Formation of a Differential Equation from a General Solution

Q.

A solution of the differential equation ${\left(\frac{dy}{dx}\right)}^2-x\left(\frac{dy}{dx}\right)+y=0$

1. $y=2x$

2. $y=–2x$

3. $y=2x–4$

4. $y=2x+4$

Q.

What is the formula of ${\left(a+b\right)}^2$?

Q. The differential equation of the family of parabolas with focus at the origin and the x–axis as axis is

1. $y{\left(\frac{dy}{dx}\right)}^2+4x\frac{dy}{dx}=4y$
   
2. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}=y$
   
3. $y{\left(\frac{dy}{dx}\right)}^2+y=2xy\frac{dy}{dx}$
   
4. $y{\left(\frac{dy}{dx}\right)}^2+2xy\frac{dy}{dx}+y=0$

Q.

Integration of $si{n}^{7}xdx$.

Q. If $x={\log }_{24}12,$ $y={\log }_{36}24$ and $z={\log }_{48}36,$ then $(1+xyz)$ equals

1. $2$
2. $2yz$
3. $2xz$
4. $2xy$

Q.

Find the value(s) of $p$ in the following pair of equations: $-x+py=1$ and $px-y=1$, if the pair of equations has no solution.

Q.

If $x=asin\theta +bcos\theta$ and $y=acos\theta -bsin\theta ,$ prove that $x^2+y^2=a^2+b^2$

Q.

The differential equation for all the straight lines which are at a unit distance from the origin is

1. 
   

2. 
   

3. 
   

4. 
   

Q.

Evaluate cos $[co{s}^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi }{6}].$

Q.

If $y=\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)$, then $\frac{dy}{dx}$ at $x=1$ is

1. $20$

2. $28$

3. $1$

4. $0$

Q. Find the derivative of $f\left(x\right)=1+x+x^2+x^3+\cdots +x^{50}$ at $x=1.$

Q.

If $y=e^{\sqrt{x}}$, then $\frac{dy}{dx}$ equals

1. $\frac{e^x}{2\sqrt{x}}$

2. $\frac{\sqrt{x}}{e^{\sqrt{x}}}$

3. $\frac{x}{e^{\sqrt{x}}}$

4. $\frac{2\sqrt{x}}{e^{\sqrt{x}}}$

Q.

The difference between the degree and the order of a differential equation that represents the family of curves given by $y^2=a\left(x+\frac{\sqrt{a}}{2}\right),a>0$ is

1. $2$

2. $3$

3. $1$

4. None of these

Q.

The abscissa of the points of the curve $y=x^3$ in the interval $[–2,2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[–2,2]$, are

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

2. $\pm \sqrt{3}$

3. $\pm \frac{\sqrt{3}}{2}$

4. $0$

Q. The differential equation satisfied by the curves $e^{2y}+2bx{e}^y+b^2=0,$ where $b$ is a parameter, is

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

Q.

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is $60°$. When he retreats $20$ feet from the bank, he finds the angle to be $30°$. The breadth of the river in feet is

1. $15$

2. $15\sqrt{3}$

3. $10\sqrt{3}$

4. $10$

Q. $\mathrm{Spherical}\mathrm{rain}\mathrm{drop}\mathrm{evaporates}\mathrm{at}a\mathrm{rate}\mathrm{proportional}\mathrm{to}\mathrm{its}\mathrm{surface}\mathrm{area}.\mathrm{The}\mathrm{differential}\mathrm{equation}\mathrm{corresponding}\mathrm{to}\mathrm{the}\mathrm{rate}\mathrm{of}\mathrm{change}\mathrm{of}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{rain}\mathrm{drop}\mathrm{if}\mathrm{the}\mathrm{constant}\mathrm{of}\mathrm{proportionality}\mathrm{is}K>0\mathrm{is}$
.

1. $\frac{\mathrm{dr}}{\mathrm{dt}}+K=0$
2. $\frac{\mathrm{dr}}{\mathrm{dt}}-K=0$
3. $\frac{\mathrm{dr}}{\mathrm{dt}}+\mathrm{Kr}=0$
4. $\frac{\mathrm{dr}}{\mathrm{dt}}-\mathrm{Kr}=0$

Q.

The distance between $4x+3y=11$ and $8x+6y=15$ is:

1. $\frac{7}{2}$

2. $4$

3. $\frac{7}{10}$

4. None of these

Q.

If $y=c{e}^{si{n}^{-1}x},$ then corresponding to this the differential equation is

1. 
   

2. 
   

3. 
   

4. None of these

Q.

The differential equation of all conics whose centre lie at the origin is of order:

1. $2$

2. $3$

3. $4$

4. None of these

Q.

The differential equation of all straight lines passing through the point (1, -1)is

[MP PET 1994]

1. 
   

2. 
   

3. 
   

4. 
   

Q. The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a(x+\frac{\sqrt{a}}{2}),a>0$ is

Q.

If $x$ is real, the maximum values of expression $\frac{(x^2+14x+9)}{(x^2+2x+3)}$ will be?

Q.

The domain of the function $f\left(x\right)=\text{exp}\left(\sqrt{5x-3-2x^2}\right)$ is

1. $\left[\frac{3}{2},\infty \right]$

2. $[1,3,2\}$

3. $(-\infty ,1]$

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

Q.

How do you know if a parabola is positive or negative $?$

Q. If $f\left(x\right)=3x^2+{\int }_0^x{e}^{-t}f(x-t)dt,$ then

1. $f\left(x\right)=0$ has $3$ solutions
2. $f\left(x\right)$ is monotonically increasing
3. $f\left(x\right)=4$ has $2$ solutions
4. ${\int }_{-1}^{1}f\left(x\right)dx=2$

Q.

A ladder $10$ m long rests against a vertical wall with the lower end on the horizontal ground.

The lower end of the ladder is pulled along the ground away from the wall at the rate of $3$ cm/s.

The height of the upper end while it is descending at the rate of $4$ cm/s, is

1. $4\sqrt{3}\mathrm{m}$

2. $5\sqrt{3}\mathrm{m}$

3. $5\sqrt{2}\mathrm{m}$

4. $8\mathrm{m}$

5. $6\mathrm{m}$

Q.

The equation of the tangent to curve $y=b{e}^{\frac{-x}{a}}$ at the point where it crosses $y$ – axis is

1. $ax+by=1$

2. $ax-by=1$

3. $\frac{x}{a}–\frac{y}{b}=1$

4. $\frac{x}{a}+\frac{y}{b}=1$

Q.

If the normal to the curve $y\left(x\right)={\int }_0^x\left(2t^2-15t+10\right)dt$ at a point $\left(a,b\right)$ is parallel to the line $x+3y=-5$ and it is also given that $a>1$, then the value of $\left|a+6b\right|$ is equal to _______.

Q.

The differential equation of all straight lines passing through origin is

1. $\mathrm{y}=\sqrt{\mathrm{x}}\left(\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.\right)$

2. $\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.=\mathrm{y}+\mathrm{x}$

3. $\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.=\mathrm{y}-\mathrm{x}$

4. None of these