Let $z = 1 + a i$, be a complex number, $a > 0$, such that $z ^ { 3 }$ is a real number. Then, the sum $1 + z + z ^ { 2 } + \ldots + z ^ { 11 }$ is equal to : (1) $1365 \sqrt { 3 } i$ (2) $- 1365 \sqrt { 3 } i$ (3) $- 1250 \sqrt { 3 } i$ (4) $1250 \sqrt { 3 } i$
Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots a _ { n } , \ldots$, be in A.P. If $a _ { 3 } + a _ { 7 } + a _ { 11 } + a _ { 15 } = 72$, then the sum of its first 17 terms is equal to : (1) 306 (2) 204 (3) 153 (4) 612
A straight line through origin $O$ meets the lines $3 y = 10 - 4 x$ and $8 x + 6 y + 5 = 0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $A B$ in the ratio (1) $2 : 3$ (2) $1 : 2$ (3) $4 : 1$ (4) $3 : 4$
A ray of light is incident along a line which meets another line $7 x - y + 1 = 0$ at the point $( 0,1 )$. The ray is then reflected from this point along the line $y + 2 x = 1$. Then the equation of the line of incidence of the ray of light is : (1) $41 x - 25 y + 25 = 0$ (2) $41 x + 25 y - 25 = 0$ (3) $41 x - 38 y + 38 = 0$ (4) $41 x + 38 y - 38 = 0$
Equation of the tangent to the circle, at the point $( 1 , - 1 )$, whose center, is the point of intersection of the straight lines $x - y = 1$ and $2 x + y = 3$ is: (1) $x + 4 y + 3 = 0$ (2) $3 x - y - 4 = 0$ (3) $x - 3 y - 4 = 0$ (4) $4 x + y - 3 = 0$
$P$ and $Q$ are two distinct points on the parabola, $y ^ { 2 } = 4 x$, with parameters $t$ and $t _ { 1 }$, respectively. If the normal at $P$ passes through $Q$, then the minimum value of $t _ { 1 } ^ { 2 }$, is (1) 8 (2) 4 (3) 6 (4) 2
A hyperbola whose transverse axis is along the major axis of the conic $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac { 3 } { 2 }$, then which of the following points does not lie on the hyperbola? (1) $( \sqrt { 5 } , 2 \sqrt { 2 } )$ (2) $( 0,2 )$ (3) $( 5,2 \sqrt { 3 } )$ (4) $( \sqrt { 10 } , 2 \sqrt { 3 } )$
The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is (1) if the area of a square increases four times, then its side is not doubled. (2) if the area of a square increases four times, then its side is doubled. (3) if the area of a square does not increase four times, then its side is not doubled. (4) if the side of a square is not doubled, then its area does not increase four times.
The mean of 5 observations is 5 and their variance is 12.4 . If three of the observations are $1,2 \& 6$; then the value of the remaining two is : (1) 1,11 (2) 5,5 (3) 5,11 (4) None of these
The angle of elevation of the top of a vertical tower from a point A , due east of it is $45 ^ { \circ }$. The angle of elevation of the top of the same tower from a point B , due south of A is $30 ^ { \circ }$. If the distance between A and B is $54 \sqrt { 2 } m$, then the height of the tower (in meters), is: (1) 108 (2) $36 \sqrt { 3 }$ (3) $54 \sqrt { 3 }$ (4) 54
Let $A$, be a $3 \times 3$ matrix, such that $A ^ { 2 } - 5 A + 7 I = O$. Statement - I : $A ^ { - 1 } = \frac { 1 } { 7 } ( 5 I - A )$. Statement - II : The polynomial $A ^ { 3 } - 2 A ^ { 2 } - 3 A + I$, can be reduced to $5 ( A - 4 I )$. Then : (1) Both the statements are true (2) Both the statements are false (3) Statement - I is true, but Statement - II is false (4) Statement - I is false, but Statement - II is true
Let C be a curve given by $y ( x ) = 1 + \sqrt { 4 x - 3 } , x > \frac { 3 } { 4 }$. If $P$ is a point on C , such that the tangent at $P$ has slope $\frac { 2 } { 3 }$, then a point through which the normal at $P$ passes, is : (1) $( 1,7 )$ (2) $( 3 , - 4 )$ (3) $( 4 , - 3 )$ (4) $( 2,3 )$
For $x \in R , x \neq 0$, if $y ( x )$ is a differentiable function such that $x \int _ { 1 } ^ { x } y ( t ) d t = ( x + 1 ) \int _ { 1 } ^ { x } t y ( t ) d t$, then $y ( x )$ equals (where $C$ is a constant) (1) $C x ^ { 3 } e ^ { \frac { 1 } { x } }$ (2) $\frac { C } { x ^ { 2 } } e ^ { - \frac { 1 } { x } }$ (3) $\frac { C } { x } e ^ { - \frac { 1 } { x } }$ (4) $\frac { C } { x ^ { 3 } } e ^ { - \frac { 1 } { x } }$
The value of the integral $\int _ { 4 } ^ { 10 } \frac { \left[ x ^ { 2 } \right] } { \left[ x ^ { 2 } - 28 x + 196 \right] + \left[ x ^ { 2 } \right] } d x$, where $[ x ]$ denotes the greatest integer less than or equal to $x$, is (1) $\frac { 1 } { 3 }$ (2) 6 (3) 7 (4) 3
The solution of the differential equation $\frac { d y } { d x } + \frac { y } { 2 } \sec x = \frac { \tan x } { 2 y }$, where $0 \leq x < \frac { \pi } { 2 }$ and $y ( 0 ) = 1$, is given by (1) $y ^ { 2 } = 1 + \frac { x } { \sec x + \tan x }$ (2) $y = 1 + \frac { x } { \sec x + \tan x }$ (3) $y = 1 - \frac { x } { \sec x + \tan x }$ (4) $y ^ { 2 } = 1 - \frac { x } { \sec x + \tan x }$
$ABC$ is a triangle in a plane with vertices $A ( 2,3,5 ) , B ( - 1,3,2 )$ and $C ( \lambda , 5 , \mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right)$ is (1) 1130 (2) 1348 (3) 1077 (4) 676
Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A , B , C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac { \vec { a } + \vec { b } + \vec { c } } { 4 }$ respectively, then the position vector of the orthocentre of this triangle, is : (1) $- \left( \frac { \vec { a } + \vec { b } + \vec { c } } { 2 } \right)$ (2) $\vec { a } + \vec { b } + \vec { c }$ (3) $\frac { ( \vec { a } + \vec { b } + \vec { c } ) } { 2 }$ (4) $\overrightarrow { 0 }$
An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is (1) $\frac { 496 } { 729 }$ (2) $\frac { 192 } { 729 }$ (3) $\frac { 240 } { 729 }$ (4) $\frac { 256 } { 729 }$