Synthesis of Amides

Q.

General solution of the differential equation $\frac{dy}{dx}=\frac{[x+y+1]}{[x+y-1]}$ is given by

1. $x+y=\log |x+y|+c$

2. $x–y=\log |x+y|+c$

3. $y=x+\log |x+y|+c$

4. $y=x\log |x+y|+c$

Q.

In Gabriel synthesis, amine is always an .

1. $\mathrm{aliphatic}\mathrm{primary}\mathrm{amine}$
2. $\mathrm{aliphatic}\mathrm{secondary}\mathrm{amine}$
3. $\mathrm{aromatic}\mathrm{primary}\mathrm{amine}$
4. $\mathrm{aromatic}\mathrm{secondary}\mathrm{amine}$

Q. The solution of the equation $log\left(\frac{dy}{dx}\right)=ax+by$ is

1. $\frac{e^{by}}{b}=\frac{e^{ax}}{a}+c$
2. $\frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+c$
   
3. $\frac{e^{-by}}{a}=\frac{e^{ax}}{a}+c$
4. None of these

Q.

They yield of acid amide in the reaction, $RCOCl+N{H}_3\to RCON{H}_2$ , is maximum when

1. Acid chloride and ammonia are treated in equimolar ratio

2. Acid chloride and ammonia are treated in 1 : 2 molar ratio

3. Acid chloride and ammonia are treated in 2 : 1 molar ratio

4. All the three gives nearly similar result

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. The family of curves passing through $(0,0)$ and satisfying the differential equation $\frac{y^{\prime \prime }}{y^{\prime }}=1$ $(\text{where}{y}^{\prime }=\frac{dy}{dx}\text{and}{y}^{\prime \prime }=\frac{d^{2}y}{d{x}^2})$ is

1. $y=k$
2. $y=kx$
3. $y=k(e^x+1)$
4. $y=k(e^x-1)$

Q. The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\log }_e(y+3x)}+3=0$ is
(where $c$ is a constant of integration)

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

Q. The solution of $\frac{dy}{dx}=\frac{ax+b}{cy+d}$ represents a parabola if

1. $a=0,c=0$
2. $a=1,b=2$
3. $a=0,c\ne 0$
4. $a=1,c=1$

Q. A differentiable function $f$ satisfies the equation $f\left(x\right)=\underset{0}{\overset{x}{\int }}\left(f\right(t)\cos t-\cos (t-x\left)\right)dt$, then the correct option(s) is/are

1. $f^{\prime }\left(0\right)=-1$
2. $f\left(x\right)$ has a maximum value $1-e$
3. $f\left(x\right)$ has a minimum value $1-e^{-1}$
4. $f^{\prime \prime }\left(\frac{\pi }{2}\right)=e$

Q. Let $y=f\left(x\right)$ be a non-negative function defined on $[0,2]$ satisfying the differential equation $y^3{y}^{\prime \prime }+1=0$. If $f^{\prime }\left(1\right)=0$ and $f\left(1\right)=1$, then

1. the maximum value of $f\left(x\right)$ is $1.$
2. the minimum value of $f\left(x\right)$ is $0.$
3. the solution $y=f\left(x\right)$ of the given differential equation represents a semi-circle with centre $(1,0).$
4. the area bounded by the curve $y=f\left(x\right)$ and $x-\text{axis}$ is $\frac{\pi }{4}\text{sq. units}.$

Q. The general solution of $\frac{dy}{dx}=\frac{1-3y-3x}{1+x+y}$ is
(where ${}^{\prime }{c}^{\prime }$ is constant of integration)

1. $x+3y+2\ln |1+x+y|=c$
2. $x+y+2\ln |1-x-y|=c$
3. $3x+y+2\ln |1+x+y|=c$
4. $3x+y+2\ln |1-x-y|=c$

Q.

The solution of the equation $\frac{dy}{dx}=e^{x-y}+x^2{e}^{-y}$ is

1. $e^y=e^x+\frac{x^3}{3}+c$

2. $e^y=e^x+2x+c$

3. $e^y=e^x+x^3+c$

4. $y=e^x+c$

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.

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. Let f be a differentiable function such that $f^{\prime }\left(x\right)=f\left(x\right)+{\int }_0^{2}f\left(x\right)dx,f\left(0\right)=\frac{4-e^2}{3}$, then f(x) is

1. $e^x-\left(\frac{e^2-1}{3}\right)$
2. $e^x-\left(\frac{e^2-2}{3}\right)$
3. $e^x+\frac{e-[arg|z-1|]}{3}$
4. $2^{ex}-\frac{e^2-1}{\left[arg\right(-|z|\left)\right]}$ (where [.] greatest integer & z is a complex number)

Q. Rearrangement of an oxime to an amide in the presence of strong acid is called

1. Curtius rearrangement

2. Fries rearrangement

3. Beckman rearrangement
4. Sandmeyer reaction

Q. If the curve $y=y\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}-\frac{2xy}{1+x^2}=0$ and $y\left(0\right)=1,$ then $y\left(1\right)$ is equal to

1. $1$
2. $\frac{1}{2}$
3. $2$
4. $0$

Q. The solution of $\frac{x^{3}dx+y{x}^{2}dy}{\sqrt{x^2+y^2}}=ydx-xdy$, $y\left(1\right)=1$ is

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