Let $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ be three sets defined as $S _ { 1 } = \{ z \in \mathbb { C } : | z - 1 | \leq \sqrt { 2 } \}$, $S _ { 2 } = \{ z \in \mathbb { C } : \operatorname { Re } ( ( 1 - i ) z ) \geq 1 \}$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Im } ( z ) \leq 1 \}$. Then, the set $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$ (1) is a singleton (2) has exactly two elements (3) has infinitely many elements (4) has exactly three elements
If the sides $A B , B C$ and $C A$ of a triangle $A B C$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to: (1) 364 (2) 240 (3) 333 (4) 360
Two tangents are drawn from a point $P$ to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$, such that the angle between these tangents is $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right)$, where $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right) \in ( 0 , \pi )$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\triangle P A B$ and $\triangle C A B$ is: (1) $11 : 4$ (2) $9 : 4$ (3) $3 : 1$ (4) $2 : 1$
Let the tangent to the circle $x ^ { 2 } + y ^ { 2 } = 25$ at the point $R ( 3,4 )$ meet $x$-axis and $y$-axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $O P Q$, then $r ^ { 2 }$ is equal to (1) $\frac { 529 } { 64 }$ (2) $\frac { 125 } { 72 }$ (3) $\frac { 625 } { 72 }$ (4) $\frac { 585 } { 66 }$
Let $L$ be a tangent line to the parabola $y ^ { 2 } = 4 x - 20$ at (6, 2). If $L$ is also a tangent to the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { b } = 1$, then the value of $b$ is equal to: (1) 11 (2) 14 (3) 16 (4) 20
If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[ \begin{array} { c c c } 3 & 4 \sqrt { 2 } & x \\ 4 & 5 \sqrt { 2 } & y \\ 5 & k & z \end{array} \right]$ is zero, then the value of $k ^ { 2 }$ is (1) 72 (2) 12 (3) 36 (4) 6
Consider the function $f : R \rightarrow R$ defined by $f ( x ) = \left\{ \begin{array} { c c } \left( 2 - \sin \left( \frac { 1 } { x } \right) \right) | x | , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$. Then $f$ is: (1) monotonic on $( - \infty , 0 ) \cup ( 0 , \infty )$ (2) not monotonic on $( - \infty , 0 )$ and $( 0 , \infty )$ (3) monotonic on $( 0 , \infty )$ only (4) monotonic on $( - \infty , 0 )$ only
If the integral $\int _ { 0 } ^ { 10 } \frac { [ \sin 2 \pi x ] } { \mathrm { e } ^ { x - [ x ] } } d x = \alpha e ^ { - 1 } + \beta e ^ { - \frac { 1 } { 2 } } + \gamma$, where $\alpha , \beta , \gamma$ are integers and $[ x ]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha + \beta + \gamma$ is equal to: (1) 0 (2) 20 (3) 25 (4) 10
Q76
First order differential equations (integrating factor)View
Let $y = y ( x )$ be the solution of the differential equation $\cos x ( 3 \sin x + \cos x + 3 ) d y = ( 1 + y \sin x ( 3 \sin x + \cos x + 3 ) ) d x , 0 \leq x \leq \frac { \pi } { 2 } , y ( 0 ) = 0$. Then, $y \left( \frac { \pi } { 3 } \right)$ is equal to: (1) $2 \log _ { \mathrm { e } } \left( \frac { 2 \sqrt { 3 } + 9 } { 6 } \right)$ (2) $2 \log _ { \mathrm { e } } \left( \frac { 2 \sqrt { 3 } + 10 } { 11 } \right)$ (3) $2 \log _ { \mathrm { e } } \left( \frac { \sqrt { 3 } + 7 } { 2 } \right)$ (4) $2 \log _ { \mathrm { e } } \left( \frac { 3 \sqrt { 3 } - 8 } { 4 } \right)$
Q77
First order differential equations (integrating factor)View
If the curve $y = y ( x )$ is the solution of the differential equation $2 \left( x ^ { 2 } + x ^ { 5 / 4 } \right) d y - y \left( x + x ^ { 1 / 4 } \right) d x = 2 x ^ { 9 / 4 } d x , x > 0$ which passes through the point $\left( 1,1 - \frac { 4 } { 3 } \log _ { \mathrm { e } } 2 \right)$, then the value of $y ( 16 )$ is equal to (1) $4 \left( \frac { 31 } { 3 } + \frac { 8 } { 3 } \log _ { e } 3 \right)$ (2) $\left( \frac { 31 } { 3 } + \frac { 8 } { 3 } \log _ { e } 3 \right)$ (3) $4 \left( \frac { 31 } { 3 } - \frac { 8 } { 3 } \log _ { \mathrm { e } } 3 \right)$ (4) $\left( \frac { 31 } { 3 } - \frac { 8 } { 3 } \log _ { e } 3 \right)$
Let $O$ be the origin. Let $\overrightarrow { O P } = x \hat { i } + y \hat { j } - \widehat { k }$ and $\overrightarrow { O Q } = - \hat { i } + 2 \hat { j } + 3 x \hat { k } , x , y \in R , x > 0$, be such that $| \overrightarrow { P Q } | = \sqrt { 20 }$ and the vector $\overrightarrow { O P }$ is perpendicular to $\overrightarrow { O Q }$. If $\overrightarrow { O R } = 3 \hat { i } + \mathrm { z } \hat { j } - 7 \hat { k } , z \in R$, is coplanar with $\overrightarrow { O P }$ and $\overrightarrow { O Q }$, then the value of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is equal to (1) 7 (2) 9 (3) 2 (4) 1
If the equation of plane passing through the mirror image of a point $( 2,3,1 )$ with respect to line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z + 2 } { - 1 }$ and containing the line $\frac { x - 2 } { 3 } = \frac { 1 - y } { 2 } = \frac { z + 1 } { 1 }$ is $\alpha x + \beta y + \gamma z = 24$ then $\alpha + \beta + \gamma$ is equal to: (1) 20 (2) 19 (3) 18 (4) 21
Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac { 1 } { 2 }$ and probability of occurrence of 0 at the odd place be $\frac { 1 } { 3 }$. Then the probability that 10 is followed by 01 is equal to: (1) $\frac { 1 } { 18 }$ (2) $\frac { 1 } { 3 }$ (3) $\frac { 1 } { 6 }$ (4) $\frac { 1 } { 9 }$
If $1 , \log _ { 10 } \left( 4 ^ { x } - 2 \right)$ and $\log _ { 10 } \left( 4 ^ { x } + \frac { 18 } { 5 } \right)$ are in arithmetic progression for a real number $x$ then the value of the determinant $\left| \begin{array} { c c c } 2 \left( x - \frac { 1 } { 2 } \right) & x - 1 & x ^ { 2 } \\ 1 & 0 & x \\ x & 1 & 0 \end{array} \right|$ is equal to:
Let the coefficients of third, fourth and fifth terms in the expansion of $\left( x + \frac { a } { x ^ { 2 } } \right) ^ { n } , x \neq 0$, be in the ratio $12 : 8 : 3$. Then the term independent of $x$ in the expansion, is equal to $\_\_\_\_$ .
Let $\tan \alpha , \tan \beta$ and $\tan \gamma ; \alpha , \beta , \gamma \neq \frac { ( 2 n - 1 ) \pi } { 2 } , n \in N$ be the slopes of the three line segments $O A , O B$ and $O C$, respectively, where $O$ is origin. If circumcentre of $\Delta A B C$ coincides with origin and its orthocentre lies on $y$-axis, then the value of $\left( \frac { \cos 3 \alpha + \cos 3 \beta + \cos 3 \gamma } { \cos \alpha \cdot \cos \beta \cdot \cos \gamma } \right) ^ { 2 }$ is equal to:
Consider a set of $3 n$ numbers having variance 4 . In this set, the mean of first $2 n$ numbers is 6 and the mean of the remaining $n$ numbers is 3 . A new set is constructed by adding 1 into each of the first $2 n$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If the variance of the new set is $k$, then $9 k$ is equal to $\_\_\_\_$ .
Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ and $B = \left[ \begin{array} { l } \alpha \\ \beta \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \end{array} \right]$ such that $A B = B$ and $a + d = 2021$, then the value of $a d - b c$ is equal to $\_\_\_\_$.
Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to
Let $I _ { n } = \int _ { 1 } ^ { e } x ^ { 19 } ( \log | x | ) ^ { n } d x$, where $n \in N$. If (20) $I _ { 10 } = \alpha I _ { 9 } + \beta I _ { 8 }$, for natural numbers $\alpha$ and $\beta$, then $\alpha - \beta$ equal to $\_\_\_\_$ .
Let $f : [ - 3,1 ] \rightarrow R$ be given as $f ( x ) = \left\{ \begin{array} { l l } \min \left\{ ( x + 6 ) , x ^ { 2 } \right\} , & - 3 \leq x \leq 0 \\ \max \left\{ \sqrt { x } , x ^ { 2 } \right\} , & 0 \leq x \leq 1 \end{array} \right.$. If the area bounded by $y = f ( x )$ and $x$-axis is $A$ sq units, then the value of $6 A$ is equal to
Let $\vec { x }$ be a vector in the plane containing vectors $\vec { a } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\vec { b } = \hat { i } + 2 \hat { j } - \hat { k }$. If the vector $\vec { x }$ is perpendicular to $( 3 \hat { i } + 2 \hat { j } - \widehat { k } )$ and its projection on $\vec { a }$ is $\frac { 17 \sqrt { 6 } } { 2 }$, then the value of $| \vec { x } | ^ { 2 }$ is equal to $\_\_\_\_$ .
Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $x + y + z = 0 , l x - n z = 0$ and $x - 2 y + z = 0$ equal to 9 units. If the locus of the point $P$ is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$, then the value of $l - n$ is equal to