Suppose $ABCD$ is a parallelogram and $P, Q$ are points on the sides $BC$ and $CD$ respectively, such that $PB = \alpha BC$ and $DQ = \beta DC$. If the area of the triangles $ABP$, $ADQ$, $PCQ$ are 15, 15 and 4 respectively, then the area of $APQ$ is
(a) 14
(b) 15
(c) 16
(d) 18.

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance '$a$' from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$, is
(A) $\frac{F}{2m}\frac{a}{\sqrt{a^2-x^2}}$
(B) $\frac{F}{2m}\frac{x}{\sqrt{a^2-x^2}}$
(C) $\frac{F}{2m}\frac{x}{a}$
(D) $\frac{F}{2m}\frac{\sqrt{a^2-x^2}}{x}$

40. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of $0.3 \mathrm {~m} / \mathrm { s } ^ { 2 }$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is $( P / 10 )$. The value of $P$ is
[Figure]
ANSWER: 4

1. A block is moving on an inclined plane making an angle $45 ^ { \circ }$ with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define $\mathrm { N } = 10 \mu$, then N is

ANSWER: 5

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?
(1) moment of inertia
(2) angular momentum
(3) angular velocity
(4) rotational kinetic energy.

One solid sphere $A$ and another hollow sphere $B$ are of same mass and same outer radii. Their moment of inertia about their diameters are respectively $\mathrm { I } _ { \mathrm { A } }$ and $\mathrm { I } _ { \mathrm { B } }$ such that
(1) $I _ { A } = I _ { B }$
(2) $I _ { A } > I _ { B }$
(3) $I _ { A } < I _ { B }$
(4) $I _ { A } / I _ { B } = d _ { A } / d _ { B }$ Where $d _ { A }$ and $d _ { B }$ are their densities.

$A$ and $B$ are two like parallel forces. A couple of moment H lies in the plane of $A$ and $B$ and is contained with them. The resultant of $A$ and $B$ after combining is displaced through a distance
(1) $\frac{2\mathrm{H}}{\mathrm{A}-\mathrm{B}}$
(2) $\frac{H}{A+B}$
(3) $\frac{H}{2(A+B)}$
(4) $\frac{H}{A-B}$

A thin circular disk is in the $x y$ plane as shown in the figure. The ratio of its moment of inertia about $z$ and $z'$ axes will be:
(1) $1 : 4$
(2) $1 : 5$
(3) $1 : 3$
(4) $1 : 2$

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians):
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 8 }$
(4) $\frac { \pi } { 4 }$

Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is
(1) $\frac { 209 } { 15 } M R ^ { 2 }$.
(2) $\frac { 152 } { 15 } M R ^ { 2 }$.
(3) $\frac { 137 } { 15 } M R ^ { 2 }$.
(4) $\frac { 17 } { 5 } M R ^ { 2 }$.

Two masses $m$ and $\frac { m } { 2 }$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of torsional constant $k$, the restoring torque is $\tau = k \theta$ for angular displacement $\theta$. If the rod is rotated by $\theta _ { 0 }$ and released, the tension in it when it passes through its mean position will be:
(1) $k \theta _ { 0 } ^ { 2 }$
(2) $\frac { 3 k \theta _ { 0 } { } ^ { 2 } } { l _ { 0 } }$
(3) $\frac { 2 k \theta _ { 0 } { } ^ { 2 } } { l }$
(4) $\frac { k \theta _ { 0 } { } ^ { 2 } } { l }$

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be
(1) $\frac { g } { 2l }$
(2) $\frac { 7g } { 3l }$
(3) $\frac { g } { 3l }$
(4) $\frac { g } { 13l }$

Consider two uniform discs of the same thickness and different radii $R _ { 1 } = R$ and $R _ { 2 } = \alpha R$ made of the same material. If the ratio of their moments of inertia $I _ { 1 }$ and $I _ { 2 }$, respectively, about their axes is $I _ { 1 } : I _ { 2 } = 1 : 16$ then the value of $\alpha$ is:
(1) $2 \sqrt { 2 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 4

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and $\mathrm { O } ^ { \prime }$ (corner point) is:
(1) $2/3$
(2) $1/4$
(3) $1/8$
(4) $1/2$

$ABC$ is a plane lamina of the shape of an equilateral triangle. $D , E$ are mid-points of $AB , AC$ and $G$ is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I _ { 0 }$. If part $ADE$ is removed, the moment of inertia of the remaining part about the same axis is $\frac { N I _ { 0 } } { 16 }$ where $N$ is an integer. Value of $N$ is:

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is $\mu = 0.4$, the maximum possible value of $100 \times \frac { b } { a }$ for a box not to topple before moving is $\_\_\_\_$

A triangular plate is shown. A force $\vec { F } = 4 \hat { \mathrm { i } } - 3 \hat { \mathrm { j } }$ is applied at point $P$. The torque at point $P$ with respect to point $O$ and $Q$ are:
(1) $- 15 - 20 \sqrt { 3 } , 15 - 20 \sqrt { 3 }$
(2) $15 + 20 \sqrt { 3 } , 15 - 20 \sqrt { 3 }$
(3) $15 - 20 \sqrt { 3 } , 15 + 20 \sqrt { 3 }$
(4) $- 15 + 20 \sqrt { 3 } , 15 + 20 \sqrt { 3 }$

A force $\vec { F } = 4 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 4 \widehat { \mathrm { k } }$ is applied on an intersection point of $x = 2$ plane and $x$-axis. The magnitude of torque of this force about a point $( 2,3,4 )$ is $\_\_\_\_$. (Round off to the Nearest Integer)

A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in the vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of the rod when it passes through its lowest position is $\_\_\_\_$ $\mathrm { m } \mathrm { s } ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be $x \times 10 ^ { - 2 } \mathrm {~kg}$. The value of $x$ is $\_\_\_\_$ .

A force of $- P \hat { k }$ acts on the origin of the coordinate system. The torque about the point $( 2 , - 3 )$ is $P ( a \hat { i } + b \hat { j } )$ , The ratio of $\frac { a } { b }$ is $\frac { x } { 2 }$. The value of $x$ is

A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :
(1) $\mathrm { W } \cos \theta$
(2) $\frac { W } { 2 }$
(3) W
(4) $W \sin \theta$

Q5. A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :
(1) $\mathrm { W } \cos \theta$
(2) $\frac { W } { 2 }$
(3) W
(4) $W \sin \theta$

A rod of mass $m$ and length $\boldsymbol{l}$ is attached to two ideal strings. Find tension in left string just after right string is cut.
(A) $\frac{2}{3}\mathrm{mg}$ (B) $\frac{\mathrm{mg}}{4}$ (C) $\frac{\mathrm{mg}}{5}$ (D) $\frac{\mathrm{mg}}{2}$

Consider a triangle ABC where $\angle \mathrm{ BAC } = 60 ^ { \circ }$.
Let D be the point of intersection of the bisector of $\angle \mathrm{ BAC }$ and the side BC. Let DE and DF be the line segments perpendicular to sides AB and AC, respectively. Let us set
$$x = \frac { \mathrm{ AB } } { \mathrm{ AC } } , \quad k = \frac { \triangle \mathrm{ DEF } } { \triangle \mathrm{ ABC } } .$$
Note that $\triangle \mathrm{ ABC }$ denotes the area of the triangle ABC, and similarly for other triangles.
(1) We are to represent $k$ in terms of $x$. Since $\triangle \mathrm{ ABD } + \triangle \mathrm{ ACD } = \triangle \mathrm{ ABC }$, when we set $b = \mathrm{ AB }$, $c = \mathrm{ AC }$ and $d = \mathrm{ AD }$, we have
$$d = \frac { \sqrt { \mathbf { A } } \, b c } { b + c } .$$
Next, since $\mathrm{ DE } = \mathrm{ DF } = \dfrac { \mathbf { B } } { \mathbf { C } } d$, we have
$$\triangle \mathrm{ DEF } = \frac { \sqrt { \mathbf { D } } } { \mathbf { EF } } d ^ { 2 } .$$
From (1) and (2), we see that
$$k = \frac { d ^ { 2 } } { \mathbf { G } \, b c } = \frac { \mathbf { H } \, b c } { \mathbf { I } ( b + c ) ^ { 2 } } .$$
Since $x = \dfrac { b } { c }$, we have
$$k = \frac { \mathbf { J } \, x } { \mathbf { K } ( x + \mathbf { L } ) ^ { 2 } } .$$
(2) If $\mathrm{ BD } = 8$ and $\mathrm{ BC } = 10$, then $x = \mathbf { M }$ and $k = \dfrac { \mathbf { N } } { \mathbf { O P } }$.

Consider a rhombus ABCD with sides of length $a$, where $a$ is a constant. Let $r$ be the radius of the circle O inscribed in the rhombus ABCD, and $\mathrm{K}, \mathrm{L}$, $\mathrm{M}, \mathrm{N}$ be the points of tangency of the circle O and the rhombus. Let $S$ denote the area of the part of the rhombus outside circle O.
We are to find the range of the values of $r$, and the maximum value of $S$.
(1) For each of $\mathbf{A}$ $\sim$ $\mathbf{C}$ below, choose the correct answer from among (0) $\sim$ (9).
Let $\angle \mathrm{ABO} = \theta$. We have $\mathrm{OB} = \mathbf{A}$, and hence $\mathrm{OK} = \mathbf{B}$. Hence, since $( \cos \theta - \sin \theta ) ^ { 2 } \geqq 0$, the range of the values taken by $r$ is
$$0 < r \leqq \mathbf { C } .$$
(0) $a$ (1) $\frac { a } { 2 }$ (2) $\frac { a } { 3 }$ (3) $a \sin \theta$ (4) $a \cos \theta$ (5) $a \tan \theta$ (6) $a \sin ^ { 2 } \theta$ (7) $a \cos ^ { 2 } \theta$ (8) $a \sin \theta \cos \theta$ (9) $a \tan ^ { 2 } \theta$
(2) For each of $\mathbf { D } \sim$ $\mathbf{F}$ below, choose the correct answer from among (0) $\sim$ (9).
When the area $S$ is expressed in terms of $r$, we have
$$S = \mathbf { D } .$$
Here, we observe that when $r = \mathbf { E }$, the value of $\mathbf{D}$ is maximized, and this value for $r$ satisfies (1). Thus, at $r = \mathbf { E }$, $S$ takes the maximum value $\mathbf { F }$.
(0) $2 a r - \pi r ^ { 2 }$ (1) $a r - \frac { \pi } { 2 } r ^ { 2 }$ (2) $\frac { a } { 2 } r - \pi r ^ { 2 }$ (3) $\frac { a r - \pi r ^ { 2 } } { 2 }$ (4) $\frac { 2 a } { \pi }$ (5) $\frac { a } { \pi }$ (6) $\frac { a } { 2 \pi }$ (7) $\frac { 4 a ^ { 2 } } { \pi }$ (8) $\frac { a ^ { 2 } } { \pi }$ (9) $\frac { a ^ { 2 } } { 4 \pi }$