Let $a \in R$ and let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 60 ^ { \frac { 1 } { 4 } } x + a = 0$. If $\alpha ^ { 4 } + \beta ^ { 4 } = - 30$, then the product of all possible values of $a$ is $\_\_\_\_$.
Let z be a complex number such that $\left| \frac { z - 2 i } { z + i } \right| = 2 , z \neq - i$. Then $z$ lies on the circle of radius 2 and centre (1) $( 2,0 )$ (2) $( 0,2 )$ (3) $( 0,0 )$ (4) $( 0 , - 2 )$
Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is $\_\_\_\_$.
Let $f ( x ) = 2 x ^ { n } + \lambda , \lambda \in \mathbb { R } , \mathrm { n } \in \mathbb { N }$, and $f ( 4 ) = 133 , f ( 5 ) = 255$. Then the sum of all the positive integer divisors of $( f ( 3 ) - f ( 2 ) )$ is (1) 61 (2) 60 (3) 58 (4) 59
For the two positive numbers $a , b$, if $a , b$ and $\frac { 1 } { 18 }$ are in a geometric progression, while $\frac { 1 } { a } , 10$ and $\frac { 1 } { b }$ are in an arithmetic progression, then $16 a + 12 b$ is equal to $\_\_\_\_$.
If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.
A triangle is formed by $X$-axis, $Y$-axis and the line $3 x + 4 y = 60$. Then the number of points $P ( a , b )$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is $\_\_\_\_$.
Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.
The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y ^ { 2 } = 6 x$. The locus of its circumcentre is: (1) $4 y ^ { 2 } - 18 y - 3 x - 18 = 0$ (2) $4 y ^ { 2 } + 18 y + 3 x + 18 = 0$ (3) $4 y ^ { 2 } - 18 y + 3 x + 18 = 0$ (4) $4 y ^ { 2 } - 18 y - 3 x + 18 = 0$
Let $A , B , C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $( S 1 ) A ^ { 13 } B ^ { 26 } - B ^ { 26 } A ^ { 13 }$ is symmetric (S2) $A ^ { 26 } C ^ { 13 } - C ^ { 13 } A ^ { 26 }$ is symmetric Then, (1) Only $S 2$ is true (2) Only S1 is true (3) Both $S 1$ and $S 2$ are false (4) Both $S 1$ and $S 2$ are true
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \log _ { \sqrt { m } } \{ \sqrt { 2 } ( \sin x - \cos x ) + m - 2 \}$, for some $m$, such that the range of $f$ is $[ 0,2 ]$. Then the value of $m$ is $\_\_\_\_$. (1) 5 (2) 3 (3) 2 (4) 4
Let the function $f ( x ) = 2 x ^ { 3 } + ( 2 p - 7 ) x ^ { 2 } + 3 ( 2 p - 9 ) x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is (1) $\left( \frac { 9 } { 2 } , \infty \right)$ (2) $\left( 0 , \frac { 9 } { 2 } \right)$ (3) $\left( - \infty , \frac { 9 } { 2 } \right)$ (4) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$
If $\int _ { \frac { 1 } { 3 } } ^ { 3 } \left| \log _ { e } x \right| dx = \frac { m } { n } \log _ { e } \left( \frac { n ^ { 2 } } { e } \right)$, where m and n are coprime natural numbers, then $m ^ { 2 } + n ^ { 2 } - 5$ is equal to $\_\_\_\_$.
Let T and C respectively, be the transverse and conjugate axes of the hyperbola $16 x ^ { 2 } - y ^ { 2 } + 64 x + 4 y + 44 = 0$. Then the area of the region above the parabola $x ^ { 2 } = y + 4$, below the transverse axis T and on the right of the conjugate axis C is: (1) $4 \sqrt { 6 } + \frac { 44 } { 3 }$ (2) $4 \sqrt { 6 } + \frac { 28 } { 3 }$ (3) $4 \sqrt { 6 } - \frac { 44 } { 3 }$ (4) $4 \sqrt { 6 } - \frac { 28 } { 3 }$
Q83
First order differential equations (integrating factor)View
Let $y = y ( t )$ be a solution of the differential equation $\frac { d y } { d t } + \alpha y = \gamma e ^ { - \beta t }$ where $\alpha > 0 , \beta > 0$ and $\gamma > 0$. Then $\lim _ { t \rightarrow \infty } y ( t )$ (1) is 0 (2) does not exist (3) is 1 (4) is $-1$
If the shortest distance between the line joining the points $( 1,2,3 )$ and $( 2,3,4 )$, and the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 2 } { 0 }$ is $\alpha$, then $28 \alpha ^ { 2 }$ is equal to $\_\_\_\_$.
Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2 , \sqrt { 3 N } , N + 2$ are in geometric progression be $\frac { k } { 48 }$. Then the value of $k$ is (1) 2 (2) 4 (3) 16 (4) 8
$25\%$ of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac { k } { 10 }$. Then the value of $k$ is $\_\_\_\_$.