If $z = \frac { 1 } { 2 } - 2 i$, is such that $| z + 1 | = \alpha z + \beta ( 1 + i ) , i = \sqrt { - 1 }$ and $\alpha , \beta \in \mathrm { R }$, then $\alpha + \beta$ is equal to (1) - 4 (2) 3 (3) 2 (4) - 1
In an A.P., the sixth term $\mathbf { a } _ { 6 } = 2$. If the $\mathbf { a } _ { 1 } \mathbf { a } _ { 4 } \mathbf { a } _ { 5 }$ is the greatest, then the common difference of the A.P., is equal to (1) $\frac { 3 } { 2 }$ (2) $\frac { 8 } { 5 }$ (3) $\frac { 2 } { 3 }$ (4) $\frac { 5 } { 8 }$
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to (1) 7 (2) 4 (3) 5 (4) 6
Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is (1) 60 (2) 53 (3) 62 (4) 30
In a $\triangle \mathrm { ABC }$, suppose $\mathrm { y } = \mathrm { x }$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$. If $2 A B = B C$ and the point $A$ and $B$ are respectively $( 4,6 )$ and $( \alpha , \beta )$, then $\alpha + 2 \beta$ is equal to (1) - 4 (2) 42 (3) 2 (4) - 1
If $f ( x ) = \left\{ \begin{array} { l } 2 + 2 x , - 1 \leq x < 0 \\ 1 - \frac { x } { 3 } , 0 \leq x \leq 3 \end{array} ; g ( x ) = \left\{ \begin{array} { l } - x , - 3 \leq x \leq 0 \\ x , 0 < x \leq 1 \end{array} \right. \right.$, then range of $( f \circ g ( x ) )$ is (1) $( 0,1 ]$ (2) $[ 0,3 )$ (3) $[ 0,1 ]$ (4) $[ 0,1 )$
Consider the function $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathrm { R }$ defined by $f ( x ) = 4 \sqrt { 2 } x ^ { 3 } - 3 \sqrt { 2 } x - 1$. Consider the statements (I) The curve $y = f ( x )$ intersects the $x$-axis exactly at one point (II) The curve $y = f ( x )$ intersects the $x$-axis at $x = \cos \frac { \pi } { 12 }$ Then (1) Only (II) is correct (2) Both (I) and (II) are incorrect (3) Only (I) is correct (4) Both (I) and (II) are correct
Let $O$ be the origin and the position vector of $A$ and $B$ be $2 \hat { i } + 2 \hat { j } + \widehat { k }$ and $2 \hat { i } + 4 \hat { j } + 4 \widehat { k }$ respectively. If the internal bisector of $\angle A O B$ meets the line $A B$ at $C$, then the length of $O C$ is (1) $\frac { 2 } { 3 } \sqrt { 31 }$ (2) $\frac { 2 } { 3 } \sqrt { 34 }$ (3) $\frac { 3 } { 4 } \sqrt { 34 }$ (4) $\frac { 3 } { 2 } \sqrt { 31 }$
Let $P Q R$ be a triangle with $R ( - 1,4,2 )$. Suppose $M ( 2,1,2 )$ is the mid point of $P Q$. The distance of the centroid of $\triangle P Q R$ from the point of intersection of the line $\frac { x - 2 } { 0 } = \frac { y } { 2 } = \frac { z + 3 } { - 1 }$ and $\frac { x - 1 } { 1 } = \frac { y + 3 } { - 3 } = \frac { z + 1 } { 1 }$ is (1) 69 (2) 9 (3) $\sqrt { 69 }$ (4) $\sqrt { 99 }$
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is (1) $\frac { 5 } { 6 }$ (2) $\frac { 1 } { 6 }$ (3) $\frac { 5 } { 11 }$ (4) $\frac { 6 } { 11 }$
All the letters of the word $G T W E N T Y$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $G T W E N T Y$ IS
Equations of two diameters of a circle are $2 x - 3 y = 5$ and $3 x - 4 y = 7$. The line joining the points $\left( - \frac { 22 } { 7 } , - 4 \right)$ and $\left( - \frac { 1 } { 7 } , 3 \right)$ intersects the circle at only one point $P ( \alpha , \beta )$. Then $17 \beta - \alpha$ is equal to
If the points of intersection of two distinct conics $x ^ { 2 } + y ^ { 2 } = 4 b$ and $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ lie on the curve $y ^ { 2 } = 3 x ^ { 2 }$, then $3 \sqrt { 3 }$ times the area of the rectangle formed by the intersection points is $\_\_\_\_$ .
If the mean and variance of the data $65,68,58,44,48,45,60 , \alpha , \beta , 60$ where $\alpha > \beta$ are 56 and 66.2 respectively, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to
Let $\mathrm { f } ( \mathrm { x } ) = 2 ^ { \mathrm { x } } - \mathrm { x } ^ { 2 } , \mathrm { x } \in \mathrm { R }$. If m and n are respectively the number of points at which the curves $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ and $\mathrm { y } = \mathrm { f } ^ { \prime } ( \mathrm { x } )$ intersects the x-axis, then the value of $\mathrm { m } + \mathrm { n }$ is
The area (in sq. units) of the part of circle $x ^ { 2 } + y ^ { 2 } = 169$ which is below the line $5 x - y = 13$ is $\frac { \pi \alpha } { 2 \beta } - \frac { 65 } { 2 } + \frac { \alpha } { \beta } \sin ^ { - 1 } \left( \frac { 12 } { 13 } \right)$ where $\alpha , \beta$ are coprime numbers. Then $\alpha + \beta$ is equal to
Q89
First order differential equations (integrating factor)View
If the solution curve $y = y ( x )$ of the differential equation $\left( 1 + y ^ { 2 } \right) \left( 1 + \log _ { e } x \right) d x + x d y = 0 , x > 0$ passes through the point $( 1,1 )$ and $y ( e ) = \frac { \alpha - \tan \left( \frac { 3 } { 2 } \right) } { \beta + \tan \left( \frac { 3 } { 2 } \right) }$, then $\alpha + 2 \beta$ is
A line with direction ratio $2,1,2$ meets the lines $\mathrm { x } = \mathrm { y } + 2 = \mathrm { z }$ and $\mathrm { x } + 2 = 2 \mathrm { y } = 2 \mathrm { z }$ respectively at the point P and Q . if the length of the perpendicular from the point $( 1,2,12 )$ to the line PQ is $l$, then $l ^ { 2 }$ is