Let $f ( x )$ be a quadratic polynomial such that $f ( - 1 ) + f ( 2 ) = 0$. If one of the roots of $f ( x ) = 0$ is 3 , then its other root lies in (1) $( - 1,0 )$ (2) $( 1,3 )$ (3) $( - 3 , - 1 )$ (4) $( 0,1 )$
Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of $n$ is (1) 201 (2) 200 (3) 101 (4) 199
If the sum of first 11 terms of an A.P. , $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\ldots$ is $0 \left( a _ { 1 } \neq 0 \right)$ then the sum of the A.P $a _ { 1 } , a _ { 3 } , a _ { 5 } , \ldots\ldots a _ { 23 }$ is $k a _ { 1 }$ where $k$ is equal to (1) $- \frac { 121 } { 10 }$ (2) $\frac { 121 } { 10 }$ (3) $\frac { 72 } { 5 }$ (4) $- \frac { 72 } { 5 }$
Let $S$ be the sum of the first 9 term of the series : $\{ x + k a \} + \left\{ x ^ { 2 } + ( k + 2 ) a \right\} + \left\{ x ^ { 3 } + ( k + 4 ) a \right\} + \left\{ x ^ { 4 } + ( k + 6 ) a \right\} + \ldots$ where $a \neq 0$ and $x \neq 1$. If $S = \frac { x ^ { 10 } - x + 45 a ( x - 1 ) } { x - 1 }$, then $k$ is equal to (1) - 5 (2) 1 (3) - 3 (4) 3
The area (in sq. units) of an equilateral triangle inscribed in the parabola $y ^ { 2 } = 8 x$, with one of its vertices on the vertex of this parabola is (1) $64 \sqrt { 3 }$ (2) $256 \sqrt { 3 }$ (3) $192 \sqrt { 3 }$ (4) $128 \sqrt { 3 }$
Let $A = \left\{ X = ( x , y , z ) ^ { T } : P X = 0 \text{ and } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \right\}$ where $P = \left[ \begin{array} { c c c } 1 & 2 & 1 \\ - 2 & 3 & - 4 \\ 1 & 9 & - 1 \end{array} \right]$ then the set $A$ (1) Is a singleton. (2) Is an empty set. (3) Contains more than two elements (4) Contains exactly two elements
Let $a , b , c \in R$ be all non-zero and satisfies $a ^ { 3 } + b ^ { 3 } + c ^ { 3 } = 2$. If the matrix $A = \left[ \begin{array} { c c c } a & b & c \\ b & c & a \\ c & a & b \end{array} \right]$ satisfies $A ^ { T } A = I$, then a value of $a b c$ can be (1) $- \frac { 1 } { 3 }$ (2) $\frac { 1 } { 3 }$ (3) 3 (4) $\frac { 2 } { 3 }$
Let $f : R \rightarrow R$ be a function which satisfies $f ( x + y ) = f ( x ) + f ( y ) , \forall x , y \in R$. If $f ( 1 ) = 2$ and $g ( n ) = \sum _ { k = 1 } ^ { ( n - 1 ) } f ( k ) , n \in N$ then the value of $n$, for which $g ( n ) = 20$, is (1) 5 (2) 20 (3) 4 (4) 9
The equation of the normal to the curve $y = ( 1 + x ) ^ { 2 y } + \cos ^ { 2 } \left( \sin ^ { - 1 } x \right)$, at $x = 0$ is (1) $y + 4 x = 2$ (2) $y = 4 x + 2$ (3) $x + 4 y = 8$ (4) $2 y + x = 4$
Let $f : ( - 1 , \infty ) \rightarrow R$ be defined by $f ( 0 ) = 1$ and $f ( x ) = \frac { 1 } { x } \log _ { e } ( 1 + x ) , x \neq 0$. Then the function $f$ (1) Decreases in $( - 1,0 )$ and increases in $( 0 , \infty )$ (2) Increases in $( - 1 , \infty )$ (3) Increases in $( - 1,0 )$ and decreases in $( 0 , \infty )$ (4) Decreases in $( - 1 , \infty )$