The statement among the following that is a tautology is:
(1) $A \vee A \wedge B$
(2) $A \wedge A \vee B$
(3) $B \rightarrow A \wedge A \rightarrow B$
(4) $A \wedge A \rightarrow B \rightarrow B$

Let $F _ { 1 } ( A , B , C ) = ( A \wedge \sim B ) \vee [ \sim C \wedge ( A \vee B ) ] \vee \sim A$ and $F _ { 2 } ( A , B ) = ( A \vee B ) \vee ( B \rightarrow \sim A )$ be two logical expressions. Then :
(1) $F _ { 1 }$ is a tautology but $F _ { 2 }$ is not a tautology
(2) $F _ { 1 }$ is not a tautology but $F _ { 2 }$ is a tautology
(3) Both $F _ { 1 }$ and $F _ { 2 }$ are not tautologies
(4) $F _ { 1 }$ and $F _ { 2 }$ both are tautologies

The contrapositive of the statement "If you will work, you will earn money" is:
(1) If you will not earn money, you will not work
(2) To earn money, you need to work
(3) You will earn money, if you will not work
(4) If you will earn money, you will work

If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?
(1) $( ( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow Q$
(2) $( ( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow \sim P$
(3) $( ( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow P$
(4) $( ( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow ( P \wedge Q )$

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

The value of $\lim _ { n \rightarrow \infty } \frac { [ r ] + [ 2 r ] + \ldots + [ n r ] } { n ^ { 2 } }$, where $r$ is non-zero real number and $[ r ]$ denotes the greatest integer less than or equal to $r$, is equal to:
(1) $\frac { r } { 2 }$
(2) $r$
(3) $2 r$
(4) 0

Let ${ } ^ { * } , \square \in \{ \wedge , \vee \}$ be such that the Boolean expression $\left( p ^ { * } \sim q \right) \Rightarrow ( p \square q )$ is a tautology. Then :
(1) ${ } ^ { * } = \vee , \square = \wedge$
(2) $* = \vee , \square = \vee$
(3) ${ } ^ { * } = \wedge , \square = \vee$
(4) ${ } ^ { * } = \wedge , \square = \wedge$

Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $f : A \rightarrow A$ be defined as $$f ( k ) = \left\{ \begin{array} { c l } k + 1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even } \end{array} \right.$$ Then the number of possible functions $g : A \rightarrow A$ such that $g o f = f$ is:
(1) ${ } ^ { 10 } \mathrm { C } _ { 5 }$
(2) $5 ^ { 5 }$
(3) 5 !
(4) $10 ^ { 5 }$

If the Boolean expression $( p \Rightarrow q ) \Leftrightarrow \left( q ^ { * } ( \sim p ) \right)$ is a tautology, then the Boolean expression $p ^ { * } ( \sim q )$ is equivalent to:
(1) $q \Rightarrow p$
(2) $\sim q \Rightarrow p$
(3) $p \Rightarrow \sim q$
(4) $p \Rightarrow q$

The value of the limit $\lim _ { \theta \rightarrow 0 } \frac { \tan \left( \pi \cos ^ { 2 } \theta \right) } { \sin \left( 2 \pi \sin ^ { 2 } \theta \right) }$ is equal to:
(1) $- \frac { 1 } { 2 }$
(2) $- \frac { 1 } { 4 }$
(3) 0
(4) $\frac { 1 } { 4 }$

The number of real roots of the equation $\tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \frac { \pi } { 4 }$ is:
(1) 1
(2) 2
(3) 4
(4) 0

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:
(1) The match will not be played and weather is not good and ground is wet.
(2) If the match will not be played, then either weather is not good or ground is wet.
(3) The match will be played and weather is not good or ground is wet.
(4) The match will not be played or weather is good and ground is not wet.

The statement $A \rightarrow ( B \rightarrow A )$ is equivalent to :
(1) $A \rightarrow ( A \wedge B )$
(2) $A \rightarrow ( A \vee B )$
(3) $A \rightarrow ( A \leftrightarrow B )$
(4) $A \rightarrow ( A \rightarrow B )$

If the Boolean expression $( \mathrm { p } \wedge \mathrm { q } ) \circledast ( \mathrm { p } \otimes \mathrm { q } )$ is a tautology, then $\circledast$ and $\otimes$ are respectively given by
(1) $\rightarrow , \rightarrow$
(2) $\wedge , \vee$
(3) $\vee , \rightarrow$
(4) $\wedge , \rightarrow$

Which of the following is not correct for relation $R$ on the set of real numbers?
(1) $( x , y ) \in \mathrm { R } \Leftrightarrow | x | - | y | \leq 1$ is reflexive but not symmetric.
(2) $( x , y ) \in \mathrm { R } \Leftrightarrow | x - y | \leq 1$ is reflexive and symmetric.
(3) $( x , y ) \in \mathrm { R } \Leftrightarrow 0 < | x - y | \leq 1$ is symmetric and transitive.
(4) $( x , y ) \in \mathrm { R } \Leftrightarrow 0 < | x | - | y | \leq 1$ is not transitive but symmetric.

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as ``$ARB$ iff there exists a non-singular matrix $P$ such that $PAP^{-1} = B$''. Then which of the following is true?

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 2 } { 7 }$
(4) $\frac { 7 } { 10 }$

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be (Take radius of earth $= 6400 \mathrm{~km}$ and $g = 10 \mathrm{~ms}^{-2}$)
(1) 800 km
(2) 1600 km
(3) 2133 km
(4) 4800 km

In the given figure, the block of mass $m$ is dropped from the point $|A|$. The expression for kinetic energy of block when it reaches point $|B|$ is
(1) $mgy_0$
(2) $\frac{1}{2}mgy_0^2$
(3) $\frac{1}{2}mgy^2$
(4) $mg(y - y_0)$

The approximate height from the surface of earth at which the weight of the body becomes $\frac { 1 } { 3 }$ of its weight on the surface of earth is : [Radius of earth $\mathrm { R } = 6400 \mathrm {~km}$ and $\sqrt { 3 } = 1.732$ ]
(1) 3840 km
(2) 4685 km
(3) 2133 km
(4) 4267 km

The time period of a satellite revolving around earth in a given orbit is 7 hours. If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be
(1) 36 hours
(2) 40 hours
(3) 30 hours
(4) 25 hours

A spherical shell of 1 kg mass and radius $R$ is rolling with angular speed $\omega$ on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin $O$ is $\frac { a } { 3 } R ^ { 2 } \omega$. The value of $a$ will be
(1) 2
(2) 3
(3) 5
(4) 4

The escape velocity of a body on a planet $A$ is $12 \mathrm {~km} \mathrm {~s} ^ { - 1 }$. The escape velocity of the body on another planet $B$, whose density is four times and radius is half of the planet $A$, is
(1) $12 \mathrm {~km} \mathrm {~s} ^ { - 1 }$
(2) $24 \mathrm {~km} \mathrm {~s} ^ { - 1 }$
(3) $36 \mathrm {~km} \mathrm {~s} ^ { - 1 }$
(4) $6 \mathrm {~km} \mathrm {~s} ^ { - 1 }$

A wire of length $L$ is hanging from a fixed support. The length changes to $L _ { 1 }$ and $L _ { 2 }$ when masses 1 kg and 2 kg are suspended respectively from its free end. Then the value of $L$ is equal to
(1) $\sqrt { L _ { 1 } L _ { 2 } }$
(2) $\frac { L _ { 1 } + L _ { 2 } } { 2 }$
(3) $2 L _ { 1 } - L _ { 2 }$
(4) $3 L _ { 1 } - L _ { 2 }$

Two point charges $Q$ each are placed at a distance $d$ apart. A third point charge $q$ is placed at a distance $x$ from mid-point on the perpendicular bisector. The value of $x$ at which charge $q$ will experience the maximum Coulomb's force is:
(1) $d$
(2) $\frac{d}{2}$
(3) $\frac{d}{\sqrt{2}}$
(4) $\frac{d}{2\sqrt{2}}$