If the maximum load carried by an elevator is 1400 kg ( 600 kg -Passengers + 800 kg -elevator) , which is moving up with a uniform speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the frictional force acting on it is 2000 N , then the maximum power used by the motor is $\_\_\_\_$ $\mathrm { kW } . \quad g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
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 rectangular block of mass 5 kg attached to a horizontal spiral spring executes simple harmonic motion of amplitude 1 m and time period 3.14 s . The maximum force exerted by spring on block is $\_\_\_\_$ N.
Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is (1) 1120 (2) 3360 (3) 1680 (4) 560
Let the number $( 22 ) ^ { 2022 } + ( 2022 ) ^ { 22 }$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7 . Then $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$ is equal to (1) 20 (2) 13 (3) 5 (4) 10
If the coefficients of $x$ and $x ^ { 2 }$ in $( 1 + x ) ^ { p } ( 1 - x ) ^ { q }$ are 4 and $-5$ respectively, then $2p + 3q$ is equal to (1) 60 (2) 69 (3) 66 (4) 63
Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x ^ { 2 } + y ^ { 2 } = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$ is the point $C( \alpha , \beta )$, then the length of the line segment $AC$ is (1) $\frac { 3 \sqrt { 5 } } { 5 }$ (2) $\frac { 4 \sqrt { 5 } } { 5 }$ (3) $\frac { 2 \sqrt { 5 } } { 5 }$ (4) $\frac { 6 \sqrt { 5 } } { 5 }$
Let a circle of radius 4 be concentric to the ellipse $15 x ^ { 2 } + 19 y ^ { 2 } = 285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 4 }$ (3) $\frac { \pi } { 6 }$ (4) $\frac { \pi } { 12 }$
Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution
$X _ { i }$
0
1
2
3
4
5
$f _ { i }$
$k + 2$
$2k$
$k ^ { 2 } - 1$
$k ^ { 2 } - 1$
$k ^ { 2 } + 1$
$k - 3$
where $\Sigma f _ { i } = 62$. If $\lfloor x \rfloor$ denotes the greatest integer $\leq x$, then $\lfloor \mu ^ { 2 } + \sigma ^ { 2 } \rfloor$ is equal to (1) 9 (2) 8 (3) 7 (4) 6