only ( $S _ { 1 }$ ) is correct.
(3) only ( $S _ { 2 }$ ) is correct.
(4) both $\left( S _ { 1 } \right)$ and $\left( S _ { 2 } \right)$ are correct.

The negation of the Boolean expression $x \leftrightarrow \sim y$ is equivalent to:
(1) $( \sim x \wedge y ) \vee ( \sim x \wedge \sim y )$
(2) $( x \wedge y ) \vee ( \sim x \wedge \sim y )$
(3) $( x \wedge \sim y ) \vee ( \sim x \wedge y )$
(4) $( x \wedge y ) \wedge ( \sim x \vee \sim y )$

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

Let $A , B , C$ and $D$ be four non-empty sets. The contrapositive statement of "If $A \subseteq B$ and $B \subseteq D$, then $A \subseteq C$" is
(1) If $A \nsubseteq C$, then $A \subseteq B$ and $B \subseteq D$
(2) If $A \subseteq C$, then $B \subset A$ and $D \subset B$
(3) If $A \nsubseteq C$, then $A \nsubseteq B$ and $B \subseteq D$
(4) If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$

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

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

Let $[ t ]$ denote the greatest integer $\leq t$ and $\lim _ { x \rightarrow 0 } x \left[ \frac { 4 } { x } \right] = A$. Then the function, $f ( x ) = \left[ x ^ { 2 } \right] \sin ( \pi x )$ is discontinuous, when $x$ is equal to:
(1) $\sqrt { A + 1 }$
(2) $\sqrt { A + 5 }$
(3) $\sqrt { A + 21 }$
(4) $\sqrt { A }$

If $f ( x + y ) = f ( x ) f ( y )$ and $\sum_{x=1}^{n} f(x) = 2$, then the value of $\sum_{x=1}^{n} f(x)$ is given. [Content truncated in source]

The angle between vector $(\vec{A})$ and $(\vec{A} - \vec{B})$ is :
(1) $\tan^{-1}\left(\frac{B\cos\theta}{A - B\sin\theta}\right)$
(2) $\tan^{-1}\left(\frac{\sqrt{3}B}{2A - B}\right)$
(3) $\tan^{-1}\left(\frac{-\frac{B}{2}}{A - B\frac{\sqrt{3}}{2}}\right)$
(4) $\tan^{-1}\left(\frac{A}{0.7B}\right)$