If $a \in R$ and the equation $- 3 ( x - [ x ] ) ^ { 2 } + 2 ( x - [ x ] ) + a ^ { 2 } = 0$ (where $[ x ]$ denotes the greatest integer $\leq x )$ has no integral solution, then all possible values of $a$ lie in the interval (1) $( - 2 , - 1 )$ (2) $( - \infty , - 2 ) \cup ( 2 , \infty )$ (3) $( - 1,0 ) \cup ( 0,1 )$ (4) $( 1,2 )$
If $z$ is a complex number such that $| z | \geq 2$, then the minimum value of $\left| z + \frac { 1 } { 2 } \right|$: (1) Is strictly greater than $\frac { 5 } { 2 }$ (2) Is strictly greater than $\frac { 3 } { 2 }$ but less than $\frac { 5 } { 2 }$ (3) Is equal to $\frac { 5 } { 2 }$ (4) Lies in the interval $( 1,2 )$
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is: (1) $2 - \sqrt { 3 }$ (2) $2 + \sqrt { 3 }$ (3) $\sqrt { 2 } + \sqrt { 3 }$ (4) $3 + \sqrt { 2 }$
Let $P S$ be the median of the triangle with vertices $P ( 2,2 ) , Q ( 6 , - 1 )$ and $R ( 7,3 )$. The equation of the line passing through $( 1 , - 1 )$ and parallel to $P S$ is (1) $4 x + 7 y + 3 = 0$ (2) $2 x - 9 y - 11 = 0$ (3) $4 x - 7 y - 11 = 0$ (4) $2 x + 9 y + 7 = 0$
Let $a , b , c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x + 2 a y + c = 0$ \& $5 b x + 2 b y + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then (1) $3 b c - 2 a d = 0$ (2) $3 b c + 2 a d = 0$ (3) $2 b c - 3 a d = 0$ (4) $2 b c + 3 a d = 0$
Let $C$ be the circle with center at $( 1,1 )$ and radius $= 1$. If $T$ is the circle centered at $( 0 , y )$, passing through the origin and touching the circle $C$ externally, then the radius of $T$ is equal to (1) $\frac { 1 } { 2 }$ (2) $\frac { 1 } { 4 }$ (3) $\frac { \sqrt { 3 } } { \sqrt { 2 } }$ (4) $\frac { \sqrt { 3 } } { 2 }$
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point $O$ on the ground is $45 ^ { \circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to $30 ^ { \circ }$. Then the speed (in m/s) of the bird is (1) $20 \sqrt { 2 }$ (2) $20 ( \sqrt { 3 } - 1 )$ (3) $40 ( \sqrt { 2 } - 1 )$ (4) $40 ( \sqrt { 3 } - \sqrt { 2 } )$
If $A$ is a $3 \times 3$ non-singular matrix such that $A A ^ { \prime } = A ^ { \prime } A$ and $B = A ^ { - 1 } A ^ { \prime }$, then $B B ^ { \prime }$ equals, where $X ^ { \prime }$ denotes the transpose of the matrix $X$. (1) $B ^ { - 1 }$ (2) $\left( B ^ { - 1 } \right) ^ { \prime }$ (3) $I + B$ (4) $I$
If $g$ is the inverse of a function $f$ and $f ^ { \prime } ( x ) = \frac { 1 } { 1 + x ^ { 5 } }$, then $g ^ { \prime } ( x )$ is equal to (1) $\frac { 1 } { 1 + \{ g ( x ) \} ^ { 5 } }$ (2) $1 + \{ g ( x ) \} ^ { 5 }$ (3) $1 + x ^ { 5 }$ (4) $5 x ^ { 4 }$
The integral $\int \left( 1 + x - \frac { 1 } { x } \right) e ^ { x + \frac { 1 } { x } } d x$, is equal to (1) $( x + 1 ) e ^ { x + \frac { 1 } { x } } + c$ (2) $- x e ^ { x + \frac { 1 } { x } } + c$ (3) $( x - 1 ) e ^ { x + \frac { 1 } { x } } + c$ (4) $x e ^ { x + \frac { 1 } { x } } + c$
Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac { d p ( t ) } { d t } = \frac { 1 } { 2 } \{ p ( t ) - 400 \}$. If $p ( 0 ) = 100$, then $p ( t )$ equals (1) $600 - 500 e ^ { \frac { t } { 2 } }$ (2) $400 - 300 e ^ { \frac { - t } { 2 } }$ (3) $400 - 300 e ^ { t / 2 }$ (4) $300 - 200 e ^ { \frac { - t } { 2 } }$
Let $A$ and $B$ be two events such that $P ( \overline { A \cup B } ) = \frac { 1 } { 6 } , P ( A \cap B ) = \frac { 1 } { 4 }$ and $P ( \bar { A } ) = \frac { 1 } { 4 }$, where $\bar { A }$ stands for the complement of the event $A$. Then the events $A$ and $B$ are (1) Independent but not equally likely. (2) Independent and equally likely. (3) Mutually exclusive and independent. (4) Equally likely but not independent.