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

Let $\triangle , \nabla \in \{ \wedge , \vee \}$ be such that $( \mathrm { p } \rightarrow \mathrm { q } ) \triangle ( \mathrm { p } \nabla \mathrm { q } )$ is a tautology. Then
(1) $\triangle = \wedge , \nabla = \vee$
(2) $\triangle = \vee , \nabla = \wedge$
(3) $\triangle = \vee , \nabla = \vee$
(4) $\triangle = \wedge , \nabla = \wedge$

If $p , q$ and $r$ are three propositions, then which of the following combination of truth values of $p , q$ and r makes the logical expression $\{ ( p \vee q ) \wedge ( ( \sim p ) \vee r ) \} \rightarrow ( ( \sim q ) \vee r )$ false ?
(1) $p = \mathrm { T } , q = \mathrm { F } , r = \mathrm { T }$
(2) $p = \mathrm { T } , q = \mathrm { T } , r = \mathrm { F }$
(3) $p = \mathrm { F } , q = \mathrm { T } , r = \mathrm { F }$
(4) $p = \mathrm { T } , q = \mathrm { F } , r = \mathrm { F }$

The statement $B \Rightarrow ( ( \sim A ) \vee B )$ is not equivalent to : (1) $B \Rightarrow ( A \Rightarrow B )$ (2) $A \Rightarrow ( A \Leftrightarrow B )$ (3) $A \Rightarrow ( ( \sim A ) \Rightarrow B )$ (4) $B \Rightarrow ( ( \sim A ) \Rightarrow B )$

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

Among the relations $S = \left\{(a,b) : a, b \in R - \{0\},\ 2 + \frac{a}{b} > 0\right\}$ and $T = \left\{(a,b) : a, b \in R,\ a^2 - b^2 \in Z\right\}$,
(1) $S$ is transitive but $T$ is not
(2) both $S$ and $T$ are symmetric
(3) neither $S$ nor $T$ is transitive
(4) $T$ is symmetric but $S$ is not

The number of relations, on the set $\{ 1,2,3 \}$ containing $( 1,2 )$ and $( 2,3 )$ which are reflexive and transitive but not symmetric, is $\_\_\_\_$ .

Let $R$ be a relation defined on $\mathbb { N }$ as a $R$ b is $2 a + 3 b$ is a multiple of $5 , a , b \in \mathbb { N }$. Then $R$ is (1) not reflexive (2) transitive but not symmetric (3) symmetric but not transitive (4) an equivalence relation

Let $D _ { k } = \left| \begin{array} { c c c } 1 & 2 k & 2 k - 1 \\ n & n ^ { 2 } + n + 2 & n ^ { 2 } \\ n & n ^ { 2 } + n & n ^ { 2 } + n + 2 \end{array} \right|$. If $\sum _ { k = 1 } ^ { n } D _ { k } = 96$, then $n$ is equal to $\_\_\_\_$ .

A fair $n ( n > 1 )$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac { n } { 9 }$, then $n$ is equal to

A physical quantity $Q$ is found to depend on quantities $a , b , c$ by the relation $Q = \frac { a ^ { 4 } b ^ { 3 } } { c ^ { 2 } }$. The percentage error in $a , b$ and $c$ are $3 \% , 4 \%$ and $5 \%$ respectively. Then, the percentage error in $Q$ is:
(1) $66 \%$
(2) $43 \%$
(3) $34 \%$
(4) $14 \%$

The angle between vector $\vec{Q}$ and the resultant of $(2\vec{Q} + 2\vec{P})$ and $(2\vec{Q} - 2\vec{P})$ is:
(1) $\tan^{-1}\frac{(2\vec{Q} - 2\vec{P})}{2\vec{Q} + 2\vec{P}}$
(2) $0^{\circ}$
(3) $\tan^{-1}(\mathrm{P}/\mathrm{Q})$
(4) $\tan^{-1}(2Q/P)$

A particle is moving in a straight line. The variation of position $x$ as a function of time $t$ is given as $x = \left( t ^ { 3 } - 6 t ^ { 2 } + 20 t + 15 \right) \mathrm { m }$. The velocity of the body when its acceleration becomes zero is:
(1) $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as $4.62\mathrm{~s}, 4.632\mathrm{~s}, 4.6\mathrm{~s}$ and $4.64\mathrm{~s}$. The arithmetic mean of these readings in correct significant figure is:
(1) 5 s
(2) 4.623 s
(3) 4.6 s
(4) 4.62 s

A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m making 10 rpm. The tension in the string, when the stone is at the lowest point is (if $\pi ^ { 2 } = 9.8$ and $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) 97 N
(2) 9.8 N
(3) 8.82 N
(4) 17.8 N

A particle moving in a circle of radius $R$ with uniform speed takes time $T$ to complete one revolution. If this particle is projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is equal to $4R$. The angle of projection $\theta$ is then given by :
(1) $\sin ^ { - 1 } \frac { 2 g T ^ { 2 } } { \pi ^ { 2 } R }$
(2) $\sin ^ { - 1 } \frac { \pi ^ { 2 } R } { 2 g T ^ { 2 } }$
(3) $\cos ^ { - 1 } \frac { 2 g T ^ { 2 } } { \pi ^ { 2 } R }$
(4) $\cos ^ { - 1 } { \frac { \pi R } { 2 g T ^ { 2 } } } ^ { \frac { 1 } { 2 } }$

If G be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the $\sqrt{\mathrm{uG}}$:
(1) pressure gradient per unit mass
(2) Gravitational potential
(3) Energy per unit mass
(4) Force per unit mass

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is $\sqrt{8/x}$. The value of $x$ is:
(1) 51
(2) 34
(3) 17
(4) 67

If $R$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $g = \pi ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$, then the length of the second's pendulum at a height $h = 2R$ from the surface of earth will be:
(1) $\frac { 2 } { 9 } \mathrm {~m}$
(2) $\frac { 1 } { 9 } \mathrm {~m}$
(3) $\frac { 4 } { 9 } \mathrm {~m}$
(4) $\frac { 8 } { 9 } \mathrm {~m}$

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :
(1) $\frac { \mathrm { GmM } \pi ^ { 2 } } { \mathrm {~L} ^ { 2 } }$
(2) $\frac { \mathrm { GMm } \pi } { 2 \mathrm {~L} ^ { 2 } }$
(3) 0
(4) $\frac { 2 \mathrm { GmM } \pi } { \mathrm { L } ^ { 2 } }$

A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_1 = 4\mathrm{~s}$. $T_2$ would be its time period if it is brought to a point which is at a height 2R from earth surface. Choose the correct relation [$R =$ radius of earth]:
(1) $2\mathrm{~T}_1 = \mathrm{T}_2$
(2) $2\mathrm{~T}_1 = 3\mathrm{T}_2$
(3) $\mathrm{T}_1 = \mathrm{T}_2$
(4) $3\mathrm{T}_1 = 2\mathrm{T}_2$

A sphere of relative density $\sigma$ and diameter $D$ has concentric cavity of diameter $d$. The ratio of $\frac { D } { d }$, if it just floats on water in a tank is :
(1) $\left( \frac { \sigma - 2 } { \sigma + 2 } \right) ^ { 1 / 3 }$
(2) $\left( \frac { \sigma } { \sigma - 1 } \right) ^ { 1 / 3 }$
(3) $\left( \frac { \sigma - 1 } { \sigma } \right) ^ { 1 / 3 }$
(4) $\left( \frac { \sigma + 1 } { \sigma - 1 } \right) ^ { 1 / 3 }$

A sample of 1 mole gas at temperature $T$ is adiabatically expanded to double its volume. If adiabatic constant for the gas is $\gamma = \frac { 3 } { 2 }$, then the work done by the gas in the process is:
(1) $\frac { R } { T } [ 2 - \sqrt { 2 } ]$
(2) $\frac { T } { R } [ 2 + \sqrt { 2 } ]$
(3) RT $[ 2 - \sqrt { 2 } ]$
(4) $\mathrm { RT } [ 2 + \sqrt { 2 } ]$

The pressure and volume of an ideal gas are related as $P V ^ { \frac { 3 } { 2 } } = K$ (Constant). The work done when the gas is taken from state $A\left(P _ { 1 } , V _ { 1 } , T _ { 1 }\right)$ to state $B\left(P _ { 2 } , V _ { 2 } , T _ { 2 }\right)$ is:
(1) $2 \left( P _ { 1 } V _ { 1 } - P _ { 2 } V _ { 2 } \right)$
(2) $2 \left( P _ { 2 } V _ { 2 } - P _ { 1 } V _ { 1 } \right)$
(3) $2 \sqrt { P _ { 1 } } V _ { 1 } - \sqrt { P _ { 2 } } V _ { 2 }$
(4) $2 P _ { 2 } \sqrt { V _ { 2 } } - P _ { 1 } \sqrt { V _ { 1 } }$

Match List I with List II:
List I
(A) Kinetic energy of planet
(B) Gravitation Potential energy of sun-planet system
(C) Total mechanical energy of planet
(D) Escape energy at the surface of planet for unit mass object
List II (I) $-\mathrm{GMm}/\mathrm{a}$ (II) $\mathrm{GMm}/2\mathrm{a}$ (III) $\frac{\mathrm{Gm}}{\mathrm{r}}$ (IV) $-\mathrm{GMm}/2\mathrm{a}$ (Where $\mathbf{a} =$ radius of planet orbit, $\mathbf{r} =$ radius of planet, $\mathrm{M} =$ mass of Sun, $\mathrm{m} =$ mass of planet) Choose the correct answer from the options given below:
(1) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(2) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(3) (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
(4) (A)-(I), (B)-(IV), (C)-(II), (D)-(III)