For a first order reaction, $\mathrm { A } \rightarrow \mathrm { P }$, $\mathrm { t } _ { 1/2 }$ (half-life) is 10 days. The time required for $\frac { 1 } { 4 } ^ { \text {th} }$ conversion of A (in days) is: $( \ln 2 = 0.693 , \ln 3 = 1.1 )$.
(1) 3.2
(2) 2.5
(3) 4.1
(4) 5

If $x$ gram of gas is adsorbed by $m$ gram of adsorbent at pressure $P$ the plot of $\log \frac { x } { \mathrm {~m} }$ versus $\log \mathrm { P }$ is linear. The slope of the plot is: (m and k are constants and $n > 1$)
(1) $\log \mathrm { k }$
(2) $\frac { 1 } { n }$
(3) 2 k
(4) $n$

If $( p \wedge \sim q ) \wedge ( p \wedge r ) \rightarrow \sim p \vee q$ is false, then the truth values of $p , q$ and $r$ are respectively
(1) $T , T , T$
(2) $F , T , F$
(3) $T , F , T$
(4) $F , F , F$

If $( \mathrm { p } \wedge \sim \mathrm { q } ) \wedge ( \mathrm { p } \wedge \mathrm { r } ) \rightarrow \sim p \vee q$ is false, then the truth values of $\mathrm { p } , \mathrm { q }$ and r are respectively
(1) $F , T , F$
(2) T,F,T
(3) F, F, F
(4) T, T, T

Consider the following two binary relations on the set $A = \{ a , b , c \} : R _ { 1 } = \{ ( c , a ) , ( b , b ) , ( a , c ) , ( c , c ) , ( b , c ) , ( a , a ) \}$ and $R _ { 2 } = \{ ( a , b ) , ( b , a ) , ( c , c ) , ( c , a ) , ( a , a ) , ( b , b ) , ( a , c ) \}$, then :
(1) $R _ { 2 }$ is symmetric but it is not transitive
(2) both $R _ { 1 }$ and $R _ { 2 }$ are not symmetric
(3) both $R _ { 1 }$ and $R _ { 2 }$ are transitive
(4) $R _ { 1 }$ is not symmetric but it is transitive

Consider the following two binary relations on the set $A = \{ a , b , c \} : R _ { 1 } = \{ ( \mathrm { c } , a ) ( b , b ) , ( \mathrm { a } , c ) , ( c , c ) , ( b , c ) , ( a , a ) \}$ and $\mathrm { R } _ { 2 } = \{ ( \mathrm { a } , \mathrm { b } ) , ( \mathrm { b } , \mathrm { a } ) , ( \mathrm { c } , \mathrm { c } ) , ( \mathrm { c } , \mathrm { a } ) , ( \mathrm { a } , \mathrm { a } ) , ( \mathrm { b } , \mathrm { b } ) , ( \mathrm { a } , \mathrm { c } ) \}$. Then
(1) $R _ { 2 }$ is symmetric but it is not transitive
(2) Both $R _ { 1 }$ and $R _ { 2 }$ are transitive
(3) Both $R _ { 1 }$ and $R _ { 2 }$ are not symmetric
(4) $R _ { 1 }$ is not symmetric but it is transitive

A particle of mass $m$ is moving along a trajectory given by $x = x _ { 0 } + \mathrm { a } \cos \omega _ { 1 } \mathrm { t }$ $y = y _ { 0 } + \mathrm { b } \sin \omega _ { 2 } \mathrm { t }$ The torque, acting on the particle about the origin, at $\mathrm { t } = 0$ is:
(1) $+ \mathrm { m } y _ { 0 } \mathrm { a } \omega _ { 1 } ^ { 2 } \widehat { \mathrm { k } }$
(2) $- \mathrm { m } \left( x _ { 0 } \mathrm {~b} \omega _ { 2 } ^ { 2 } - y _ { 0 } \mathrm { a } \omega _ { 1 } ^ { 2 } \right) \widehat { \mathrm { k } }$
(3) Zero
(4) $\mathrm { m } \left( - x _ { 0 } \mathrm {~b} + y _ { 0 } \mathrm { a } \right) \omega _ { 1 } ^ { 2 } \widehat { \mathrm { k } }$

A spring whose unstretched length is $l$ has a force constant $k$. The spring is cut into two pieces of unstretched lengths $l _ { 1 }$ and $l _ { 2 }$ where, $l _ { 1 } = n l _ { 2 }$ and $n$ is an integer. The ratio $k _ { 1 } / k _ { 2 }$ of the corresponding force constants, $k _ { 1 }$ and $k _ { 2 }$ will be:
(1) $\frac { 1 } { n ^ { 2 } }$
(2) $n ^ { 2 }$
(3) $n$
(4) $\frac { 1 } { n }$

A thin disc of mass M and radius R has mass per unit area $\sigma ( \mathrm { r } ) = \mathrm { kr } ^ { 2 }$ where r is the distance from its centre. Its moment inertia about an axis going through its centre of mass and perpendicular to its plane is:
(1) $\frac { M R ^ { 2 } } { 3 }$
(2) $\frac { M R ^ { 2 } } { 2 }$
(3) $\frac { M R ^ { 2 } } { 6 }$
(4) $\frac { 2 M R ^ { 2 } } { 3 }$

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is:
(1) 16 cm
(2) 14 cm
(3) 12 cm
(4) 18 cm

A person of mass $M$ is sitting on a swing of length $L$ and swinging with an angular amplitude $\theta_0$. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $l$ ($l \ll L$), is close to:
(1) $Mgl\left(1 - \theta_0^2\right)$
(2) $Mgl\left(1 + \frac{\theta_0^2}{2}\right)$
(3) $Mgl$
(4) $Mgl\left(1 + \theta_0^2\right)$

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string)
(1) $20 \mathrm { rad } / \mathrm { s } ^ { 2 }$
(2) $16 \mathrm { rad } / \mathrm { s } ^ { 2 }$
(3) $12 \mathrm { rad } / \mathrm { s } ^ { 2 }$
(4) $10 \mathrm { rad } / \mathrm { s } ^ { 2 }$

Two coaxial discs, having moments of inertia $I _ { 1 }$ and $\frac { I _ { 1 } } { 2 }$, are rotating with respective angular velocities $\omega _ { 1 }$ and $\frac { \omega _ { 1 } } { 2 }$, about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E _ { f }$ and $E _ { i }$ are the final and initial total energies, then $\left( E _ { f } - E _ { i } \right)$ is:
(1) $\frac { I _ { 1 } \omega _ { 1 } ^ { 2 } } { 6 }$
(2) $\frac { 3 } { 8 } I _ { 1 } \omega _ { 1 } ^ { 2 }$
(3) $- \frac { I _ { 1 } \omega _ { 1 } ^ { 2 } } { 12 }$
(4) $- \frac { I _ { 1 } \omega _ { 1 } ^ { 2 } } { 24 }$

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The first part has a mass of $\frac { 7M } { 8 }$ and is converted into uniform disc of radius $2R$. The second part is converted into a uniform solid sphere. Let $I _ { 1 }$ be the moment of inertia of the disc about its axis and $\mathrm { I } _ { 2 }$ be the moment of inertia of the new sphere about its axis. The ratio $I _ { 1 } / I _ { 2 }$ is given by:
(1) 140
(2) 185
(3) 65
(4) 285

An equilateral triangle ABC is cut from a thin solid sheet of wood. D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is $\mathrm { I } _ { 0 }$. If the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is $I$. Then
(1) $I = \frac { 15 } { 16 } I _ { 0 }$
(2) $\mathrm { I } = \frac { 3 } { 4 } \mathrm { I } _ { 0 }$
(3) $I = \frac { 9 } { 16 } I _ { 0 }$
(4) $\mathrm { I } = \frac { \mathrm { I } _ { 0 } } { 4 }$

A satellite of mass M is in a circular orbit of radius R about the center of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastic. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be:
(1) In an elliptical orbit
(2) Such that it escapes to infinity
(3) In a circular orbit of a different radius
(4) In the same circular orbit of radius $R$

A uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T(x)$ at a distance $x$ from the axis, then which of the following graphs depicts it most closely?

A circular disc $D _ { 1 }$ of mass $M$ and radius $R$ has two identical discs $D _ { 2 }$ and $D _ { 3 }$ of the same mass $M$ and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis $\mathrm { OO } ^ { \prime }$, passing through the centre of $D _ { 1 }$, as shown in the figure, will be
(1) $\mathrm { MR } ^ { 2 }$
(2) $3 \mathrm { MR } ^ { 2 }$
(3) $\frac { 4 } { 5 } \mathrm { MR } ^ { 2 }$
(4) $\frac { 2 } { 3 } \mathrm { MR } ^ { 2 }$

The value of acceleration due to gravity at Earth's surface is $9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, is close to: (Radius of earth $= 6.4 \times 10 ^ { 6 } \mathrm {~m}$)
(1) $1.6 \times 10 ^ { 6 } \mathrm {~m}$
(2) $2.6 \times 10 ^ { 6 } \mathrm {~m}$
(3) $6.4 \times 10 ^ { 6 } \mathrm {~m}$
(4) $9.0 \times 10 ^ { 6 } \mathrm {~m}$

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick $AB$ of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s , is close to:
(1) $1.6 \times 10 ^ { - 5 } \mathrm {~N} \mathrm {~m}$
(2) $2.0 \times 10 ^ { - 5 } \mathrm {~N} \mathrm {~m}$
(3) $7.9 \times 10 ^ { - 6 } \mathrm {~N} \mathrm {~m}$
(4) $4.0 \times 10 ^ { - 6 } \mathrm {~N} \mathrm {~m}$

A satellite is revolving in a circular orbit at a height $h$ from the earth surface, such that $h \ll R$ where $R$ is the radius of the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is
(1) $\sqrt { 2gR }$
(2) $\sqrt { gR }$
(3) $\sqrt { \frac { gR } { 2 } }$
(4) $\sqrt { gR } ( \sqrt { 2 } - 1 )$

A straight rod of length $L$ extends from $x = a$ to $x = L + a$. The gravitational force it exerts on a point mass '$m$' at $x = 0$, if the mass per unit length of the rod is $A + B x ^ { 2 }$, is given by:
(1) $G m \left[ A \left( \frac { 1 } { a + L } - \frac { 1 } { a } \right) + B L \right]$
(2) $G m \left[ A \left( \frac { 1 } { a + L } - \frac { 1 } { a } \right) - B L \right]$
(3) $G m \left[ A \left( \frac { 1 } { a } - \frac { 1 } { a + L } \right) - B L \right]$
(4) $G m \left[ A \left( \frac { 1 } { a } - \frac { 1 } { a + L } \right) + B L \right]$

A circular disc of radius $b$ has a hole of radius $a$ at its centre. If the mass per unit area of the disc varies as $\frac{\sigma_0}{r}$ then, the radius of gyration of the disc about its axis passing through the center is
(1) $\frac{a+b}{3}$
(2) $\sqrt{\frac{a^2+b^2+ab}{3}}$
(3) $\frac{a+b}{2}$
(4) $\sqrt{\frac{a^2+b^2+ab}{2}}$

The ratio of the weights of a body on Earth's surface to that on the surface of a planet is $9 : 4$. The mass of the planet is $\frac { 1 } { 9 }$th of that of the Earth. If $R$ is the radius of the Earth, what is the radius of the planet? (Take the planets to have the same mass density)
(1) $\frac { R } { 4 }$
(2) $\frac { R } { 2 }$
(3) $\frac { R } { 3 }$
(4) $\frac { R } { 9 }$

The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is 2 s. The period of oscillation of the same pendulum on the planet would be:
(1) $\frac { \sqrt { 3 } } { 2 } \mathrm {~s}$
(2) $\frac { 2 } { \sqrt { 3 } } \mathrm {~s}$
(3) $\frac { 3 } { 2 } s$
(4) $2 \sqrt { 3 } \mathrm {~s}$