If $z _ { 1 } , z _ { 2 }$ are two distinct complex number such that $\left| \frac { z _ { 1 } - 2 z _ { 2 } } { \frac { 1 } { 2 } - z _ { 1 } \bar { z } _ { 2 } } \right| = 2$, then (1) $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ and $z _ { 2 }$ lies on a circle of radius 1. (2) both $z _ { 1 }$ and $z _ { 2 }$ lie on the same circle. (3) either $z _ { 1 }$ lies on a circle of radius $\frac { 1 } { 2 }$ or $z _ { 2 }$ lies on a circle of radius 1. (4) either $z _ { 1 }$ lies on a circle of radius 1 or $z _ { 2 }$ lies on a circle of radius $\frac { 1 } { 2 }$.
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315 ^ { \text {th} }$ position in this arrangement is: (1) NRAGUP (2) NRAPUG (3) NRAPGU (4) NRAGPU
Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then: (1) $\mathrm { P } ^ { 2 } = 6 \sqrt { 3 } \mathrm { Q }$ (2) $\mathrm { P } ^ { 2 } = 36 \sqrt { 3 } \mathrm { Q }$ (3) $P = 36 \sqrt { 3 } Q ^ { 2 }$ (4) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$
A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to: (1) 150 (2) 180 (3) 160 (4) 125
If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to: (1) 37 (2) 437 (3) $- 27$ (4) 5
Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to: (1) 25 (2) 24 (3) 26 (4) 23
Suppose for a differentiable function $h , h ( 0 ) = 0 , h ( 1 ) = 1$ and $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 1 ) = 2$. If $\mathrm { g } ( x ) = h \left( \mathrm { e } ^ { x } \right) \mathrm { e } ^ { h ( x ) }$, then $g ^ { \prime } ( 0 )$ is equal to: (1) 5 (2) 4 (3) 8 (4) 3
If the area of the region $\left\{ ( x , y ) : \frac { \mathrm { a } } { x ^ { 2 } } \leq y \leq \frac { 1 } { x } , 1 \leq x \leq 2,0 < \mathrm { a } < 1 \right\}$ is $\left( \log _ { \mathrm { e } } 2 \right) - \frac { 1 } { 7 }$ then the value of $7 \mathrm { a } - 3$ is equal to: (1) 0 (2) 2 (3) $- 1$ (4) 1
Suppose the solution of the differential equation $\frac { d y } { d x } = \frac { ( 2 + \alpha ) x - \beta y + 2 } { \beta x - 2 \alpha y - ( \beta \gamma - 4 \alpha ) }$ represents a circle passing through origin. Then the radius of this circle is: (1) 2 (2) $\sqrt { 17 }$ (3) $\frac { 1 } { 2 }$ (4) $\frac { \sqrt { 17 } } { 2 }$
Let $\mathrm { P } ( \alpha , \beta , \gamma )$ be the image of the point $\mathrm { Q } ( 3 , - 3,1 )$ in the line $\frac { x - 0 } { 1 } = \frac { y - 3 } { 1 } = \frac { z - 1 } { - 1 }$ and R be the point $( 2,5 , - 1 )$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda ^ { 2 } = 14 K$, then $K$ is equal to: (1) 36 (2) 81 (3) 72 (4) 18
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is: (1) $\frac { 18 } { 25 }$ (2) $\frac { 12 } { 25 }$ (3) $\frac { 6 } { 25 }$ (4) $\frac { 4 } { 25 }$
The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at $( x _ { 0 } , y _ { 0 } )$. If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$
If the system of equations $$2 x + 7 y + \lambda z = 3$$ $$3 x + 2 y + 5 z = 4$$ $$x + \mu y + 32 z = - 1$$ has infinitely many solutions, then $( \lambda - \mu )$ is equal to $\_\_\_\_$
Let $[ \mathrm { t } ]$ denote the greatest integer less than or equal to t. Let $f : [ 0 , \infty ) \rightarrow \mathbf { R }$ be a function defined by $f ( x ) = \left[ \frac { x } { 2 } + 3 \right] - [ \sqrt { x } ]$. Let $S$ be the set of all points in the interval $[ 0,8 ]$ at which $f$ is not continuous. Then $\sum _ { \mathrm { a } \in \mathrm { S } } \mathrm { a }$ is equal to $\_\_\_\_$
Let $[ t ]$ denote the largest integer less than or equal to $t$. If $$\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$$ where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$
Q88
First order differential equations (integrating factor)View
If the solution $y ( x )$ of the given differential equation $\left( \mathrm { e } ^ { y } + 1 \right) \cos x \mathrm {~d} x + \mathrm { e } ^ { y } \sin x \mathrm {~d} y = 0$ passes through the point $\left( \frac { \pi } { 2 } , 0 \right)$, then the value of $\mathrm { e } ^ { y \left( \frac { \pi } { 6 } \right) }$ is equal to $\_\_\_\_$
If the shortest distance between the lines $\frac { x - \lambda } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { 1 }$ and $\frac { x + 2 } { - 3 } = \frac { y + 5 } { 2 } = \frac { z - 4 } { 4 }$ is $\frac { 44 } { \sqrt { 30 } }$, then the largest possible value of $| \lambda |$ is equal to $\_\_\_\_$
From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ is equal to $\_\_\_\_$