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

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) $p \vee ( \sim q )$

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 $\_\_\_\_$ .

$\lim_{n \to \infty} \left(\frac{1}{1+n} + \frac{1}{2+n} + \frac{1}{3+n} + \ldots + \frac{1}{2n}\right)$ is equal to:
(1) 0
(2) $\log_e 2$
(3) $\log_e \frac{3}{2}$
(4) $\log_e \frac{2}{3}$

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 $\_\_\_\_$ .

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is $\_\_\_\_$.

The remainder when $19^{200} + 23^{200}$ is divided by 49, is $\_\_\_\_$.

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) $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle of mass $m$ projected with a velocity $u$ making an angle of $30 ^ { \circ }$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is :
(1) $\frac { \sqrt { 3 } } { 16 } \frac { m u ^ { 3 } } { g }$
(2) $\frac { \sqrt { 3 } } { 2 } \frac { m u ^ { 2 } } { g }$
(3) $\frac { m u ^ { 3 } } { \sqrt { 2 } g }$
(4) zero

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 particle moving in a straight line covers half the distance with speed $6 \mathrm {~m} / \mathrm { s }$. The other half is covered in two equal time intervals with speeds $9 \mathrm {~m} / \mathrm { s }$ and $15 \mathrm {~m} / \mathrm { s }$ respectively. The average speed of the particle during the motion is :
(1) $10 \mathrm {~m} / \mathrm { s }$
(2) $8 \mathrm {~m} / \mathrm { s }$
(3) $9.2 \mathrm {~m} / \mathrm { s }$
(4) $8.8 \mathrm {~m} / \mathrm { s }$

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

All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is:
(1) $g$
(2) $\frac { g } { 3 }$
(3) $\frac { g } { 2 }$
(4) $\frac { g } { 4 }$

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) $\cos ^ { - 1 } { \frac { \pi R } { 2 g T ^ { 2 } } } ^ { \frac { 1 } { 2 } }$

A body travels 102.5 m in $\mathrm { n } ^ { \text {th} }$ second and 115.0 m in $( \mathrm { n } + 2 ) ^ { \text {th} }$ second. The acceleration is :
(1) $6.25 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(2) $12.5 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(3) $9 \mathrm {~m} / \mathrm { s } ^ { 2 }$
(4) $5 \mathrm {~m} / \mathrm { s } ^ { 2 }$

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

A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution?
(1) 25
(2) 50
(3) 100
(4) 20