The Boolean expression $( p \wedge q ) \Rightarrow ( ( r \wedge q ) \wedge p )$ is equivalent to: (1) $( p \wedge r ) \Rightarrow ( p \wedge q )$ (2) $( q \wedge r ) \Rightarrow ( p \wedge q )$ (3) $( p \wedge q ) \Rightarrow ( r \wedge q )$ (4) $( p \wedge q ) \Rightarrow ( r \vee q )$

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

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) ${ } ^ { * } = \wedge , \square = \wedge$

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 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.

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac { 1 } { 2 }$ and probability of occurrence of 0 at the odd place be $\frac { 1 } { 3 }$. Then the probability that 10 is followed by 01 is equal to:
(1) $\frac { 1 } { 18 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 9 }$

Let $\tan \alpha , \tan \beta$ and $\tan \gamma ; \alpha , \beta , \gamma \neq \frac { ( 2 n - 1 ) \pi } { 2 } , n \in N$ be the slopes of the three line segments $O A , O B$ and $O C$, respectively, where $O$ is origin. If circumcentre of $\Delta A B C$ coincides with origin and its orthocentre lies on $y$-axis, then the value of $\left( \frac { \cos 3 \alpha + \cos 3 \beta + \cos 3 \gamma } { \cos \alpha \cdot \cos \beta \cdot \cos \gamma } \right) ^ { 2 }$ is equal to:

$3 \times 7 ^ { 22 } + 2 \times 10 ^ { 22 } - 44$ when divided by 18 leaves the remainder

Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $x + y + z = 0 , l x - n z = 0$ and $x - 2 y + z = 0$ equal to 9 units. If the locus of the point $P$ is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$, then the value of $l - n$ is equal to

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

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) $\frac{d}{2\sqrt{2}}$

A capacitor is discharging through a resistor $R$. Consider in time $t_1$, the energy stored in the capacitor reduces to half of its initial value and in time $t_2$, the charge stored reduces to one eighth of its initial value. The ratio $\frac{t_1}{t_2}$ will be
(1) $\frac{1}{2}$
(2) $\frac{1}{3}$
(3) $\frac{1}{4}$
(4) $\frac{1}{6}$

The combination of two identical cells, whether connected in series or parallel combination provides the same current through an external resistance of $2\Omega$. The value of internal resistance of each cell is
(1) $2\Omega$
(2) $4\Omega$
(3) $6\Omega$
(4) $8\Omega$

For $z = a^2 x^3 y^{\frac{1}{2}}$, where '$a$' is a constant. If percentage error in measurement of '$x$' and '$y$' are $4\%$ and $12\%$, respectively, then the percentage error for '$z$' will be $\_\_\_\_$ $\%$.

As per the given figure, two plates $A$ and $B$ of thermal conductivity $K$ and $2K$ are joined together to form a compound plate. The thickness of plates are 4.0 cm and 2.5 cm respectively and the area of cross-section is $120 \mathrm {~cm} ^ { 2 }$ for each plate. The equivalent thermal conductivity of the compound plate is $\left( 1 + \frac { 5 } { \alpha } \right) K$, then the value of $\alpha$ will be $\_\_\_\_$ .