If the difference between the roots of the equation $x ^ { 2 } + a x + 1 = 0$ is less than $\sqrt { 5 }$, then the set of possible values of $a$ is (1) $( - 3,3 )$ (2) $( - 3 , \infty )$ (3) $( 3 , \infty )$ (4) $( - \infty , - 3 )$
In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals (1) $\frac { 1 } { 2 } ( 1 - \sqrt { 5 } )$ (2) $\frac { 1 } { 2 } \sqrt { 5 }$ (3) $\sqrt { 5 }$ (4) $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$
If $p$ and $q$ are positive real numbers such that $p ^ { 2 } + q ^ { 2 } = 1$, then the maximum value of ( $p + q$ ) is (1) 2 (2) $1 / 2$ (3) $\frac { 1 } { \sqrt { 2 } }$ (4) $\sqrt { 2 }$
In the binomial expansion of $( a - b ) ^ { n } , n \geq 5$, the sum of $5 ^ { \text {th } }$ and $6 ^ { \text {th } }$ terms is zero, then $\frac { a } { b }$ equals (1) $\frac { 5 } { n - 4 }$ (2) $\frac { 6 } { n - 5 }$ (3) $\frac { n - 5 } { 6 }$ (4) $\frac { n - 4 } { 5 }$
Let $A ( h , k ) , B ( 1,1 )$ and $C ( 2,1 )$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' k ' can take is given by (1) $\{ 1,3 \}$ (2) $\{ 0,2 \}$ (3) $\{ - 1,3 \}$ (4) $\{ - 3 , - 2 \}$
If one of the lines of $m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0$ is a bisector of the angle between the lines $x y = 0$, then $m$ is (1) $- 1 / 2$ (2) - 2 (3) 1 (4) 2
Consider a family of circles which are passing through the point $( - 1,1 )$ and are tangent to $x -$ axis. If $( h , k )$ are the co-ordinates of the centre of the circles, then the set of values of $k$ is given by the interval (1) $0 < \mathrm { k } < 1 / 2$ (2) $k \geq 1 / 2$ (3) $- 1 / 2 \leq k \leq 1 / 2$ (4) $k \leq 1 / 2$
The equation of a tangent to the parabola $y ^ { 2 } = 8 x$ is $y = x + 2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is (1) $( - 1,1 )$ (2) $( 0,2 )$ (3) $( 2,4 )$ (4) $( - 2,0 )$
The function $f : R \sim \{ 0 \} \rightarrow R$ given by $f ( x ) = \frac { 1 } { x } - \frac { 2 } { e ^ { 2 x } - 1 }$ can be made continuous at $x = 0$ by defining $f ( 0 )$ as (1) 2 (2) - 1 (3) 0 (4) 1
The average marks of boys in a class is 52 and that of girls is 42 . The average marks of boys and girls combined is 50 . The percentage of boys in the class is (1) 40 (2) 20 (3) 80 (4) 60
A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $A B ( = a )$ subtends an angle of $60 ^ { \circ }$ at the foot of the tower, and the angle of elevation of the top of the tower from $A$ or $B$ is $30 ^ { \circ }$. The height of the tower is (1) $\frac { 2 a } { \sqrt { 3 } }$ (2) $2 a \sqrt { 3 }$ (3) $\frac { a } { \sqrt { 3 } }$ (4) $a \sqrt { 3 }$
If $D = \left| \begin{array} { c c c } 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{array} \right|$ for $x \neq 0 , y \neq 0$ then $D$ is (1) divisible by neither $x$ nor $y$ (2) divisible by both $x$ and $y$ (3) divisible by $x$ but not $y$ (4) divisible by $y$ but not $x$
Let $f : R \rightarrow R$ be a function defined by $f ( x ) = \operatorname { Min } \{ x + 1 , | x | + 1 \}$. Then which of the following is true? (1) $f ( x ) \geq 1$ for all $x \in R$ (2) $f ( x )$ is not differentiable at $x = 1$ (3) $f ( x )$ is differentiable everywhere (4) $f ( x )$ is not differentiable at $x = 0$
The normal to a curve at $P ( x , y )$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a (1) ellipse (2) parabola (3) circle (4) pair of straight lines
A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f ( x ) = \log _ { \mathrm { e } } x$ on the interval $[ 1,3 ]$ is (1) $2 \log _ { 3 } e$ (2) $\frac { 1 } { 2 } \log _ { e } 3$ (3) $\log _ { 3 } e$ (4) $\log _ { e } 3$
Let $F ( x ) = f ( x ) + f \left( \frac { 1 } { x } \right)$, where $f ( x ) = \int _ { 1 } ^ { x } \frac { \log t } { 1 + t } d t$. Then $F ( e )$ equals (1) $\frac { 1 } { 2 }$ (2) 0 (3) 1 (4) 2
Q110
Integration using inverse trig and hyperbolic functionsView
The solution for $x$ of the equation $\int _ { \sqrt { 2 } } ^ { x } \frac { d t } { t \sqrt { t ^ { 2 } - 1 } } = \frac { \pi } { 2 }$ is (1) 2 (2) $\pi$ (3) $\frac { \sqrt { 3 } } { 2 }$ (4) None of these
The differential equation of all circles passing through the origin and having their centres on the $x$-axis is (1) $x ^ { 2 } = y ^ { 2 } + x y \frac { d y } { d x }$ (2) $x ^ { 2 } = y ^ { 2 } + 3 x y \frac { d y } { d x }$ (3) $y ^ { 2 } = x ^ { 2 } + 2 x y \frac { d y } { d x }$ (4) $y ^ { 2 } = x ^ { 2 } - 2 x y \frac { d y } { d x }$
The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be $\sqrt { 19 } \mathrm {~N}$. The value of P is (1) 5 N (2) 6 N (3) 3 N (4) 4 N
If $\hat { u }$ and $\hat { v }$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat { u } \times 3 \hat { v }$ is a unit vector for (1) exactly two values of $\theta$ (2) more than two values of $\theta$ (3) no value of $\theta$ (4) exactly one value of $\theta$
Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$. If $L$ makes an angles $\alpha$ with the positive $x$-axis, then $\cos \alpha$ equals (1) $\frac { 1 } { \sqrt { 3 } }$ (2) $\frac { 1 } { 2 }$ (3) 1 (4) $\frac { 1 } { \sqrt { 2 } }$
If a line makes an angle of $\frac { \pi } { 4 }$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is (1) $\frac { \pi } { 6 }$ (2) $\frac { \pi } { 3 }$ (3) $\frac { \pi } { 4 }$ (4) $\frac { \pi } { 2 }$
If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are (1) $( 4,9 , - 3 )$ (2) $( 4 , - 3,3 )$ (3) $( 4,3,5 )$ (4) $( 4,3 , - 3 )$
A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is (1) $1 / 729$ (2) $8 / 9$ (3) $8 / 729$ (4) $8 / 243$
Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2 , respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is (1) 0.06 (2) 0.14 (3) 0.2 (4) None of these