Integrating Factor

Q. If $y=y\left(x\right)$ is the solution curve of the differential equation $x^{2}dy+(y-\frac{1}{x})dx=0;x>0,$ and $y\left(1\right)=1,$ then $y\left(\frac{1}{2}\right)$ is equal to:

1. $3-e$
2. $\frac{3}{2}-\frac{1}{\sqrt{e}}$
3. $3+\frac{1}{\sqrt{e}}$
4. $3+e$

Q. The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2{y}^2}+xy\frac{dy}{dx}=0$ is (where $C$ is a constant of integration)

1. $\sqrt{1+y^2}+\sqrt{1+x^2}=\frac{1}{2}{\log }_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}\right)+C$
2. $\sqrt{1+y^2}-\sqrt{1+x^2}=\frac{1}{2}{\log }_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2+1}}\right)+C$
3. $\sqrt{1+y^2}+\sqrt{1+x^2}=\frac{1}{2}{\log }_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right)+C$
4. $\sqrt{1+y^2}-\sqrt{1+x^2}=\frac{1}{2}{\log }_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right)+C$

Q. If ${\cos }^{-1}\left(a\right)+{\cos }^{-1}\left(b\right)+{\cos }^{-1}\left(c\right)=3\pi$ and $f$ be a function such that $f\left(1\right)=2$ and $f(x+y)=f\left(x\right)\cdot f\left(y\right)$ for all $x,y\in R,$ then the value of $a^{2f\left(1\right)}+b^{2f\left(2\right)}+c^{2f\left(3\right)}+\frac{a+b+c}{a^{2f\left(1\right)}+b^{2f\left(2\right)}+c^{2f\left(3\right)}}$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. Consider the parabola with vertex $(\frac{1}{2},\frac{3}{4})$ and the directrix $y=\frac{1}{2}$. Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point $Q$, then $(PQ{)}^2$ is equal to

1. $\frac{15}{2}$
2. $\frac{125}{16}$
3. $\frac{75}{8}$
4. $\frac{25}{2}$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $e^x\sqrt{1-y^2}dx+\left(\frac{y}{x}\right)dy=0,$ $y\left(1\right)=-1.$ Then the value of $\left(y\right(3){)}^2$ is equal to

1. $1+4e^6$
2. $1-4e^6$
3. $1-4e^3$
4. $1+4e^3$

Q. Let $n$ denote the number of solutions of the equation $z^2+3\overline{z}=0,$ where $z$ is a complex number. Then the value of $\sum _{k=0}^{\infty }\frac{1}{n^k}$ is equal to

1. $1$
2. $2$
3. $\frac{4}{3}$
4. $\frac{3}{2}$

Q. Consider the differential equation, $y^{2}dx+(x-\frac{1}{y})dy=0.$ If value of $y$ is $1$ when $x=1,$ then the value of $x$ for which $y=2,$ is:

1. $\frac{3}{2}-\sqrt{e}$
2. $\frac{5}{2}+\frac{1}{\sqrt{e}}$
3. $\frac{3}{2}-\frac{1}{\sqrt{e}}$
4. $\frac{1}{2}+\frac{1}{\sqrt{e}}$

Q.

The order and degree of the differential equation $\sqrt{\sin \left(x\right)}\left(dx+dy\right)=\sqrt{\cos \left(x\right)}\left(dx-dy\right)$ is

1. $\left(1,2\right)$

2. $\left(2,2\right)$

3. $\left(1,1\right)$

4. $\left(2,1\right)$

Q. If $f,g$ are invertible functionsgiven by $f\left(x\right)=3x-2$ and $\left(gof{)}^{-1}\right(x)=x-2$, then

1. $g\left(x\right)=\frac{x+8}{2}$
2. $g\left(x\right)=\frac{x-8}{3}$
3. $g\left(x\right)=\frac{x+8}{3}$
4. $g\left(x\right)=\frac{2x+8}{3}$

Q.

The solution of the differential equation $\frac{dy}{dx}+\frac{2xy}{1+x^2}={\left(\frac{1}{1+x^2}\right)}^2$ is

1. $y(1-x^2)={\tan }^{-1}x+c$

2. $y(1+x^2)={\tan }^{-1}x+c$

3. $y{(1+x^2)}^2={\tan }^{-1}x+c$

4. $y{(1-x^2)}^2={\tan }^{-1}x+c$

Q. If $\underset{n\to \infty }{lim}{\left(\frac{{}^{3n}{C}_n}{{}^{2n}{C}_n}\right)}^{1/n}=\frac{A}{B}$, where $A,B$ are co-prime, then $A+B$ is divisible by

1. $41$
2. $43$
3. $17$
4. $19$

Q. The solution of the differential equation $\frac{dy}{dx}+\frac{y}{2}\sec x=\frac{\tan x}{2y},$ where $0\le x\le \frac{\pi }{2},$ and $y\left(0\right)=1,$ is given by:

1. $y^2=1+\frac{x}{\sec x+\tan x}$
2. $y=1-\frac{x}{\sec x+\tan x}$
3. $y=1+\frac{x}{\sec x+\tan x}$
4. $y^2=1-\frac{x}{\sec x+\tan x}$

Q. The general solution of the differential equation $(1+e^{\frac{x}{y}})dx+(1-\frac{x}{y})e^{\frac{x}{y}}dy=0$ is ($c$ is an arbitary constant)

1. $y+x{e}^{\frac{x}{y}}=c$
2. $y-x{e}^{\frac{x}{y}}=c$
3. $x+y{e}^{\frac{x}{y}}=c$
4. $x-y{e}^{\frac{x}{y}}=c$

Q. The solution of the differential equation $(e^{x^2}+e^{y^2})y\frac{dy}{dx}+e^{x^2}(x{y}^2-x)=0$ is

1. $e^{x^2}(y^2-1)+e^{y^2}=C$
2. $e^{y^2}(x^2-1)+e^{x^2}=C$
3. $e^{y^2}(y^2-1)+e^{x^2}=C$
4. $e^{x^2}(y-1)+e^{y^2}=C$

Q.

If $f\left(x\right)=x^5-20x^3+240x$, then $f\left(x\right)$ satisfies which of the following

1. It is monotonically decreasing everywhere

2. It is monotonically decreasing only in $\left(0,\infty \right)$

3. It is monotonically increasing everywhere

4. It is monotonically decreasing only in $\left(-\infty ,0\right)$

Q. The degree of the differential equation ${[1+{\left(\frac{dy}{dx}\right)}^2]}^2=\frac{d^{2}y}{d{x}^2}$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. Let $P(x_0,y_0)$ be a point on the curve $C:(x^2-11)(y+1)+4=0,$ where $x_0,y_0\in N.$ Then the area (in sq. units) of the triangle formed by the normal drawn to the curve $C$ at $P$ and the co-ordinate axes is

1. $\frac{27}{2}$
2. $\frac{27}{4}$
3. $\frac{31}{4}$
4. $\frac{31}{2}$

Q. An integrating factor of the differential equation $x\frac{dy}{dx}+y\log x=x{e}^x{x}^{-\frac{1}{2}\log x},(x>0)$ is

1. $(\sqrt{x}{)}^{\log x}$
2. $x^{\log x}$
3. $(\sqrt{e}{)}^{\log x}$
4. $e^{x^2}$

Q. Integrating factor of differential equation $\cos x\frac{dy}{dx}+y\sin x=1$ is

1. $|\cos x|$
2. $|\tan x|$
3. $|\sec x|$
4. $\sin x$

Q.

In the expansion of $(1+x+x^2)e^{-x}$, the coefficient of $x^2$ is

1. $1$

2. $-1$

3. $\frac{1}{2}$

4. $-\frac{1}{2}$

Q. If $3^{x-y}=27,3^{x+y}=243,$ what is the value of $x?$

1. $1$
2. $2$
3. $3$
4. $4$

Q. The area bounded by the curve $x{y}^2=a^2(a-x),a>0$ and $y-$axis is

1. $\frac{\pi a^2}{2}$ sq. units
2. $\pi a^2$ sq. units
3. $2{\pi }^{2}a$ sq. units
4. $\pi a$ sq. units

Q. If for $x\ge 0$ , $y=y\left(x\right)$ is the solution of the differential equation $(1+x)dy=\left[\right(1+x{)}^2+y-3]dx,y(2)=0,$ then y(3) is equal to

Q. Let $y=y\left(t\right)$ be a solution of the differential equation $y^{\prime }+2ty=t^2$, then $16\underset{t\to \infty }{lim}\frac{y}{t}$ is

Q. The solution of the differential equation $\frac{dy}{dx}=\frac{\sin y+x}{\sin 2y-x\cos y}$ is:
(where $c$ is constant of integration)

1. ${\sin }^{2}y=x\sin y+\frac{x^2}{2}+c$
2. ${\sin }^{2}y=x\sin y-\frac{x^2}{2}+c$
3. ${\sin }^{2}y=x+\sin y+\frac{x^2}{2}+c$
4. ${\sin }^{2}y=x-\sin y+\frac{x^2}{2}+c$

Q.

Find the value of $\tan 30°$ without using a calculator.

Q. The general solution of $\frac{dy}{dx}={(x+y)}^2,$ where $c$ is the constant of integration, is

1. $x+y=\tan (x+c)$
2. $x+c=\tan (x+y)$
3. $y=\tan x+c$
4. ${(x+y)}^2=\tan (x+c)$

Q. Consider the curve $f(x,y)=0$ which satisfies the differential equation $\frac{dy}{dx}+\frac{1}{x-y^2+4}=0$ such that $y\left(1\right)=-1$. If $f(x,y)$ represents a conic, then the length of its latus rectum is

Q. A differential equation associated to the primitive $y=a+b{e}^{5x}+c{e}^{-7x}$ is (where $y_n$ is $n^{th}$ derivative w.r.t. $x$)

1. $y_3+2y_2-y_1=0$
2. $4y_3+5y_2-20y_1=0$
3. $y_3+2y_2-35y_1=0$
4. $y_3+5y_2-20y_1=0$

Q. The solution of the differential equation $\left[\right(x+1)\frac{y}{x}+\sin y]dx+[x+\ln x+x\cos y]dy=0$is
(where $c$ is integration constant)

1. $xy+x\ln y+x\sin y=c$
2. $xy+y\ln x+x\cos y=c$
3. $xy+y\ln x+x\sin y=c$
4. $xy+x\ln y+x\cos y=c$