Variable Separable Method

Q.

Prove that $\frac{\text{cos}x}{(1-sinx)}=tan(\frac{\pi }{4}+\frac{x}{2})$

Q.

The solution of the differential equation $\frac{dy}{dx}=secx(secx+tanx)$ is

1. y = sec x + tan x + c.
   

2. y = sec x + cot x + c.
   

3. y = sec x - tan x + c

4. None of these

Q.

The differential equations satisfied by the system of parabolas $y^2=4a\left(x+a\right)$ is:

1. $y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$

2. $y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$

3. $y\left(\frac{dy}{dx}\right)-2x\left(\frac{dy}{dx}\right)-y=0$

4. $y\left(\frac{dy}{dx}\right)-2x\left(\frac{dy}{dx}\right)+y=0$

Q. The Cartesian coordinates (x, y) of a point on a curve are given by $x:y:1=t^3:t^2-3:t-1$ where t is a parameter, then the points given by t = a, b, c are collinear, if

1. abc + 3(a + b + c) = ab + bc + ca
2. 3abc + 2(a + b + c) = ab + bc + ca
3. abc + 2(a + b + c) = 3(ab + bc + ca)
4. None of the above

Q. Given that the slope of the tangent to a curve $y=y\left(x\right)$ at any point $(x,y)$ is $\frac{2y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2x-2y=0$, then its equation is :

1. $x{\log }_e|y|=2(x-1)$
2. $x{\log }_e|y|=(x-1)$
3. $x{\log }_e|y|=-2(x-1)$
4. $x^2{\log }_e|y|=-2(x-1)$

Q. The equation of the curve which passes through the point (2a, a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a is

1. $y(x-a)=a^2$
2. $y(x+a)=a^2$
3. $x(y-a)=a^2$
4. $x(y+a)=a^2$

Q. The equation of the curve passing through the origin and satisfying the differential equation ${\left(\frac{dy}{dx}\right)}^2=(x-y{)}^2$, is

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

Q. If $x{e}^{xy}-y-{\sin }^{2}x=0$ then $\frac{dy}{dx}$ at $x=0$ is

Q.

The solution of the differential equation $\frac{dy}{dx}=1+x+y+xy$ is

1. $log(1+y)=x+\frac{x^2}{2}+c$

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

4. None of these

Q. If the curve $y=y\left(x\right)$ satisfies the differential equation $y-x\frac{dy}{dx}=a(y^2+\frac{dy}{dx})$ and always passes through a fixed point $(1,1),$ then the total number of possible values of $a$ is
(Assume the constant of integration to be zero)

Q. Minimum distance between the curves $y^2=4x\&x^2+y^2-12x+31=0$ is

1. $\sqrt{21}$
2. $\sqrt{26}-\sqrt{5}$
3. $\sqrt{20}-\sqrt{5}$
4. $\sqrt{21}-\sqrt{5}$

Q. Let $f$ be a twice differentiable function on $R$ such that $t^{2}f\left(x\right)-2t{f}^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)=0$ has two equal values of $t$ for all $x$ and $f\left(0\right)=1,f^{\prime }\left(0\right)=2.$ Then the value of $3\underset{x\to 0}{lim}(\frac{f\left(x\right)-1}{x}-\frac{t}{2})$ is

Q. If $\frac{dy}{dx}+\frac{3}{{\cos }^{2}x}y=\frac{1}{{\cos }^{2}x},x\in (\frac{-\pi }{3},\frac{\pi }{3}),$ and $y\left(\frac{\pi }{4}\right)=\frac{4}{3}$, then $y(-\frac{\pi }{4})$ equals :

1. $-\frac{4}{3}$
2. $\frac{1}{3}+e^6$
3. $\frac{1}{3}+e^3$
4. $\frac{1}{3}$

Q. Let $C_1$ be the curve obtained by solution of differential equation $2xy\frac{dy}{dx}=y^2-x^2,$ $x>0.$ Let the curve $C_2$ be the solution of $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}.$ If both the curves pass through $(1,1),$ then the area enclosed by the curves $C_1$ and $C_2$ is equal to :

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

Q. The general solution of the differential equation $ydx-xdy+\ln xdx=0$ is
(where $C$ is constant of integration)

1. $y+\ln x-1=C$
2. $y+\ln x-1=Cx$
3. $y+\ln x+1=Cx$
4. $y-\ln x-1=C$

Q. Which of the following differential equations can be solved using variable separable method.
$1.(x^2+y^2)dx-2xydy=0$
$2.(x^2+\sin x)dx+\left(\sin y\right)dy=0$

1. Only 1
2. Only 2
3. Both 1 and 2
4. None of the above

Q. If $|z-2+2i|=1,$ then

1. maximum value of $|z|$ is $\sqrt{8}+1$
2. maximum value of $|z|$ is $\sqrt{10}+1$
3. minimum value of $|z|$ is $\sqrt{8}-1$
4. minimum value of $|z|$ is $\sqrt{6}-1$

Q. Let $y=y\left(x\right)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1$.
If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to:

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

Q. The population $P=P\left(t\right)$ at time $‘t^{\prime }$ of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450.$ If $P\left(0\right)=850,$ then the time at which population becomes zero is :

1. $\frac{1}{2}{\log }_{e}18$
2. $2{\log }_{e}18$
3. ${\log }_{e}9$
4. ${\log }_{e}18$