Variable Separable Method

Q.

What is the integral of ${\tan }^{2}x$?

Q. The solution of the differential equation, $\frac{dy}{dx}=(x-y{)}^2$, when $y\left(1\right)=1$, is :

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

Q.

The solution of $\left(\frac{dy}{dx}\right)+1=e^{x+y}$is

1. $e^{-(x+y)}+x+c=0$

2. $e^{-(x+y)}–x+c=0$

3. $e^{x+y}+x+c=0$

4. $e^{x+y}–x+c=0$

Q. Let the solution curve of the differential equation $xdy=(\sqrt{x^2+y^2}+y)dx,x>0$, intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$. Then the value of $\alpha$ is

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

Q.

If ${\cos }^{-1}\left(x\right)+{\cos }^{-1}\left(y\right)+{\cos }^{-1}\left(z\right)=3\pi$, then $xy+yz+zx=?$

1. $0$

2. $1$

3. $2$

4. $3$

Q. Let $y=y\left(x\right)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{x^2-1}y={\left(\frac{x-1}{x+1}\right)}^{1/2},x>1$ passing through the point $(2,\sqrt{\frac{1}{3}})$. Then $\sqrt{7}y\left(8\right)$ is

Q.

The rate of growth of bacteria in culture is proportional to the number of bacteria present and the bacteria count is $1000$ at the initial time $t=0$.

The number of bacteria has increased by $20\%$ in $2hours$. If the population of bacteria is $2000$ after $\frac{k}{{\log }_e\left({\displaystyle \frac{6}{5}}\right)}hours$, then${\left(\frac{k}{{\log }_{e}2}\right)}^2$ is equal to

1. $4$

2. $2$

3. $16$

4. $8$

Q.

The cost of 4kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of each item per kg by matrix method

Q. Let $f:R\to R$ be a function such that $f\left(2\right)=4$ and $f^{\prime }\left(2\right)=1.$ Then, the value of $\underset{x\to 2}{lim}\frac{x^{2}f\left(2\right)-4f\left(x\right)}{x-2}$ is equal to

1. $16$
2. $8$
3. $4$
4. $12$

Q.

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

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

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

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

4. None of these.

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.

Subtract $-5{\mathrm{y}}^2$ from ${\mathrm{y}}^2$.

Q. Let the solution curve $y=f\left(x\right)$ of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}},x\in (-1,1)$ pass through the origin. Then ${\int }_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f\left(x\right)dx$ is

Q.

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

1. $\mathrm{x}+\mathrm{y}=\log \left(\frac{\mathrm{cy}}{\mathrm{x}}\right)$

2. $\mathrm{x}+\mathrm{y}=\log \left(cxy\right)$

3. $\mathrm{x}-\mathrm{y}=\log \left(\frac{cx}{y}\right)$

4. $\mathrm{y}-\mathrm{x}=\log \left(\frac{cx}{y}\right)$

Q.

$\int \frac{\cos 4x+1}{\cot x-\tan x}dx=k\cos 4x+C$, then

1. $k=-\frac{1}{2}$

2. $k=-\frac{1}{8}$

3. $k=-\frac{1}{4}$

4. None of these

Q.

Let $y=y\left(x\right)$be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left[(y+1)e^{\frac{x^2}{2}}-x\right],0<x<2.1,$ with $y\left(2\right)=0$. Then the value of $\frac{dy}{dx}$ at$x=1$ is equal to:

1. $\frac{\left[e^{5/2}\right]}{{[1+e^2]}^2}$

2. $\frac{\left[5e^{1/2}\right]}{{[1+e^2]}^2}$

3. $\frac{-2e^2}{{[1+e^2]}^2}$

4. $\frac{-e^{3/2}}{{[1+e^2]}^2}$

Q.

The solution of the differential equation $\mathrm{ydx}-\mathrm{xdy}+\left(\mathrm{logx}\right)\mathrm{dx}=0$ is

1. $\mathrm{y}=\log \mathrm{x}+\mathrm{cx}$

2. $\mathrm{y}+1+\mathrm{logx}=\mathrm{cx}$

3. $\mathrm{y}+\mathrm{cx}=\log \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

4. None of these

Q.

If ${\log }_{\cos \left(x\right)}\tan \left(x\right)+{\log }_{\sin \left(x\right)}cot\left(x\right)=0$, then the most general solution of $x$ is

1. $2n\mathrm{\pi }-\frac{3\mathrm{\pi }}{4},n\in \mathrm{\mathbb{Z}}$

2. $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{4},n\in \mathrm{\mathbb{Z}}$

3. $n\mathrm{\pi }+\frac{\mathrm{\pi }}{4},n\in \mathrm{\mathbb{Z}}$

4. None of these

Q. Let the solution curve $y=y\left(x\right)$ of the differential equation
$[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}]x\frac{dy}{dx}=x+[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}]y$ pass through the points $(1,0)$ and $(2\alpha ,\alpha ),\alpha >0$. Then $\alpha$ is equal to

Q. Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}\frac{\lambda |x^2-5x+6|}{\mu (5x-x^2-6)},& x<2\\ & \\ e^{\frac{\tan (x-2)}{x-\left[x\right]}},& x>2\\ \mu ,& x=2\end{array}$
where $\left[x\right]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x=2,$ then $\lambda +\mu$ is equal to

1. $1$
2. $e(e-2)$
3. $e(-e+1)$
4. $2e-1$

Q.

If $f\left(x\right)=3x-5$, then $f^{-1}\left(x\right)$ is:

1. $\frac{1}{3x-5}$

2. $\frac{x+5}{3}$

3. Does not exist because $f$ is not one −one.

4. Does not exist because $f$ is not onto.

Q. Tangent is drawn at any point $P(x,y)$ on a curve, which passes through $(1,1)$. The tangent cuts $X$-axis and $Y$-axis at $A$ and $B$ respectively. If $AP:BP=3:1$, then

1. the differential equation of the curve is $3x\frac{dy}{dx}+y=0$
2. the differential equation of the curve is $3x\frac{dy}{dx}-y=0$
3. the curve passes through $(\frac{1}{8},2)$
4. the normal at $(1,1)$ is $x+3y=4$

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

1. $Noneofthese$
2. $y=c(x-a)(1+ay)$
   
3. $y=c(x+a)(1+ay)$
   
4. $y=c(x+a)(1-ay)$
   

Q.

The area bounded by the curves $y={(x-1)}^2,y={(x+1)}^{2}andy=\frac{1}{4}$ is

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

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

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

4. $\frac{1}{5}$

Q.

Solve the following:

$8x–7–3x=6x–2x–3$

Q. Let $f$ be a twice differentiable function defined on $R$ such that $f\left(0\right)=1,$ $f^{\prime }\left(0\right)=2$ and $f^{\prime }\left(x\right)\ne 0$ for all $x\in R.$ If $\left|\begin{array}{cc}f\left(x\right)& f^{\prime }\left(x\right)\\ f^{\prime }\left(x\right)& f^{\prime \prime }\left(x\right)\end{array}\right|=0,$ for all $x\in R,$ then the value of $f\left(1\right)$ lies in the interval

1. $(9,12)$
2. $(6,9)$
3. $(3,6)$
4. $(0,3)$

Q.

The general solution of the differential equation $(1+y^2)dx+(1+x^2)dy=0$ is

1. $x–y=C(1–xy)$

2. $x–y=C(1+xy)$

3. $x+y=C(1–xy)$

4. $x+y=C(1+xy)$

Q.

The equation of a plane passing through the line of intersection of the planes$x+2y+3z=2,x-y+z=3$ and at a distance $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1)$ is

1. $5x–11y+z=17$

2. $\sqrt{2}x+y=3\sqrt{2}–1$

3. $x+y+z=\sqrt{3}$

4. $x–\sqrt{2}y=1–\sqrt{2}$

Q.

The triangle formed by the tangent to the curve $f\left(x\right)=x^2+bx-b$ at the point $\left(1,1\right)$ and the co-ordinate axes, lies in the first quadrant. If its area is $2$ then the value of $b$ is

1. $-1$

2. $3$

3. $-3$

4. $1$

Q.

The solution of the differential equation $\left[\mathrm{y}\right(1+{\mathrm{x}}^{-1})+\mathrm{siny}]\mathrm{dx}+[\mathrm{x}+\mathrm{logx}+\mathrm{xcos}\mathrm{y}]\mathrm{dy}=0$ is

1. $\mathrm{xy}+\mathrm{ylogx}=\mathrm{c}$

2. $\mathrm{xy}+\mathrm{xsiny}=\mathrm{c}$

3. $\mathrm{xy}+\mathrm{ylogx}+\mathrm{xsiny}=\mathrm{c}$

4. None of these