The set of all values of $\lambda$ for which the system of linear equations $$\begin{aligned} & x - 2 y - 2 z = \lambda x \\ & x + 2 y + z = \lambda y \\ & - x - y = \lambda z \end{aligned}$$ has a non-trivial solution :
(1) is an empty set
(2) contains more than two elements
(3) is a singleton
(4) contains exactly two elements

$\lim _ { n \rightarrow \infty } \left( \frac { n } { n ^ { 2 } + 1 ^ { 2 } } + \frac { n } { n ^ { 2 } + 2 ^ { 2 } } + \frac { n } { n ^ { 2 } + 3 ^ { 2 } } + \ldots\ldots + \frac { 1 } { 5 n ^ { 2 } } \right)$ is equal to
(1) $\frac { \pi } { 4 }$
(2) $\tan ^ { - 1 } ( 2 )$
(3) $\frac { \pi } { 2 }$
(4) $\tan ^ { - 1 } ( 3 )$

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three unit vectors, out of which vectors $\vec { b }$ and $\vec { c }$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec { a }$ makes with vectors $\vec { b }$ and $\vec { c }$ respectively and $\vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { 1 } { 2 } \vec { b }$, then $| \alpha - \beta |$ is equal to :
(1) $90 ^ { \circ }$
(2) $60 ^ { \circ }$
(3) $45 ^ { \circ }$
(4) $30 ^ { \circ }$

If an angle between the line, $\frac { x + 1 } { 2 } = \frac { y - 2 } { 1 } = \frac { z - 3 } { - 2 }$ and the plane, $x - 2 y - k z = 3$ is $\cos ^ { - 1 } \left( \frac { 2 \sqrt { 2 } } { 3 } \right)$, then a value of $k$ is
(1) $\sqrt { \frac { 5 } { 3 } }$
(2) $\sqrt { \frac { 3 } { 5 } }$
(3) $- \frac { 3 } { 5 }$
(4) $- \frac { 5 } { 3 }$

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $\left( - \lambda ^ { 2 } , 1,1 \right) , \left( 1 , - \lambda ^ { 2 } , 1 \right)$ and $\left( 1,1 , - \lambda ^ { 2 } \right)$ also passes through the point $( - 1 , - 1,1 )$. Then $S$ is equal to :
(1) $\{ \sqrt { 3 } \}$
(2) $\{ 3 , - 3 \}$
(3) $\{ 1 , - 1 \}$
(4) $\{ \sqrt { 3 } , - \sqrt { 3 } \}$

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :
(1) $\frac { 400 } { 3 }$ gain
(2) $\frac { 400 } { 9 }$ gain
(3) $\frac { 400 } { 3 }$ loss
(4) 0

A 60 HP electric motor lifts an elevator having a maximum total load capacity of 2000 kg. If the frictional force on the elevator is 4000 N, the speed of the elevator at full load is close to: $\left( 1 \mathrm { HP } = 746 \mathrm {~W} , g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$
(1) $1.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $1.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $2.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A simple pendulum is being used to determine the value of gravitational acceleration $g$ at a certain place. The length of the pendulum is 25.0 cm and a stopwatch with 1 s resolution measures the time taken for 40 oscillations to be 50 s. The accuracy in g is:
(1) $5.40\%$
(2) $3.40\%$
(3) $4.40\%$
(4) $2.40\%$

Moment of inertia of a cylinder of mass m, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M \left( \frac { R ^ { 2 } } { 4 } + \frac { L ^ { 2 } } { 12 } \right)$. If such a cylinder is to be made for a given mass of a material, the ratio $\frac { L } { R }$ for it to have minimum possible $I$ is:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 3 } { 2 }$
(3) $\sqrt { \frac { 3 } { 2 } }$
(4) $\sqrt { \frac { 2 } { 3 } }$

Three point particles of masses $1.0 \mathrm {~kg} , 1.5 \mathrm {~kg}$ and 2.5 kg are placed at three corners of a right angle triangle of sides $4.0 \mathrm {~cm} , 3.0 \mathrm {~cm}$ and 5.0 cm as shown in the figure. The centre of mass of the system is at a point:
(1) 0.6 cm right and 2.0 cm above 1 kg mass.
(2) 1.5 cm right and 1.2 cm above 1 kg mass.
(3) 2.0 cm right and 0.9 cm above 1 kg mass.
(4) 0.9 cm right and 2.0 cm above 1 kg mass.

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of $3 \mathrm {~m} / \mathrm { s }$. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator ($\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$) must be at least:
(1) 56300 W
(2) 62360 W
(3) 48000 W
(4) 66000 W

Consider a force $\vec { F } = - x \hat { i } + y \hat { j }$. The work done by this force in moving a particle from point $A ( 1,0 )$ to $B ( 0,1 )$ along the line segment is : (all quantities are in SI units)
(1) 2
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 3 } { 2 }$

A block of mass $\mathrm { m } = 1 \mathrm {~kg}$ slides with velocity $\mathrm { v } = 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. if the rod has mass $M = 2 \mathrm {~kg}$, and length $\ell = 1 \mathrm {~m}$, the value of $\theta$ is approximately (take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) $63 ^ { \circ }$
(2) $55 ^ { \circ }$
(3) $69 ^ { \circ }$
(4) $49 ^ { \circ }$

The velocity $(v)$ and time $(t)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at 4.333 seconds. The total distance covered by the body in 6 s is:
(1) $\frac{37}{3}$ m
(2) 12 m
(3) 11 m
(4) $\frac{49}{4}$ m

A particle of mass $m$ is fixed to one end of a light spring having force constant $k$ and unstretched length $l$. The other end is fixed. The system is given an angular speed $\omega$ about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is:
(1) $\frac { m l \omega ^ { 2 } } { k - \omega m }$
(2) $\frac { m l \omega ^ { 2 } } { k - m \omega ^ { 2 } }$
(3) $\frac { m l \omega ^ { 2 } } { k + m \omega ^ { 2 } }$
(4) $\frac { m l \omega ^ { 2 } } { k + m \omega }$

As shown in the figure, a bob of mass $m$ is tied to a massless string whose other end portion is wound on a fly wheel (disc) of radius $r$ and mass $m$. When released from rest the bob starts falling vertically. When it has covered a distance of $h$, the angular speed of the wheel will be:
(1) $\frac { 1 } { \mathrm { r } } \sqrt { \frac { 4 \mathrm { gh } } { 3 } }$
(2) $r \sqrt { \frac { 3 } { 2 g h } }$
(3) $\frac { 1 } { \mathrm { r } } \sqrt { \frac { 2 \mathrm { gh } } { 3 } }$
(4) $r \sqrt { \frac { 3 } { 4 g h } }$

A mass of 10 kg is suspended by a rope of length 4 m, from the ceiling. A force F is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of $45 ^ { \circ }$ with the vertical. Then F equals: (Take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the rope to be massless)
(1) 100 N
(2) 90 N
(3) 70 N
(4) 75 N

A particle moves such that its position vector $\vec{r}(t) = \cos\omega t\,\hat{i} + \sin\omega t\,\hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}(t)$ and acceleration $\vec{a}(t)$ of the particle:
(1) $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed away from the origin
(2) $\vec{v}$ and $\vec{a}$ both are perpendicular to $\vec{r}$
(3) $\vec{v}$ and $\vec{a}$ both are parallel to $\vec{r}$
(4) $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed towards the origin

Two particles of equal mass $m$ have respective initial velocities $u \hat { i }$ and $u \left( \frac { \hat { i } + \hat { j } } { 2 } \right)$. They collide completely inelastically. The energy lost in the process is:
(1) $\frac { 1 } { 3 } m u ^ { 2 }$
(2) $\frac { 1 } { 8 } m u ^ { 2 }$
(3) $\frac { 3 } { 4 } m u ^ { 2 }$
(4) $\sqrt { \frac { 2 } { 3 } } m u ^ { 2 }$

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius $\mathrm { R } _ { \mathrm { e } }$. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become $\sqrt { \frac { 3 } { 2 } }$ times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is $R$. Value of $R$ is:
(1) $4 \mathrm { R } _ { \mathrm { e } }$
(2) $2.5 \mathrm { R } _ { \mathrm { e } }$
(3) $3 R _ { e }$
(4) $2 \mathrm { R } _ { \mathrm { e } }$

Starting from the origin at time $\mathrm { t } = 0$, with initial velocity $5 \widehat { \mathrm { j } } \mathrm { ms } ^ { - 1 }$, a particle moves in the $x - y$ plane with a constant acceleration of $( 10 \widehat { \mathrm { i } } + 4 \widehat { \mathrm { j } } ) \mathrm { ms } ^ { - 2 }$. At time t , its coordinates are $\left( 20 \mathrm {~m} , \mathrm { y } _ { 0 } \mathrm {~m} \right)$. The values of t and $\mathrm { y } _ { 0 }$ are, respectively:
(1) 2 s and 18 m
(2) 4 s and 52 m
(3) 2 s and 24 m
(4) 5 s and 25 m

A small ball of mass $m$ is thrown upward with velocity $u$ from the ground. The ball experiences a resistive force $m k v ^ { 2 }$ where $v$ is it speed. The maximum height attained by the ball is:
(1) $\frac { 1 } { 2 k } \tan ^ { - 1 } \frac { k u ^ { 2 } } { g }$
(2) $\frac { 1 } { k } \ln \left( 1 + \frac { k u ^ { 2 } } { 2 g } \right)$
(3) $\frac { 1 } { k } \tan ^ { - 1 } \frac { k u ^ { 2 } } { 2 g }$
(4) $\frac { 1 } { 2 k } \ln \left( 1 + \frac { k u ^ { 2 } } { g } \right)$

A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{dM(t)}{dt} = bv^2(t)$, where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is:
(1) $-bv^3(t)$
(2) $\frac{-bv^3}{M(t)}$
(3) $-\frac{2bv^3}{M(t)}$
(4) $-\frac{bv^3}{2M(t)}$

The coordinates of the centre of mass of a uniform flag-shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in the figure) are:
(1) $( 1.25 \mathrm {~m} , 1.50 \mathrm {~m} )$
(2) $( 0.75 \mathrm {~m} , 1.75 \mathrm {~m} )$
(3) $( 0.75 \mathrm {~m} , 0.75 \mathrm {~m} )$
(4) $( 1 \mathrm {~m} , 1.75 \mathrm {~m} )$

The radius of gyration of a uniform rod of length $l$, about an axis passing through a point $\frac { l } { 4 }$ away from the centre of the rod, and perpendicular to it, is:
(1) $\frac { 1 } { 4 } l$
(2) $\frac { 1 } { 8 } l$
(3) $\sqrt { \frac { 7 } { 48 } } l$
(4) $\sqrt { \frac { 3 } { 8 } } l$