The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is (1) 16800 (2) 33600 (3) 18000 (4) 14800
The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is (1) 720 (2) $126 ( 5 ! ) ^ { 2 }$ (3) $7 ( 360 ) ^ { 2 }$ (4) $7 ( 720 ) ^ { 2 }$
Let $S _ { K } = \frac { 1 + 2 + \ldots + K } { K }$ and $\sum _ { j = 1 } ^ { n } S ^ { 2 } { } _ { j } = \frac { n } { A } \left( B n ^ { 2 } + C n + D \right)$ where $A , B , C , D \in N$ and $A$ has least value then (1) $A + C + D$ is not divisible by $D$ (2) $A + B = 5 ( D - C )$ (3) $A + B + C + D$ is divisible by 5 (4) $A + B$ is divisible by $D$
If the coefficients of three consecutive terms in the expansion of $( 1 + x ) ^ { n }$ are in the ratio $1 : 5 : 20$ then the coefficient of the fourth term is (1) 2436 (2) 5481 (3) 1827 (4) 3654
Let $[ t ]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left( 3 x ^ { 2 } - \frac { 1 } { 2 x ^ { 5 } } \right) ^ { 7 }$ is $\alpha$ then $[ \alpha ]$ is equal to $\_\_\_\_$
Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to (1) 18 (2) 17 (3) 15 (4) 16
Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$
Let $R$ be the focus of the parabola $y ^ { 2 } = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$. Let the point $G ( 10 , 10 )$ be the centroid of the triangle $P Q R$. If $c - m = 6$, then $P Q ^ { 2 }$ is (1) 296 (2) 325 (3) 317 (4) 346
Let the mean and variance of 8 numbers $x , y , 10 , 12 , 6 , 12 , 4 , 8$ be 9 and 9.25 respectively. If $x > y$, then $3 x - 2 y$ is equal to $\_\_\_\_$
Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is (1) 752 (2) 782 (3) 792 (4) 772
Let $A = \{ 0 , 3 , 4 , 6 , 7 , 8 , 9 , 10 \}$ and $R$ be the relation defined on $A$ such that $R = \{ ( x , y ) \in A \times A : x - y$ is odd positive integer or $x - y = 2 \}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $\_\_\_\_$
Let $P = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $Q = P A P ^ { T }$. If $P ^ { T } Q ^ { 2007 } P = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ then $2 a + b - 3 c - 4 d$ is equal to (1) 2004 (2) 2005 (3) 2007 (4) 2006
If $a _ { \alpha }$ is the greatest term in the sequence $a _ { n } = \frac { n ^ { 3 } } { n ^ { 4 } + 147 } , n = 1 , 2 , 3 \ldots$, then $\alpha$ is equal to $\_\_\_\_$
Let $I ( x ) = \int \frac { x + 1 } { x \left( 1 + x e ^ { x } \right) ^ { 2 } } d x , x > 0$. If $\lim _ { x \rightarrow \infty } I ( x ) = 0$ then $I ( 1 )$ is equal to (1) $\frac { e + 2 } { e + 1 } - \log _ { e } ( e + 1 )$ (2) $\frac { e + 1 } { e + 2 } + \log _ { e } ( e + 1 )$ (3) $\frac { e + 1 } { e + 2 } - \log _ { e } ( e + 1 )$ (4) $\frac { e + 2 } { e + 1 } + \log _ { e } ( e + 1 )$
If the solution curve of the differential equation $\left( y - 2 \log _ { e } x \right) d x + \left( x \log _ { e } x ^ { 2 } \right) d y = 0 , x > 1$ passes through the points $\left( e , \frac { 4 } { 3 } \right)$ and $\left( e ^ { 4 } , \alpha \right)$, then $\alpha$ is equal to $\_\_\_\_$
Let $\vec { a } = 6 \hat { i } + 9 \hat { j } + 12 \hat { k } , \vec { b } = \alpha \hat { i } + 11 \hat { j } - 2 \hat { k }$ and $\vec { c }$ be vectors such that $\vec { a } \times \vec { c } = \vec { a } \times \vec { b }$. If $\vec { a } \cdot \vec { c } = - 12$, and $\vec { c } \cdot ( \hat { i } - 2 \hat { j } + \hat { k } ) = 5$ then $\vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } )$ is equal to $\_\_\_\_$
If the equation of the plane containing the line $x + 2 y + 3 z - 4 = 0 = 2 x + y - z + 5$ and perpendicular to the plane $\vec { r } = ( \hat { i } - \hat { j } ) + \lambda ( \hat { i } + \hat { j } + \hat { k } ) + \mu ( \hat { i } - 2 \hat { j } + 3 \hat { k } )$ is $a x + b y + c z = 4$ then $( a - b + c )$ is equal to (1) 18 (2) 22 (3) 20 (4) 24
Let $\lambda _ { 1 } , \lambda _ { 2 }$ be the values of $\lambda$ for which the points $\left( \frac { 5 } { 2 } , 1 , \lambda \right)$ and $( - 2 , 0 , 1 )$ are at equal distance from the plane $2 x + 3 y - 6 z + 7 = 0$. If $\lambda _ { 1 } > \lambda _ { 2 }$, then the distance of the point $\left( \lambda _ { 1 } - \lambda _ { 2 } , \lambda _ { 2 } , \lambda _ { 1 } \right)$ from the line $\frac { x - 5 } { 1 } = \frac { y - 1 } { 2 } = \frac { z + 7 } { 2 }$ is $\_\_\_\_$
In a bolt factory, machines $A , B$ and $C$ manufacture respectively $20 \% , 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective then the probability that it is manufactured by the machine $C$ is (1) $\frac { 5 } { 14 }$ (2) $\frac { 9 } { 28 }$ (3) $\frac { 3 } { 7 }$ (4) $\frac { 2 } { 7 }$