The equation $\sqrt { 3 x ^ { 2 } + x + 5 } = x - 3$, where $x$ is real, has (1) no solution (2) exactly four solutions (3) exactly one solution (4) exactly two solutions
Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval (1) $( 11,13 ]$ (2) $( 14,17 )$ (3) $[ 10,12 )$ (4) $[ 8,9 ]$
Let $f ( n ) = \left[ \frac { 1 } { 3 } + \frac { 3 n } { 100 } \right] n$, where $[ n ]$ denotes the greatest integer less than or equal to $n$. Then $\sum _ { n = 1 } ^ { 56 } f ( n )$ is equal to (1) 56 (2) 1287 (3) 1399 (4) 689
The number of terms in an $A.P$. is even, the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac { 1 } { 2 }$, then the number of terms in the $A.P$. is (1) 4 (2) 8 (3) 16 (4) 12
If a line $L$ is perpendicular to the line $5 x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5 sq units, then the distance of the line $L$ from the line $x + 5 y = 0$ is (1) $\frac { 7 } { \sqrt { 13 } }$ units (2) $\frac { 7 } { \sqrt { 5 } }$ units (3) $\frac { 5 } { \sqrt { 13 } }$ units (4) $\frac { 5 } { \sqrt { 7 } }$ units
The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $\left( a ^ { 2 } + 1 , a ^ { 2 } + 1 \right)$ and $( 2 a , - 2 a ) , a \neq 0$. Then for any $a$, the orthocentre of this triangle lies on the line (1) $y - \left( a ^ { 2 } + 1 \right) x = 0$ (2) $y - 2 a x = 0$ (3) $y + x = 0$ (4) $( a - 1 ) ^ { 2 } x - ( a + 1 ) ^ { 2 } y = 0$
The equation of the circle described on the chord $3 x + y + 5 = 0$ of the circle $x ^ { 2 } + y ^ { 2 } = 16$ as the diameter is (1) $x ^ { 2 } + y ^ { 2 } + 3 x + y + 1 = 0$ (2) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0$ (3) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0$ (4) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0$
A chord is drawn through the focus of the parabola $y ^ { 2 } = 6 x$ such that its distance from the vertex of this parabola is $\frac { \sqrt { 5 } } { 2 }$, then its slope can be (1) $\frac { \sqrt { 5 } } { 2 }$ (2) $\frac { 2 } { \sqrt { 3 } }$ (3) $\frac { \sqrt { 3 } } { 2 }$ (4) $\frac { 2 } { \sqrt { 5 } }$
The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$, meets the $x$-axis and $y$-axis at $A$ and $B$, respectively. Then $OA ^ { 2 } - OB ^ { 2 }$, where $O$ is the origin, equals (1) $- \frac { 20 } { 9 }$ (2) $\frac { 16 } { 9 }$ (3) 4 (4) $- \frac { 4 } { 3 }$
Let $\bar { x } , M$ and $\sigma ^ { 2 }$ be respectively the mean, mode and variance of $n$ observations $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $d _ { i } = - x _ { i } - a , i = 1,2 , \ldots , n$, where $a$ is any number. Statement I: Variance of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ is $\sigma ^ { 2 }$. Statement II: Mean and mode of $d _ { 1 } , d _ { 2 } , \ldots , d _ { n }$ are $- \bar { x } - a$ and $- M - a$, respectively. (1) Statement I and Statement II are both true (2) Statement I and Statement II are both false (3) Statement I is true and Statement II is false (4) Statement I is false and Statement II is true
Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB - BA$ is (1) skew symmetric (2) $I$ or $- I$, where $I$ is an identity matrix (3) symmetric (4) neither symmetric nor skew symmetric
If $\Delta _ { r } = \left| \begin{array} { c c c } r & 2 r - 1 & 3 r - 2 \\ \frac { n } { 2 } & n - 1 & a \\ \frac { 1 } { 2 } n ( n - 1 ) & ( n - 1 ) ^ { 2 } & \frac { 1 } { 2 } ( n - 1 ) ( 3 n + 4 ) \end{array} \right|$, then the value of $\sum _ { r = 1 } ^ { n - 1 } \Delta _ { r }$ (1) Is independent of both $a$ and $n$ (2) Depends only on $a$ (3) Depends only on $n$ (4) Depends both on $a$ and $n$
The function $f ( x ) = | \sin 4 x | + | \cos 2 x |$, is a periodic function with a fundamental period (1) $\pi$ (2) $2 \pi$ (3) $\frac { \pi } { 4 }$ (4) $\frac { \pi } { 2 }$
Let $f : R \rightarrow R$ be defined by $f ( x ) = \frac { | x | - 1 } { | x | + 1 }$, then $f$ is (1) one-one but not onto (2) neither one-one nor onto (3) both one-one and onto (4) onto but not one-one
Let $f : R \rightarrow R$ be a function such that $| f ( x ) | \leq x ^ { 2 }$, for all $x \in R$. Then, at $x = 0 , f$ is (1) differentiable but not continuous (2) neither continuous nor differentiable (3) continuous as well as differentiable (4) continuous but not differentiable
If the volume of a spherical ball is increasing at the rate of $4 \pi \mathrm { cc } / \mathrm { sec }$ then the rate of increase of its radius (in $\mathrm { cm } / \mathrm { sec }$), when the volume is $288 \pi \mathrm { cc }$ is (1) $\frac { 1 } { 9 }$ (2) $\frac { 1 } { 6 }$ (3) $\frac { 1 } { 24 }$ (4) $\frac { 1 } { 36 }$
If non-zero real numbers $b$ and $c$ are such that $\min f ( x ) > \max g ( x )$, where $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 } , ( x \in R )$; then $\left| \frac { c } { b } \right|$ lies in the interval (1) $( \sqrt { 2 } , \infty )$ (2) $\left[ \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$ (3) $\left( 0 , \frac { 1 } { 2 } \right)$ (4) $\left[ \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } \right]$
If $m$ is a non-zero number and $\int \frac { x ^ { 5 m - 1 } + 2 x ^ { 4 m - 1 } } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 3 } } d x = f ( x ) + c$, then $f ( x )$ is equal to (1) $\frac { \left( x ^ { 5 m } - x ^ { 4 m } \right) } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$ (2) $\frac { 1 } { 2 m } \frac { x ^ { 4 m } } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$ (3) $\frac { x ^ { 5 m } } { 2 m \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$ (4) $\frac { 2 m \left( x ^ { 5 m } + x ^ { 4 m } \right) } { \left( x ^ { 2 m } + x ^ { m } + 1 \right) ^ { 2 } }$
Let, the function $F$ be defined as $F ( x ) = \int _ { 1 } ^ { x } \frac { e ^ { t } } { t } d t , x > 0$, then the value of the integral $\int _ { 1 } ^ { x } \frac { e ^ { t } } { t + a } d t$, where $a > 0$, is (1) $e ^ { a } [ F ( x ) - F ( 1 + a ) ]$ (2) $e ^ { - a } [ F ( x + a ) - F ( a ) ]$ (3) $e ^ { a } [ F ( x + a ) - F ( 1 + a ) ]$ (4) $e ^ { - a } [ F ( x + a ) - F ( 1 + a ) ]$
If $\vec { x } = 3 \hat { i } - 6 \hat { j } - \widehat { k } , \vec { y } = \hat { i } + 4 \hat { j } - 3 \widehat { k }$ and $\vec { z } = 3 \hat { i } - 4 \hat { j } - 12 \widehat { k }$, then the magnitude of the projection of $\vec { x } \times \vec { y }$ on $\vec { z }$ is (1) 14 (2) 12 (3) 15 (4) 10
If the angle between the line $2 ( x + 1 ) = y = z + 4$ and the plane $2 x - y + \sqrt { \lambda } z + 4 = 0$ is $\frac { \pi } { 6 }$, then the value of $\lambda$ is (1) $\frac { 45 } { 7 }$ (2) $\frac { 135 } { 11 }$ (3) $\frac { 135 } { 7 }$ (4) $\frac { 45 } { 11 }$
Let $A$ and $E$ be any two events with positive probabilities Statement I: $P ( E / A ) \geq P ( A / E ) P ( E )$. Statement II: $P ( A / E ) \geq P ( A \cap E )$. (1) Both the statements are false (2) Both the statements are true (3) Statement-I is false, Statement-II is true (4) Statement - I is true, Statement - II is false