All the points in the set $S = \left\{ \frac { \alpha + i } { \alpha - i } , \alpha \in R \right\} , i = \sqrt { - 1 }$ lie on a (1) straight line whose slope is - 1 (2) circle whose radius is $\sqrt { 2 }$ (3) circle whose radius is 1 (4) straight line whose slope is 1
A committee of 11 member is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then: (1) $m = n = 68$ (2) $n = m - 8$ (3) $m = n = 78$ (4) $m + n = 68$
Let the sum of the first $n$ terms of a non-constant A.P., $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n }$ be $50 n + \frac { n ( n - 7 ) } { 2 } A$, where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $\left( d , a _ { 50 } \right)$ is equal to (1) $( 50,50 + 46 A )$ (2) $( A , 50 + 45 A )$ (3) $( 50,50 + 45 A )$ (4) $( A , 50 + 46 A )$
If the fourth term in the Binomial expansion of $\left( \frac { 2 } { x } + x ^ { \log _ { 8 } x } \right) ^ { 6 } , ( x > 0 )$ is $20 \times 8 ^ { 7 }$, then a value of $x$ is (1) $8 ^ { - 2 }$ (2) 8 (3) $8 ^ { 3 }$ (4) $8 ^ { 2 }$
If a tangent to the circle $x ^ { 2 } + y ^ { 2 } = 1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is: (1) $x ^ { 2 } + y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$ (2) $x ^ { 2 } + y ^ { 2 } - 4 x ^ { 2 } y ^ { 2 } = 0$ (3) $x ^ { 2 } + y ^ { 2 } - 2 x y = 0$ (4) $x ^ { 2 } + y ^ { 2 } - 2 x ^ { 2 } y ^ { 2 } = 0$
Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to (1) $y ^ { 3 }$ (2) $y \left( y ^ { 2 } - 1 \right)$ (3) $y ^ { 3 } - 1$ (4) $y \left( y ^ { 2 } - 3 \right)$
Let $f ( x ) = 15 - | x - 10 | ; x \in R$. Then the set of all values of $x$, at which the function $g ( x ) = f ( f ( x ) )$ is not differentiable, is: (1) $\{ 5,10,15 \}$ (2) $\{ 10 \}$ (3) $\{ 10,15 \}$ (4) $\{ 5,10,15,20 \}$
Let $\sum _ { k = 1 } ^ { 10 } f ( a + k ) = 16 \left( 2 ^ { 10 } - 1 \right)$, where the function $f$ satisfies $f ( x + y ) = f ( x ) f ( y )$ for all natural numbers $x , y$ and $f ( 1 ) = 2$. Then the natural number ' $a$ ' is: (1) 3 (2) 16 (3) 4 (4) 2
If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly (1) Two irrational and two rational numbers (2) Four rational numbers (3) Two irrational and one rational number (4) Four irrational numbers
If the tangent to the curve, $y = x ^ { 3 } + a x - b$ at the point $( 1 , - 5 )$ is perpendicular to the line, $- x + y + 4 = 0$, then which one of the following points lies on the curve? (1) $( 2 , - 2 )$ (2) $( 2 , - 1 )$ (3) $( - 2,1 )$ (4) $( - 2,2 )$
Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x$ at ( $x , y$ ) is parallel to the line segment joining the points $( 1 , f ( 1 ) )$ and $( - 1 , f ( - 1 ) )$, then $S$ is equal to (1) $\left\{ - \frac { 1 } { 3 } , - 1 \right\}$ (2) $\left\{ - \frac { 1 } { 3 } , 1 \right\}$ (3) $\left\{ \frac { 1 } { 3 } , 1 \right\}$ (4) $\left\{ \frac { 1 } { 3 } , - 1 \right\}$
Four persons can hit a target correctly with probabilities $\frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 8 }$ respectively. If all hit at the target independently, then the probability that the target would be hit, is (1) $\frac { 25 } { 192 }$ (2) $\frac { 7 } { 32 }$ (3) $\frac { 1 } { 192 }$ (4) $\frac { 25 } { 32 }$