If $A = \{ x \in R : | x | < 2 \}$ and $B = \{ x \in R : | x - 2 | \geq 3 \}$; then (1) $A \cap B = ( - 2 , - 1 )$ (2) $B - A = R - ( - 2,5 )$ (3) $A \cup B = R - ( 2,5 )$ (4) $A - B = [ - 1,2 )$
Let $a , b \in R , a \neq 0$ be such that the equation, $a x ^ { 2 } - 2 b x + 5 = 0$ has a repeated root $\alpha$, which is also a root of the equation, $x ^ { 2 } - 2 b x - 10 = 0$. If $\beta$ is the other root of this equation, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to: (1) 25 (2) 26 (3) 28 (4) 24
Let $a _ { n }$ be the $n ^ { \text {th } }$ term of a G.P. of positive terms. If $\sum _ { n = 1 } ^ { 100 } a _ { 2 n + 1 } = 200$ and $\sum _ { n = 1 } ^ { 100 } a _ { 2 n } = 100$, then $\sum _ { n = 1 } ^ { 200 } a _ { n }$ is equal to: (1) 300 (2) 225 (3) 175 (4) 150
In the expansion of $\left( \frac { x } { \cos \theta } + \frac { 1 } { x \sin \theta } \right) ^ { 16 }$, if $l _ { 1 }$ is the least value of the term independent of $x$ when $\frac { \pi } { 8 } \leq \theta \leq \frac { \pi } { 4 }$ and $l _ { 2 }$ is the least value of the term independent of $x$ when $\frac { \pi } { 16 } \leq \theta \leq \frac { \pi } { 8 }$, then the ratio $l _ { 2 } : l _ { 1 }$ is equal to: (1) $1 : 8$ (2) $16 : 1$ (3) $8 : 1$ (4) $1 : 16$
If one end of a focal chord $AB$ of the parabola $y ^ { 2 } = 8 x$ is at $A \left( \frac { 1 } { 2 } , - 2 \right)$, then the equation of the tangent to it at $B$ is: (1) $2 x + y - 24 = 0$ (2) $x - 2 y + 8 = 0$ (3) $x + 2 y + 8 = 0$ (4) $2 x - y - 24 = 0$
The following system of linear equations $7 x + 6 y - 2 z = 0$ $3 x + 4 y + 2 z = 0$ $x - 2 y - 6 z = 0$, has (1) infinitely many solutions, ( $x , y , z$ ) satisfying $y = 2z$ (2) no solution (3) infinitely many solutions, $( x , y , z )$ satisfying $x = 2z$ (4) only the trivial solution
Let $a - 2 b + c = 1$. If $f ( x ) = \left| \begin{array} { l l l } x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{array} \right|$, then: (1) $f ( - 50 ) = 501$ (2) $f ( - 50 ) = - 1$ (3) $f ( 50 ) = - 501$ (4) $f ( 50 ) = 1$
Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a , b \in R , g ^ { \prime } ( a ) = 5$ and $g ( a ) = b$, then $f ^ { \prime } ( b )$ is equal to: (1) $\frac { 1 } { 5 }$ (2) 1 (3) 5 (4) $\frac { 2 } { 5 }$
Let a function $f : [ 0,5 ] \rightarrow R$ be continuous, $f ( 1 ) = 3$ and $F$ be defined as: $F ( x ) = \int _ { 1 } ^ { x } t ^ { 2 } g ( t ) d t$, where $g ( t ) = \int _ { 1 } ^ { t } f ( u ) d u$. Then for the function $F ( x )$, the point $x = 1$ is: (1) a point of local minima (2) not a critical point (3) a point of local maxima (4) a point of inflection