A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N . If the maximum speed with which the stone can revolve is $\frac { K } { \pi } \mathrm { rev } \min ^ { - 1 }$. The value of $K$ is : (Assume the string is massless and un-stretchable) (1) 400 (2) 300 (3) 600 (4) 800
A block of mass 10 kg starts sliding on a surface with an initial velocity of $9.8 \mathrm {~ms} ^ { - 1 }$. The coefficient of friction between the surface and block is 0.5 . The distance covered by the block before coming to rest is :[use $\mathrm { g } = 9.8 \mathrm {~ms} ^ { - 2 }$ ] (1) 9.8 m (2) 4.9 m (3) 12.5 m (4) 19.6 m
A particle experiences a variable force $\overrightarrow { \mathrm { F } } = \left( 4 x \hat { i } + 3 y ^ { 2 } \hat { j } \right)$ in a horizontal $x - y$ plane. Assume distance in meters and force is newton. If the particle moves from point $( 1,2 )$ to point $( 2,3 )$ in the $x - y$ plane, then Kinetic Energy changes by : (1) 25 J (2) 50 J (3) 12.5 J (4) 0 J
From the top of a tower, a ball is thrown vertically upward which reaches the ground in 6 s . A second ball thrown vertically downward from the same position with the same speed reaches the ground in 1.5 s . A third ball released, from the rest from the same location, will reach the ground in $\_\_\_\_$ s.
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 $\_\_\_\_$ .
If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3 x ^ { 2 } + \lambda x - 1 = 0$ is 15 , then $6 \left( \alpha ^ { 3 } + \beta ^ { 3 } \right) ^ { 2 }$ is equal to (1) 46 (2) 36 (3) 24 (4) 18
Let $A = \{ z \in \mathrm { C } : 1 \leqslant | z - ( 1 + i ) | \leqslant 2 \}$ and $B = \{ z \in A : | z - ( 1 - i ) | = 1 \}$. Then, $B$ (1) is an empty set (2) contains exactly two elements (3) contains exactly three elements (4) is an infinite set
If $\left\{ a _ { i } \right\} _ { i = 1 } ^ { \mathrm { n } }$, where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum _ { i = 1 } ^ { n } a _ { i } = 192 , \sum _ { i = 1 } ^ { \frac { n } { 2 } } a _ { 2 i } = 120$, then $n$ is equal to (1) 18 (2) 36 (3) 96 (4) 48
Let $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ be a circle passing through ( 0,6 ) and touching the parabola $y = x ^ { 2 }$ at ( 2,4 ). Then $A + C$ is equal to (1) 16 (2) $\frac { 88 } { 5 }$ (3) 72 (4) - 8
Let $S = \{ \sqrt { n } : 1 \leqslant n \leqslant 50$ and $n$ is odd $\}$. Let $a \in S$ and $A = \left[ \begin{array} { r r r } 1 & 0 & a \\ - 1 & 1 & 0 \\ - a & 0 & 1 \end{array} \right]$. If $\Sigma _ { a \in S } \operatorname { det } ( \operatorname { adj } A ) = 100 \lambda$, then $\lambda$ is equal to (1) 218 (2) 221 (3) 663 (4) 1717
The number of values of $\alpha$ for which the system of equations $x + y + z = \alpha$ $\alpha x + 2 \alpha y + 3 z = - 1$ $x + 3 \alpha y + 5 z = 4$ is inconsistent, is (1) 0 (2) 1 (3) 2 (4) 3
For the function $f ( x ) = 4 \log _ { e } ( x - 1 ) - 2 x ^ { 2 } + 4 x + 5 , x > 1$, which one of the following is NOT correct? (1) $f ( x )$ is increasing in $( 1,2 )$ and decreasing in $( 2 , \infty )$ (2) $f ( x ) = - 1$ has exactly two solutions (3) $f ^ { \prime } ( \mathrm { e } ) - f ^ { \prime \prime } ( 2 ) < 0$ (4) $f ( x ) = 0$ has a root in the interval $( e , e + 1 )$
If the tangent at the point $\left( x _ { 1 } , y _ { 1 } \right)$ on the curve $y = x ^ { 3 } + 3 x ^ { 2 } + 5$ passes through the origin, then $\left( x _ { 1 } , y _ { 1 } \right)$ does NOT lie on the curve