Let $S$ be the set of positive integral values of $a$ for which $\frac { a x ^ { 2 } + 2 a + 1 x + 9 a + 4 } { x ^ { 2 } - 8 x + 32 } < 0 , \quad \forall x \in \mathbb { R }$. Then, the number of elements in $S$ is: (1) 1 (2) 0 (3) $\infty$ (4) 3
For $0 < c < b < a$, let $( a + b - 2 c ) x ^ { 2 } + ( b + c - 2 a ) x + ( c + a - 2 b ) = 0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in ( -1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$. (1) Both (I) and (II) are true (2) Neither (I) nor (II) is true (3) Only (II) is true (4) Only (I) is true
Let $\alpha , \quad \beta , \quad \gamma , \quad \delta \in Z$ and let $A(\alpha , \beta)$, $B(1, 0)$, $C(\gamma , \delta)$ and $D(1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB = \sqrt { 10 }$ and the points $A$ and $C$ lie on the line $3 y = 2 x + 1$, then $2 \alpha + \beta + \gamma + \delta$ is equal to (1) 10 (2) 5 (3) 12 (4) 8
If one of the diameters of the circle $x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$, then the radius of the circle $C$ is (1) $\sqrt { 20 }$ (2) 4 (3) 6 (4) $3 \sqrt { 2 }$
$\lim _ { x \rightarrow 0 } \frac { e ^ { 2 \sin x } - 2 \sin x - 1 } { x ^ { 2 } }$ (1) is equal to $-1$ (2) does not exist (3) is equal to 1 (4) is equal to 2
Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals (1) $2 : 1 : 4$ (2) $4 : 1 : 4$ (3) $1 : 2 : 4$ (4) $1 : 1 : 4$
If the system of linear equations $$\begin{aligned}
& x - 2 y + z = - 4 \\
& 2 x + \alpha y + 3 z = 5 \\
& 3 x - y + \beta z = 3
\end{aligned}$$ has infinitely many solutions, then $12 \alpha + 13 \beta$ is equal to (1) 60 (2) 64 (3) 54 (4) 58
The solution curve of the differential equation $y \frac { d x } { d y } = x \left( \log _ { e } x - \log _ { e } y + 1 \right) , \quad x > 0 , \quad y > 0$ passing through the point $(e, 1)$ is (1) $\log _ { e } \frac { y } { x } = x$ (2) $\log _ { e } \frac { y } { x } = y ^ { 2 }$ (3) $\log _ { e } \frac { x } { y } = y$ (4) $2 \log _ { e } \frac { x } { y } = y + 1$
Q76
First order differential equations (integrating factor)View
Let $y = y(x)$ be the solution of the differential equation $\frac { d y } { d x } = \frac { \tan x + y } { \sin x \sec x - \sin x \tan x } , \quad x \in \left(0, \frac { \pi } { 2 }\right)$ satisfying the condition $y\left(\frac { \pi } { 4 }\right) = 2$. Then, $y\left(\frac { \pi } { 3 }\right)$ is (1) $\sqrt { 32 } + \log _ { e } \sqrt { 3 }$ (2) $\frac { \sqrt { 3 } } { 2} \left(2 + \log _ { e } 3\right)$ (3) $\sqrt { 3 } \left(1 + 2 \log _ { e } 3\right)$ (4) $\sqrt { 3 } \left(2 + \log _ { e } 3\right)$