Geometrical Applications of Differential Equations

Q. The solution of differential equation $(3y-7x+7)dx+(7y-3x+3)dy=0$ is:
(where $C$ is constant of integration)

1. $C={(x+y-2)}^2{(y-x+2)}^5$
2. $C={(x+y-2)}^5{(y-x+2)}^2$
3. $C={(x+y-1)}^2{(y-x+1)}^5$
4. $C={(x+y-1)}^5{(y-x+1)}^2$

Q.

The circumcenter of the triangle whose vertices are $\left(0,0\right),(3,0)$and $\left(0,4\right)$is

1. $\left(\frac{3}{2},2\right)$

2. $\left(2,\frac{3}{2}\right)$

3. $\left(0,0\right)$

4. None of these

Q.

If the distance between $(2,3)$ and $(-5,2)$ is equal to the distance between $(x,2)$ and $(1,3)$, then the values of$x$ are

1. $-6,8$

2. $6,8$

3. $-8,6$

4. $-7,7$

Q. The equation of the curve passing through the origin if the middle point of the segment of its normal form any point of the curve to the x-axis, lies on the parabola $2y^2=x$

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

Q. Let $a$ be a real number such that the function $f\left(x\right)=a{x}^2+6x-15,$ $x\in R$ is increasing in $(-\infty ,\frac{3}{4})$ and decreasing in $(\frac{3}{4},\infty )$. Then the function $g\left(x\right)=a{x}^2-6x+15,$ $x\in R$ has a

1. Local maximum at $x=\frac{3}{4}$
2. Local maximum at $x=-\frac{3}{4}$
3. Local minimum at $x=-\frac{3}{4}$
4. Local minimum at $x=\frac{3}{4}$

Q. If $y^{1/4}+y^{-1/4}=2x$, and $(x^2-1)\frac{d^{2}y}{d{x}^2}+\alpha x\frac{dy}{dx}+\beta y=0$, then $|\alpha -\beta |$ is equal to

Q.

The number of integral values of $x$ satisfying $\sqrt{-x^2+10x-16}<x-2$ is

Q. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth $y$, where the constant of proportionality $k>0$ depends on the acceleration due to gravity and the geometry of the hole. If $t$ is measured in minutes and $k=\frac{1}{15}$, then the time to drain the tank if the water is $4\text{m}$ deep to start with is

1. $30\text{min}$
2. $45\text{min}$
3. $60\text{min}$
4. $80\text{min}$

Q.

If $f$ be a real valued function defined on the interval $(0,\infty )$ by $f\left(x\right)=\ln x+$ integral from $0$ to $x\sqrt{1+\sin t}dt$. Then which of the following statement (s) is (are) correct?

1. $f\left(x\right)$ exists for all $x\in (0,\infty )$

2. $f\left(x\right)$ exists for all $x\in (0,\infty )$ and $f$ is continuous on $(0,\infty )$ but not differentiable on $(0,\infty )$

3. there exist $ɑ>1$ such that $|f(x)<1|f\left(x\right)|$ for all $x\in (ɑ,\infty )$

4. there exists $>0$ such that $|f(x\left)\right|<1|f(x\left)\right|\le$ form all $x\in (0,\infty )$ but not differentiable on $(0,\infty )$

Q.

Number of points having position vector $ai+bi+ck,a,b,c\in \left\{1,2,3,4,5\right\}$ such that $2n+3b+5c$ is divisible by $4$ is?

1. $140$

2. $70$

3. $100$

4. $75$

Q.

The triangle formed by the points $(0,7,10),(–1,6,6),(–4,9,6)$ is

1. Equilateral triangle

2. Isosceles triangle

3. Right angle triangle

4. Right-angled isosceles

Q.

The equation of the ellipse having vertices at $(\pm 5,0)$ and foci $(\pm 4,0)$ is:

1. $\frac{x^2}{25}+\frac{y^2}{16}=1$

2. $9x^2+25y^2=225$

3. $\frac{x^2}{9}+\frac{y^2}{25}=1$

4. $4x^2+5y^2=20$

Q. If the independent variable $x$ is changed to $y$, then the differential equation $x\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3-\left(\frac{dy}{dx}\right)=0$ is changed to $x\frac{d^{2}x}{d{y}^2}+{\left(\frac{dx}{dy}\right)}^2=k$ where $k$ equals

Q. The differential equation to the family of parabola having latus rectum $4a$ and axis parallel to the $x-$axis is

1. of order $1$ and degree $2$
2. of order $2$ and degree $3$
3. of order $2$ and degree $1$
4. of order $2$ and degree $2$

Q. The equation $y^2{e}^{xy}=9e^{-3}\cdot x^2$ defines $y$ as a differentiable function of $x.$ The value of $\frac{dy}{dx}$ for $x=-1$ and $y=3$ is

1. $-\frac{15}{2}$
2. $-\frac{9}{5}$
3. $3$
4. $15$

Q. Let $f\left(x\right)$ be a non-positive continuous function and $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt\forall x\ge 0$ and $f\left(x\right)\ge cF\left(x\right)$ where $c>0$ and let $g:[0,\infty )\to R$ be a function such that $\frac{dg\left(x\right)}{dx}<g\left(x\right)\forall x>0$ and $g\left(0\right)=0.$
The total number of root(s) of the equation $f\left(x\right)=g\left(x\right)$ is/are

1. $1$
2. $2$
3. $0$
4. $\infty$

Q.

In the argand plane, the distinct roots of $1+z+z^3+z^4=0$ ($z$ is a complex number) represent vertices of :

1. A square

2. An equilateral triangle

3. A rhombus

4. A rectangle

Q. If $y=y\left(x\right)$ is the solution of the differential equation $\frac{5+e^x}{2+y}\cdot \frac{dy}{dx}+e^x=0$ satisfying $y\left(0\right)=1$, then a value of $y\left({\log }_{e}13\right)$ is:

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

Q. Let ${}^{\prime }{f}^{\prime }$ be a real valued function such that $0\le f\left(x\right)\le \frac{1}{2}$ and for some fixed $a>0,f(x+a)=\frac{1}{2}-\sqrt{f\left(x\right)-\left(f\right(x){)}^2},\forall x\in R$, then the period of the function $f\left(x\right)$ is

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

Q.

Find the Circumcentre and Circumradius of the triangle whose verices are $(1,1),(2,-1)$ and $(3,2)$

Q.

The largest interval in which the function $f\left(x\right)=3\sin \left[\sqrt{\left(\frac{{\mathrm{\pi }}^2}{9}\right)-x^2}\right]$ assumes values is

1. $\left[0,3\sqrt{3}\right]$

2. $\left[\frac{-3\sqrt{3}}{2},\frac{3\sqrt{3}}{2}\right]$

3. $\left[0,\frac{3\sqrt{3}}{2}\right]$

4. $\left[\frac{-3\sqrt{3}}{2},0\right]$

Q. A curve $y=f\left(x\right)$ passes through $(1,1)$ and tangent at $P(x,y)$ cuts the $x-$axis and $y-$axis at $A$ and $B$ respectively such that $AP:PB=1:3,$ then which of the following is/are true:

1. Differential equation of curve is $x{y}^{\prime }-3y=0$
2. Differential equation of curve is $x{y}^{\prime }+3y=0$
3. Curve passes through $(2,\frac{1}{8})$
4. Curve passes through $(2,\frac{1}{4})$

Q. Differentiate the function $\frac{{\cos }^{-1}\frac{x}{2}}{\sqrt{2x+7}},-2<x<2$ w.r.t.$x$.

Q. Let $y=f\left(x\right)$ defined on $R$ satisfy the differential equation $(1+x^2)\frac{dy}{dx}+2xy=2x$ and $y\left(0\right)=2.$ Then which of the following statements is/are correct?

1. $f\left(x\right)$ is neither even nor odd function.
2. $f\left(x\right)$ increases on $(-\infty ,0)$ and decreases on $(0,\infty )$.
3. The $x-$intercept of normal on the graph of $y=f\left(x\right)$ at $x=1$ equals $\frac{1}{4}$.
4. The area bounded by $y=f\left(x\right)$ with $x-$axis between the ordinates $x=0$ and $x=1$ equals $\left(\frac{\pi +4}{4}\right)$ sq. units.

Q.

The circumcentre of the triangle with vertices $(8,6),(8,-2)$ and $(2,-2)$ is at the point

1. $(2,-1)$

2. $(1,-2)$

3. $(5,2)$

4. $(2,5)$

5. $(4,5)$

Q. The equation of a curve passing through $(1,0)$ for which the product of the abscissa of a point $P$ and the intercept made by a normal at $P$ on the $x$-axis equals twice the square of the distance of the point $P$ from the origin $O$, is

1. $x^2+y^2=x^4$
2. $x^2+y^2=2x^4$
3. $x^2+y^2=4x^4$
4. $x^2+y^2=8x^4$

Q. The value of determinant using Bagula method $\left|\begin{array}{ccc}3& -2& 3\\ 0& 2& 0\\ 2& -1& 3\end{array}\right|$ is

1. $30$
2. $-30$
3. $-6$
4. $6$

Q. In the curve $y=c{e}^{x/a},$ where $a,c\in R-\left\{0\right\},$ the

1. subtangent is constant
2. subnormal varies as the square of the ordinate
3. tangent at $(x_1,y_1)$ on the curve intersects the $x-$axis at a distance of $(x_1-a)$ from the origin
4. equation of the normal at the point where the curve cuts the $y-$axis is $cy+ax=c^2$

Q.

Find the shortest distance from a hotel located at $\left(2,4\right)$ to a straight road with equation $3x+5y+8=0$.

Q. Let $f:R\to (0,\pi )$ be defined as $f\left(x\right)={\cot }^{-1}\left(\frac{2-|x|}{2+|x|}\right).$ Then which of the following statements is(are) TRUE?

1. $f\left(x\right)$ is neither injective nor surjective.
2. $f\left(x\right)$ is continuous and differentiable on $R$.
3. $f\left(x\right)$ is both even function and non-periodic function.
4. $\underset{x\to \frac{-2}{\sqrt{3}}}{lim}f\left(x\right)=\frac{5\pi }{12}$.