If $\lambda \in \mathrm { R }$ is such that the sum of the cubes of the roots of the equation, $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is (1) 20 (2) $2 \sqrt { 5 }$ (3) $2 \sqrt { 7 }$ (4) $4 \sqrt { 2 }$
The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } z \neq 1$, is (1) $\{ 0 \}$ (2) an empty set (3) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$ (4) equal to $R$
$n$ - digit numbers are formed using only three digits 2,5 and 7 . The smallest value of $n$ for which 900 such distinct numbers can be formed, is (1) 6 (2) 8 (3) 9 (4) 7
If $b$ is the first term of an infinite G.P whose sum is five, then $b$ lies in the interval. (1) $( - \infty , - 10 )$ (2) $( 10 , \infty )$ (3) $( 0,10 )$ (4) $( - 10,0 )$
If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x ^ { 2 } - 10 x - 25 = 0$ then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cdot \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is (1) 25 (2) - 25 (3) - 10 (4) 10
In a triangle $A B C$, coordiantates of $A$ are $( 1,2 )$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then area of $\triangle A B C$ (in sq. units) is (1) 5 (2) 9 (3) 12 (4) 4
A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line, $y - 4 x + 3 = 0$, then its radius is equal to (1) $\sqrt { 5 }$ (2) 1 (3) $\sqrt { 2 }$ (4) 2
Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3 , then the equation of the common tangent to the two parabolas is? (1) $3 ( x + y ) + 4 = 0$ (2) $8 ( 2 x + y ) + 3 = 0$ (3) $4 ( x + y ) + 3 = 0$ (4) $x + 2 y + 3 = 0$
If the tangents drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $A B$ is (1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$ (2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$ (3) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$ (4) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
The mean of a set of 30 observations is 75 . If each other observation is multiplied by a nonzero number $\lambda$ and then each of them is decreased by 25 , their mean remains the same. The $\lambda$ is equal to (1) $\frac { 10 } { 3 }$ (2) $\frac { 4 } { 3 }$ (3) $\frac { 1 } { 3 }$ (4) $\frac { 2 } { 3 }$
An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it, is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is (1) 1500 (2) 750 (3) 720 (4) 1440
$f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$ (1) Exists and is equal to - 2 (2) Does not exist (3) Exist and is equal to 0 (4) Exists and is equal to 2
Let $S$ be the set of all real values of $k$ for which the system of linear equations $$\begin{aligned}
& x + y + z = 2 \\
& 2 x + y - z = 3 \\
& 3 x + 2 y + k z = 4
\end{aligned}$$ has a unique solution. Then $S$ is (1) an empty set (2) equal to $\mathrm { R } - \{ 0 \}$ (3) equal to $\{ 0 \}$ (4) equal to $R$
If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is (1) - 34 (2) - 32 (3) - 2 (4) 4
If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm , then the curved surface area ( $\mathrm { in } \mathrm { cm } ^ { 2 }$ ) of this cone is (1) $8 \sqrt { 3 } \pi$ (2) $6 \sqrt { 2 } \pi$ (3) $6 \sqrt { 3 } \pi$ (4) $8 \sqrt { 2 } \pi$
If $f \left( \frac { x - 4 } { x + 2 } \right) = 2 x + 1 , ( x \in R = \{ 1 , - 2 \} )$, then $\int f ( x ) d x$ is equal to (where $C$ is a constant of integration) (1) $12 \log _ { e } | 1 - x | - 3 x + c$ (2) $- 12 \log _ { e } | 1 - x | - 3 x + c$ (3) $- 12 \log _ { e } | 1 - x | + 3 x + c$ (4) $12 \log _ { e } | 1 - x | + 3 x + c$
The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$, is (1) $\frac { 13 } { 3 }$ (2) $\frac { 10 } { 3 }$ (3) $\frac { 5 } { 3 }$ (4) $\frac { 8 } { 3 }$
Q86
First order differential equations (integrating factor)View
Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where $$f ( x ) = \left\{ \begin{array} { l c }
1 , & x \in [ 0,1 ] \\
0 , & \text { otherwise }
\end{array} \right.$$ If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is (1) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$ (2) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$ (3) $\frac { 1 } { 2 e }$ (4) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$
A variable plane passes through a fixed point ( $3,2,1$ ) and meets $x , y$ and $z$ axes at $A , B$ and $C$ respectively. A plane is drawn parallel to $y z$ - plane through $A$, a second plane is drawn parallel $z x$ plane through $B$ and a third plane is drawn parallel to $x y$ - plane through $C$. Then the locus of the point of intersection of these three planes, is (1) $( x + y + z = 6 )$ (2) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$ (3) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$ (4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$
A box ' $A$ ' contains 2 white, 3 red and 2 black balls. Another box ' $B ^ { \prime }$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box ' $B ^ { \prime }$ is (1) $\frac { 7 } { 16 }$ (2) $\frac { 9 } { 32 }$ (3) $\frac { 7 } { 8 }$ (4) $\frac { 9 } { 16 }$