The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is : (1) 89 (2) 84 (3) 86 (4) 79
Let $a _ { 1 } = b _ { 1 } = 1$ and $a _ { n } = a _ { n - 1 } + ( n - 1 ) , b _ { n } = b _ { n - 1 } + \mathrm { a } _ { n - 1 } , \forall n \geq 2$. If $\mathrm { S } = \sum _ { \mathrm { n } = 1 } ^ { 10 } \left( \frac { b _ { n } } { 2 ^ { n } } \right)$ and $\mathrm { T } = \sum _ { n = 1 } ^ { 8 } \frac { \mathrm { n } } { 2 ^ { n - 1 } }$ then $2 ^ { 7 } ( 2 S - T )$ is equal to $\_\_\_\_$
Let $\left\{ a _ { k } \right\}$ and $\left\{ b _ { k } \right\} , k \in \mathbb { N }$, be two G.P.s with common ratio $r _ { 1 }$ and $r _ { 2 }$ respectively such that $\mathrm { a } _ { 1 } = \mathrm { b } _ { 1 } = 4$ and $\mathrm { r } _ { 1 } < \mathrm { r } _ { 2 }$. Let $\mathrm { c } _ { \mathrm { k } } = \mathrm { a } _ { \mathrm { k } } + \mathrm { b } _ { \mathrm { k } } , \mathrm { k } \in \mathbb { N }$. If $\mathrm { c } _ { 2 } = 5$ and $\mathrm { c } _ { 3 } = \frac { 13 } { 4 }$ then $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { c } _ { \mathrm { k } } - \left( 12 \mathrm { a } _ { 6 } + 8 \mathrm {~b} _ { 4 } \right)$ is equal to $\_\_\_\_$
Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $( 1 + x ) ^ { 99 }$. Let a be the middle term in the expansion of $\left( 2 + \frac { 1 } { \sqrt { 2 } } \right) ^ { 200 }$. If $\frac { { } ^ { 200 } C _ { 99 } K } { a } = \frac { 2 ^ { l } m } { n }$, where $m$ and $n$ are odd numbers, then the ordered pair $( l , \mathrm { n } )$ is equal to: (1) $( 50,51 )$ (2) $( 51,99 )$ (3) $( 50,101 )$ (4) $( 51,101 )$
A circle with centre $( 2,3 )$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S ( \alpha , \beta )$, then $4 \alpha - 7 \beta$ is equal to $\_\_\_\_$
A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$
If the tangent at a point P on the parabola $\mathrm { y } ^ { 2 } = 3 \mathrm { x }$ is parallel to the line $x + 2 y = 1$ and the tangents at the points $Q$ and $R$ on the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 1 } = 1$ are perpendicular to the line $x - y = 2$, then the area of the triangle $P Q R$ is: (1) $\frac { 9 } { \sqrt { 5 } }$ (2) $5 \sqrt { 3 }$ (3) $\frac { 3 } { 2 } \sqrt { 5 }$ (4) $3 \sqrt { 5 }$
Let $X = \{ 11,12,13 , \ldots , 40,41 \}$ and $Y = \{ 61,62,63 , \ldots , 90,91 \}$ be the two sets of observations. If $\overline { \mathrm { x } }$ and $\overline { \mathrm { y } }$ are their respective means and $\sigma ^ { 2 }$ is the variance of all the observations in $\mathrm { X } \cup \mathrm { Y }$, then $\left| \overline { \mathrm { x } } + \overline { \mathrm { y } } - \sigma ^ { 2 } \right|$ is equal to $\_\_\_\_$
The set of all values of $t \in \mathbb { R }$, for which the matrix $\left[ \begin{array} { c c c } \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) & \mathrm { e } ^ { - t } ( - 2 \sin t - \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( 2 \sin t + \cos t ) & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } \cos t & \mathrm { e } ^ { - t } \sin t \end{array} \right]$ is invertible, is (1) $\left\{ ( 2 k + 1 ) \frac { \pi } { 2 } , k \in \mathbb { Z } \right\}$ (2) $\left\{ k \pi + \frac { \pi } { 4 } , k \in \mathbb { Z } \right\}$ (3) $\{ k \pi , k \in \mathbb { Z } \}$ (4) $\mathbb { R }$
Let A be a symmetric matrix such that $| A | = 2$ and $\left[ \begin{array} { l l } 2 & 1 \\ 3 & \frac { 3 } { 2 } \end{array} \right] A = \left[ \begin{array} { l l } 1 & 2 \\ \alpha & \beta \end{array} \right]$. If the sum of the diagonal elements of A is $s$, then $\frac { \beta s } { \alpha ^ { 2 } }$ is equal to $\_\_\_\_$.
Consider a function $\mathrm { f } : \mathbb { N } \rightarrow \mathbb { R }$, satisfying $f ( 1 ) + 2 f ( 2 ) + 3 f ( 3 ) + \ldots + x f ( x ) = x ( x + 1 ) f ( x ) ; x \geq 2$ with $f ( 1 ) = 1$. Then $\frac { 1 } { f ( 2022 ) } + \frac { 1 } { f ( 2028 ) }$ is equal to (1) 8200 (2) 8000 (3) 8400 (4) 8100
Let $f$ and $g$ be twice differentiable functions on $R$ such that $f ^ { \prime \prime } ( x ) = g ^ { \prime \prime } ( x ) + 6 x$ $f ^ { \prime } ( 1 ) = 4 g ^ { \prime } ( 1 ) - 3 = 9$ $f ( 2 ) = 3 g ( 2 ) = 12$ Then which of the following is NOT true ? (1) $g ( - 2 ) - f ( - 2 ) = 20$ (2) If $- 1 < x < 2$, then $| f ( x ) - g ( x ) | < 8$ (3) $\left| f ^ { \prime } ( x ) - g ^ { \prime } ( x ) \right| < 6 \Rightarrow - 1 < x < 1$ (4) There exists $x _ { 0 } \in \left( 1 , \frac { 3 } { 2 } \right)$ such that $f \left( x _ { 0 } \right) = g \left( x _ { 0 } \right)$
If the equation of the normal to the curve $y = \frac { x - a } { ( x + b ) ( x - 2 ) }$ at the point $( 1 , - 3 )$ is $x - 4 y = 13$ then the value of $a + b$ is equal to $\_\_\_\_$
The area of the region $A = \left\{ ( x , y ) : | \cos x - \sin x | \leq y \leq \sin x , 0 \leq x \leq \frac { \pi } { 2 } \right\}$ is: (1) $1 - \frac { 3 } { \sqrt { 2 } } + \frac { 4 } { \sqrt { 5 } }$ (2) $\sqrt { 5 } + 2 \sqrt { 2 } - 4.5$ (3) $\frac { 3 } { \sqrt { 5 } } - \frac { 3 } { \sqrt { 2 } } + 1$ (4) $\sqrt { 5 } - 2 \sqrt { 2 } + 1$
Q84
First order differential equations (integrating factor)View
Let $y = y ( x )$ be the solution of the differential equation $x \log _ { e } x \frac { d y } { d x } + y = x ^ { 2 } \log _ { e } x , ( x > 1 )$. If $y ( 2 ) = 2$, then $y ( e )$ is equal to (1) $\frac { 4 + e ^ { 2 } } { 4 }$ (2) $\frac { 1 + e ^ { 2 } } { 4 }$ (3) $\frac { 2 + e ^ { 2 } } { 2 }$ (4) $\frac { 1 + e ^ { 2 } } { 2 }$
The plane $2 x - y + z = 4$ intersects the line segment joining the points $\mathrm { A } ( \mathrm { a } , - 2,4 )$ and $\mathrm { B } ( 2 , \mathrm {~b} , - 3 )$ at the point C in the ratio 2 : 1 and the distance of the point C from the origin is $\sqrt { 5 }$. If $a b < 0$ and P is the point $( \mathrm { a } - \mathrm { b } , \mathrm { b } , 2 \mathrm {~b} - \mathrm { a } )$ then $\mathrm { CP } ^ { 2 }$ is equal to: (1) $\frac { 17 } { 3 }$ (2) $\frac { 16 } { 3 }$ (3) $\frac { 73 } { 3 }$ (4) $\frac { 97 } { 3 }$
If the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { 1 }$ and $\frac { x - a } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 3 } { 1 }$ intersects at the point $P$, then the distance of the point $P$ from the plane $z = a$ is : (1) 16 (2) 28 (3) 10 (4) 22
Let $S = \left\{ w _ { 1 } , w _ { 2 } , \ldots \right\}$ be the sample space associated to a random experiment. Let $P \left( w _ { n } \right) = \frac { P \left( w _ { n - 1 } \right) } { 2 } , n \geq 2$ . Let $A = \{ 2 k + 3 l ; k , l \in \mathbb { N } \}$ and $B = \left\{ w _ { n } ; n \in A \right\}$. Then $P ( B )$ is equal to (1) $\frac { 3 } { 32 }$ (2) $\frac { 3 } { 64 }$ (3) $\frac { 1 } { 16 }$ (4) $\frac { 1 } { 32 }$