General Solution of a Differential Equation

Q.

If $\mathrm{y}={\mathrm{x}}^{\mathrm{sinx}},$ then $\frac{\mathrm{dy}}{\mathrm{dx}}=?$

1. $\frac{x^{\sin x}\left[x\cos x\log x+\sin x\right]}{x}$

2. $\frac{{\mathrm{x}}^{\mathrm{sinx}}\left[\mathrm{xcosxlogx}+\cos \mathrm{x}\right]}{\mathrm{x}}$

3. $\frac{\mathrm{y}\left[\mathrm{xcosxlogx}+\mathrm{cosx}\right]}{\mathrm{x}}$

4. None of these

Q.

If $y=\left(\frac{1+\tan x}{1-\tan x}\right)$, then $\frac{dy}{dx}=$

1. $\frac{se{c}^{2}x}{{\left(1-\tan x\right)}^2}$

2. $\frac{2se{c}^{2}x}{{\left(1-\tan x\right)}^2}$

3. $\frac{2se{c}^{2}x}{{\left(1+\tan x\right)}^2}$

4. None of these

Q.

If $x^y{y}^x=1$, then $\frac{dy}{dx}$ is:

1. $\frac{-y\left(y+x\log y\right)}{x\left(y\log x+x\right)}$

2. $\frac{y\left(x+y\log x\right)}{x\left(x\log y+y\right)}$

3. $\frac{y\left(y+x\log y\right)}{x\left(y\log x+x\right)}$

4. None of these

Q.

The solution curve of the differential equation, $(1+e^{-x})(1+y^2)\frac{dy}{dx}=y^2$, which passes through the point$(0,1)$, is

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

2. $y^2+1=y\left[{\log }_e\left(\frac{1+e^{-x}}{2}\right)+2\right]$

3. $y^2+1=y\left[{\log }_e\left(\frac{1+e^x}{2}\right)+2\right]$

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

Q. Let the function $y=f\left(x\right)$ satisfies the differential equation $x^2\frac{dy}{dx}=y^2{e}^{1/x}(x\ne 0)$ and $\underset{x\to 0^{-}}{lim}f\left(x\right)=1.$ Identify the CORRECT statement(s) ?

1. Range of $f$ is $(0,1)-\left\{\frac{1}{2}\right\}$
2. $f\left(x\right)$ is bounded
3. $\underset{x\to 0^{+}}{lim}f\left(x\right)=1$
4. $\underset{0}{\overset{e}{\int }}f\left(x\right)dx>\underset{0}{\overset{1}{\int }}f\left(x\right)dx$

Q.

The general solution of $\frac{dy}{dx}+ytanx=secx$ is
(a) y secx=tanx+C
(b) y tanx=secx+C
(c) tanx=y tanx+C
(d) x secx=tany+C

Q.

The equation of a directrix of the ellipse$\left(\frac{x^2}{16}\right)+\left(\frac{y^2}{25}\right)=1$ is

1. $3y=\pm 5$

2. $y=\pm 5$

3. $3y=\pm 25$

4. $y=\pm 3$

Q.

If $y=2^{\frac{1}{{\log }_x\left(8\right)}}$, then $x=$

1. $y$

2. $y^2$

3. $y^3$

4. None of these

Q.

The differential equation of all circles of radius $a$ is of order:

1. $2$

2. $3$

3. $4$

4. None of these

Q. Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $P{y}^{\prime \prime }+Q{y}^{\prime }+1=0$, where $P,Q$ are functions of $x,y\text{and}{y}^{\prime }(\text{here}{y}^{\prime }=\frac{dy}{dx},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}),$ then which of the following statements is (are) true ?

1. $P=y+x$
2. $P=y-x$
3. $P+Q=1-x+y+y^{\prime }+(y^{\prime }{)}^2$
4. $P-Q=x+y-y^{\prime }-(y^{\prime }{)}^2$

Q. The general solution of $\frac{dy}{dx}=y$ is .

1. $y=e^x$
2. $y=e^x+c$
3. $y=k{e}^x$
4. $y=k{e}^x+c$

Q. Which among the following functions from $R$ to $R$ is not one-one function.

1. $f\left(x\right)=x^3$
2. $g\left(x\right)=2x+1$
3. $h\left(x\right)=x^2+x$
4. None of the above

Q.

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60 per kg and food Q costs Rs. 80 per kg. Food P contains 3 units per kg of vitamin A and 5 units per kg of vitamin B while food Q contains 4 units per kg of vitamin A and 2 units per kg of vitamin B. Determine the minimum cost of the mixture.

Q.

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

1. $2x=\sin \left(y\right)\left(1+c{x}^2\right)$

2. $\frac{y}{\sin \left(x\right)}+\log \left(x\right)=c$

3. $\log \left(x\right)+x=c$

4. $\log \left(x\right)+y=c$

Q.

If $\mathrm{f}\left(\mathrm{t}\right)$ is an odd function, then ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is

1. An odd function

2. An even function

3. Neither even nor odd

4. 0

Q.

The value of ${\int }_0^1\log \left[x\right]dx$ is

1. $0$

2. $1$

3. $\log 1$

4. None of these

Q.

If $a,b,c$are in A.P., then$2^{ax+1},2^{bx+1},2^{cx+1},x\ne 0$ are in

1. $A.P$

2. $G.P$ only when $x>0$

3. $G.P$ if $x<0$

4. $G.P$ for all $x\ne 0$

Q.

Find the equation of the curve passing through the point $(0,\frac{\pi }{4})$ whose differential equation is $sinxcosydx+cosxsinydy=0$

Q. $\text{Let}f\left(x\right)=\int x^2(2x+6\tan x-2x{\tan }^{2}x){\cos }^{2}xdx$ and $f\left(x\right)$ passes through $(\pi ,0)$ then number of solutions of the equation $f\left(x\right)=x^3\text{in}xϵ[0,2\pi ]\text{is}$

Q. A normal is drawn at a point $P(x,y)$ on a curve. If it meets the $x-$axis and the $y-$axis such that $(x-\text{intercept}{)}^{-1}+(y-\text{intercept}{)}^{-1}=1,$ then the radius of the director circle of the curve passing through $(3,3)$ is

Q.

Find the particular solution of the differential equation
$(1+e^{2x})dy+(1+y^2)e^{x}dx=0$ given that y = 1 when x = 0

Q.

If $\tan h^{-1}\left(x+iy\right)=\frac{1}{2}\tan h^{-1}\left(\frac{2x}{1+x^2+y^2}\right)+\frac{i}{2}{\tan }^{-1}\left(\frac{2y}{1-x^2-y^2}\right);x,y\in R$ then $\tan h^{-1}\left(iy\right)$ is

1. $i\tan h^{-1}\left(y\right)$

2. $-i\tan h^{-1}\left(y\right)$

3. $i{\tan }^{-1}\left(y\right)$

4. $-{\tan }^{-1}\left(y\right)$

Q.

Identify the empty set$,$ the singleton set and the pair set$.$

$B=\{x:x^2-9=0,x\in Z\}$

Q.

The value of$\underset{x\to 0}{lim}(1+x{)}^{\frac{1}{x}}$ is-------

1. e

2. 1

3. 0

4. 

Q. The order of differential equation of all circles of given radius '$a$' is:

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

Q.

For the differential equation in given question find a particular solution satisfying the given condition.​​

$\frac{dy}{dx}=ytanx,y=1whenx=0$

Q. General solution of a differential equation is $y=k{e}^x$. Find the particular solution if the curve passes through (0, 1)

1. $y=e^x$
2. $y=k{e}^x$
3. $y=k{e}^x+C$
4. $y=2e^x$

Q. Find the particular solution of the differential equation $(x-y)\frac{dy}{dx}=(x+2y)$, given that $y=0$ when $x=1$.

Q. Solve:

Q. Prove that $y={ae}^{-2x}+{be}^x$ is the solution of differential equation $\frac{d^{2}y}{{dx}^2}+\frac{dy}{dx}-2y=0$