If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is $\_\_\_\_$ $\times 10^{-2}\mathrm{~MeV}$. (Given $1\mathrm{~u} = 931\mathrm{~MeV}/\mathrm{c}^2$, atomic mass of helium $= 4.002603\mathrm{~u}$)

The potential for the given half cell at 298 K is $-\ldots\ldots\ldots . . \times 10 ^ { - 2 } \mathrm {~V}$. $2 \mathrm { H } _ { (\mathrm { aq }) } ^ { + } + 2 \mathrm { e } ^ { - } \rightarrow \mathrm { H } _ { 2 } (\mathrm {~g})$ $\mathrm { H } ^ { + } = 1 \mathrm { M } , \mathrm { P } _ { \mathrm { H } _ { 2 } } = 2 \mathrm {~atm}$ (Given $2.303 RT / F = 0.06$ V, $\log 2 = 0.3$)

The ratio of $\frac { {}^{14}\mathrm { C } } { {}^{12}\mathrm { C } }$ in a piece of wood is $\frac { 1 } { 8 }$ part that of atmosphere. If half life of ${}^{ 14 } \mathrm { C }$ is 5730 years, the age of wood sample is $\_\_\_\_$ years.

Let $R$ be a relation on $Z \times Z$ defined by $( a , b ) R ( c , d )$ if and only if $a d - b c$ is divisible by 5 . Then R is
(1) Reflexive and symmetric but not transitive
(2) Reflexive but neither symmetric not transitive
(3) Reflexive, symmetric and transitive
(4) Reflexive and transitive but not symmetric

Let $A = \{ 2,3,6,8,9,11 \}$ and $B = \{ 1,4,5,10,15 \}$. Let $R$ be a relation on $A \times B$ defined by ( $a , b$ ) $R ( c , d )$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is (1) an equivalence relation. (2) reflexive and symmetric but not transitive. (3) transitive but not symmetric. (4) reflexive but not symmetric.

Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in \mathbb{R}$ and $(a,b) R_2 (c,d) \Leftrightarrow a + d = b + c$ for all $a, b, c, d \in \mathbb{N} \times \mathbb{N}$. Then
(1) Only $R_1$ is an equivalence relation
(2) Only $R_2$ is an equivalence relation
(3) $R_1$ and $R_2$ both are equivalence relations
(4) Neither $R_1$ nor $R_2$ is an equivalence relation

Let a relation R on $\mathrm { N } \times N$ be defined as: $\left( x _ { 1 } , y _ { 1 } \right) \mathrm { R } \left( x _ { 2 } , y _ { 2 } \right)$ if and only if $x _ { 1 } \leq x _ { 2 }$ or $y _ { 1 } \leq y _ { 2 }$. Consider the two statements: (I) R is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?
(1) Both (I) and (II) are correct.
(2) Only (II) is correct.
(3) Neither (I) nor (II) is correct.
(4) Only (I) is correct.

Let $S = \{ 1,2,3 , \ldots , 10 \}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = \{ ( A , B ) : A \cap B \neq \phi ; A , B \in M \}$ is:
(1) symmetric and reflexive only
(2) reflexive only
(3) symmetric and transitive only
(4) symmetric only

The function $\mathrm { f } : \mathrm { N } - \{ 1 \} \rightarrow \mathrm { N }$; defined by $f ( \mathrm { n } ) =$ the highest prime factor of $n$, is:
(1) both one-one and onto
(2) one-one only
(3) onto only
(4) neither one-one nor onto

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { 72 ^ { x } - 9 ^ { x } - 8 ^ { x } + 1 } { \sqrt { 2 } - \sqrt { 1 + \cos x } } , & x \neq 0 \\ a \log _ { e } 2 \log _ { e } 3 & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then the value of $a ^ { 2 }$ is equal to
(1) 968
(2) 1152
(3) 746
(4) 1250

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that $\vec { b }$ and $\vec { c }$ are non-collinear if $\vec { a } + 5 \vec { b }$ is collinear with $\overrightarrow { c , b } + 6 \overrightarrow { c c }$ is collinear with $\vec { a }$ and $\vec { a } + \alpha \vec { b } + \beta \vec { c } = \overrightarrow { 0 }$, then $\alpha + \beta$ is equal to
(1) 35
(2) 30
(3) - 30
(4) - 25

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is $\underline{\hspace{1cm}}$.

Q1. In an experiment to measure focal length $( f )$ of convex lens, the least counts of the measuring scales for the position of object $( \mathrm { u } )$ and for the position of image $( \mathrm { v } )$ are $\Delta \mathrm { u }$ and $\Delta \mathrm { v }$, respectively. The error in the measurement of the focal length of the convex lens will be:
(1) $2 f \left[ \frac { \Delta \mathrm { u } } { \mathrm { u } } + \frac { \Delta \mathrm { v } } { \mathrm { v } } \right]$
(2) $\frac { \Delta u } { u } + \frac { \Delta v } { v }$
(3) $f ^ { 2 } \left[ \frac { \Delta \mathrm { u } } { \mathrm { u } ^ { 2 } } + \frac { \Delta \mathrm { v } } { \mathrm { v } ^ { 2 } } \right]$
(4) $f \left[ \frac { \Delta \mathrm { u } } { \mathrm { u } } + \frac { \Delta \mathrm { v } } { \mathrm { v } } \right]$

Q1. Given below are two statements : Statement (I) : Dimensions of specific heat is $\left[ \mathrm { L } ^ { 2 } \mathrm {~T} ^ { - 2 } \mathrm {~K} ^ { - 1 } \right]$. Statement (II) : Dimensions of gas constant is $\left[ \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 1 } \mathrm {~K} ^ { - 1 } \right]$. In the light of the above statements, choose the most appropriate answer from the options given below.
(1) Both statement (I) and statement (II) are correct
(2) Statement (I) is correct but statement (II) is incorrect
(3) Both statement (I) and statement (II) are incorrect
(4) Statement (I) is incorrect but statement (II) is Statement (I) is incorrect but statement (II) is correct correct

Q1. In an expression $a \times 10 ^ { \mathrm { b } }$;
(1) $b$ is order of magnitude for $a \geq 5$
(2) $b$ is order of magnitude for $a \leq 5$
(3) $a$ is order of magnitude for $b \leq 5$
(4) $b$ is order of magnitude for $5 < a \leq 10$

Q1. If $\epsilon _ { 0 }$ is the permittivity of free space and E is the electric field, then $\epsilon _ { 0 } \mathrm { E } ^ { 2 }$ has the dimensions :
(1) $\left[ \mathrm { M } ^ { - 1 } \mathrm {~L} ^ { - 3 } \mathrm {~T} ^ { 4 } \mathrm {~A} ^ { 2 } \right]$
(2) $\left[ \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 } \right]$
(3) $\left[ \mathrm { M } ^ { \circ } \mathrm { L } ^ { - 2 } \mathrm { TA } \right]$
(4) $\left[ \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 } \right]$

Q1. The dimensional formula of latent heat is :
(1) $\left[ \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 } \right]$
(2) $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 2 } \mathrm {~T} ^ { - 2 } \right]$
(3) $\left[ \mathrm { MLT } ^ { - 2 } \right]$
(4) $\left[ \mathrm { M } ^ { \circ } \mathrm { LT } ^ { - 2 } \right]$

Q1. The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h / \sqrt { 2 m E }$. The dimensional formula for Planck's constant is :
(1) $\left[ \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 1 } \right]$
(2) $\left[ \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 } \right]$
(3) $\left[ \mathrm { MLT } ^ { - 2 } \right]$
(4) $\left[ \mathrm { M } ^ { 2 } \mathrm {~L} ^ { 2 } \mathrm {~T} ^ { - 2 } \right]$

Q2. The equation of stationary wave is : $y = 2 a \sin \left( \frac { 2 \pi n t } { \lambda } \right) \cos \left( \frac { 2 \pi x } { \lambda } \right)$. Which of the following is NOT correct :
(1) The dimensions of $n / \lambda$ is $[ T ]$
(2) The dimensions of n is $\left[ \mathrm { LT } ^ { - 1 } \right]$
(3) The dimensions of $x$ is [L] [0pt] (4) The dimensions of nt is [L]

Q2. 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 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

Q2. Young's modulus is determined by the equation given by $\mathrm { Y } = 49000 \frac { \mathrm {~m} } { 1 } \frac { \mathrm { dyn } } { \mathrm { cm } ^ { 2 } }$ where $M$ is the mass and $l$ is the extension of wire used in the experiment. Now error in Young modules $( Y )$ is estimated by taking data from $M - l$ plot in graph paper. The smallest scale divisions are 5 g and 0.02 cm along load axis and extension axis respectively. If the value of $M$ and $l$ are 500 g and 2 cm respectively then percentage error of $Y$ is :
(1) $0.5 \%$
(2) $2 \%$
(3) $0.02 \%$
(4) $0.2 \%$

Q3. 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

Q3. A clock has $75 \mathrm {~cm} , 60 \mathrm {~cm}$ long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel $x$ distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $\pi = 3.14$ ) :
(1) 140.5
(2) 118.9
(3) 139.4
(4) 220.0

Q4. A stationary particle breaks into two parts of masses $m _ { A }$ and $m _ { B }$ which move with velocities $v _ { A }$ and $v _ { B }$ respectively. The ratio of their kinetic energies ( $K _ { B } : K _ { A }$ ) is :
(1) $v _ { B } : v _ { A }$
(2) $m _ { B } : m _ { A }$
(3) $m _ { B } v _ { B } : m _ { A } v _ { A }$
(4) $1 : 1$

Q4. A thin circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. If another disc of same dimensions but of mass $\mathrm { M } / 2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :
(1) $\frac { 3 } { 2 } \omega$
(2) $\frac { 5 } { 4 } \omega$
(3) $\frac { 2 } { 3 } \omega$
(4) $\frac { 4 } { 5 } \omega$
Q5. [Figure]
A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :
(1) $\sqrt { 6 } \mathrm {~m}$
(2) $\sqrt { 5 } \mathrm {~m}$
(3) 1 m
(4) 2 m