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
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
Mass per unit area of a circular disc of radius a depends on the distance $r$ from its centre as $\sigma ( r ) = A + B r$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is: (1) $2 \pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$ (2) $2 \pi a ^ { 4 } \left( \frac { a A } { 4 } + \frac { B } { 5 } \right)$ (3) $\pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$ (4) $2 \pi \mathrm { a } ^ { 4 } \left( \frac { \mathrm {~A} } { 4 } + \frac { \mathrm { B } } { 5 } \right)$
A box weighs 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$ at the north pole and the radius of the earth $= 6400 \mathrm {~km}$): (1) 195.66 N (2) 194.32 N (3) 194.66 N (4) 195.32 N
The sum of two forces $\overrightarrow { \mathrm { P } }$ and $\overrightarrow { \mathrm { Q } }$ is $\overrightarrow { \mathrm { R } }$ such that $| \overrightarrow { \mathrm { R } } | = | \overrightarrow { \mathrm { P } } |$. Find the angle between resultant of $2 \overrightarrow { \mathrm { P } }$ and $\overrightarrow { \mathrm { Q } }$ and $\overrightarrow { \mathrm { Q } }$.
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 $\_\_\_\_$
Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } - x - 1 = 0$. If $p _ { k } = ( \alpha ) ^ { k } + ( \beta ) ^ { k } , k \geq 1$, then which one of the following statements is not true? (1) $p _ { 3 } = p _ { 5 } - p _ { 4 }$ (2) $p _ { 5 } = 11$ (3) $\left( p _ { 1 } + p _ { 2 } + p _ { 3 } + p _ { 4 } + p _ { 5 } \right) = 26$ (4) $p _ { 5 } = p _ { 2 } \cdot p _ { 3 }$
If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \ldots$. is $( 102 ) \mathrm { m }$, then m is equal to (1) 20 (2) 25 (3) 5 (4) 10
The number of ordered pairs $( r , k )$ for which $6 . { } ^ { 35 } C _ { r } = \left( k ^ { 2 } - 3 \right) . { } ^ { 36 } C _ { r + 1 }$, where $k$ is an integer is (1) 3 (2) 2 (3) 6 (4) 4
The locus of the mid-points of the perpendiculars drawn from points on the line $x = 2 y$, to the line $x = y$, is. (1) $2 x - 3 y = 0$ (2) $5 x - 7 y = 0$ (3) $3 x - 2 y = 0$ (4) $7 x - 5 y = 0$
Let the tangents drawn from the origin to the circle, $x ^ { 2 } + y ^ { 2 } - 8 x - 4 y + 16 = 0$ touch it at the points $A$ and $B$. Then $( A B ) ^ { 2 }$ is equal to (1) $\frac { 52 } { 5 }$ (2) $\frac { 56 } { 5 }$ (3) $\frac { 64 } { 5 }$ (4) $\frac { 32 } { 5 }$
If $3 x + 4 y = 12 \sqrt { 2 }$ is a tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 9 } = 1$ for some $a \in R$, then the distance between the foci of the ellipse is (1) $2 \sqrt { 7 }$ (2) 4 (3) $2 \sqrt { 5 }$ (4) $2 \sqrt { 2 }$
Let $\mathrm { A } = \left[ a _ { i j } \right]$ and $\mathrm { B } = \left[ b _ { i j } \right]$ be two $3 \times 3$ real matrices such that $b _ { i j } = ( 3 ) ^ { ( i + j - 2 ) } a _ { i j }$, where $i , j = 1,2,3$. If the determinant of B is 81, then determinant of A is (1) $\frac { 1 } { 3 }$ (2) 3 (3) $\frac { 1 } { 81 }$ (4) $\frac { 1 } { 9 }$
Let $f ( x )$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim _ { x \rightarrow 0 } \left( 2 + \frac { f ( x ) } { x ^ { 3 } } \right) = 4$, then which one of the following is not true? (1) $f$ is an odd function (2) $f ( 1 ) - 4 f ( - 1 ) = 4$ (3) $x = 1$ is a point of local minimum and $x = - 1$ is a point of local maximum (4) $x = 1$ is a point of local maxima of $f$