Logarithmic Differentiation

Q.

Factorize $x^4+x^2+1$

Q. for $x>1$, If $(2x{)}^{2y}=4e^{2x-2y}$, then
$(1+{\log }_{e}2x{)}^2\frac{dy}{dx}$ is equal to :

1. $\frac{x{\log }_{e}2x+{\log }_{e}2}{x}$
2. ${\log }_{e}2x$
3. $\frac{x{\log }_{e}2x-{\log }_{e}2}{x}$
4. $x{\log }_{e}2x$

Q. $\sqrt[3]{3y}\sqrt{x}=\sqrt[6]{6(x+y{)}^5}$, then $\frac{dy}{dx}=$

1. $x-y$
2. $\frac{x}{y}$
3. $\frac{y}{x}$
4. $x+y$

Q.

If ${(1+2x+3x^2)}^{10}=a_0+a_{1}x+a_2{x}^2+...+a_{20}{x}^{20}$, then $a_1$ equals

1. $10$

2. $20$

3. $210$

4. None of these

Q.

How do you find the integral of $\int {\sin }^{3}xdx$ ?

Q.

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

1. $\frac{4}{{\left(x+1\right)}^2}$

2. $\frac{4\left(x-1\right)}{{\left(x+1\right)}^3}$

3. $\frac{\left(x-1\right)}{{\left(x+1\right)}^3}$

4. $\frac{4}{{\left(x+1\right)}^3}$

Q. Let $y=\frac{1}{1+x+\ln x}$, Then

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

Q.

If $y=\frac{e^{2x}\cos x}{x\sin x}$, then $\frac{dy}{dx}$is equal to ?

1. $\frac{e^{2x}\left[(2x-1)cotx–x\cos e{c}^{2}x\right]}{x^2}$

2. $\frac{e^{2x}\left[(2x-1)cotx–\cos e{c}^{2}x\right]}{x^2}$

3. $\frac{e^{2x}\left[(2x-1)cotx+\cos e{c}^{2}x\right]}{x^2}$

4. none of these

Q.

Derivative of the function $f\left(x\right)={\log }_5\left({\log }_7\left(x\right)\right),x>7$is

1. $\frac{1}{x\log \left(5\right)\log \left(7\right){\log }_7\left(x\right)}$

2. $\frac{1}{x\log \left(5\right)\log \left(7\right)}$

3. $\frac{1}{x\log \left(x\right)}$

4. none of these

Q. The value of $\frac{1}{3}+\frac{1}{8}+\frac{1}{15}+\frac{1}{24}+\cdots +\infty$ is

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

Q.

If $1,a_1,a_2,...a_{n-1}$ are the $nth$ roots of unity, then the value of$(1-a_1)(1-a_2)(1-a_3)...(1-a_{n-1})$ is

1. $\sqrt{3}$

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

3. $n$

4. $0$

Q.

If $f$ be the greatest integer function defined as $f\left(x\right)=\left[x\right]$ and $g$ be the modulus function defined as $g\left(x\right)=\left|x\right|$, then the value of gof $\left(-\frac{5}{4}\right)$ is

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

2. $0$

3. $2$

4. does not exist

Q.

Domain of the function $f\left(x\right)=\frac{3}{(4-x^2)}+{\log }_{10}\left(x^3-x\right)$ is

1. $(1,2)$

2. $(-1,0)\cup (1,2)$

3. $(1,2)\cup (2,\infty )$

4. $(-1,0)\cup (1,2)\cup (2,\infty )$

Q.

The conjugate of the complex number $\frac{{\left(1+i\right)}^2}{\left(1-i\right)}$ is

1. $1-i$

2. $1+i$

3. $-1+i$

4. $-1-i$

Q.

$\text{The angles of a triangle are in the ratio 3 : 5 : 10. Then, the ratio of the smallest side to the greatest side is}$

1. 1 : 2 sin 10°

2. 1 : sin 10°

3. 1 : cos 10°

4. 1 : 2 cos 10°

Q.

If $A+C=2B$, then $\frac{\cos C-\cos A}{\sin A-\sin C}$ is equal to

1. $cotB$

2. $cot2B$

3. $\tan 2B$

4. $\tan B$

Q.

The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=6^x+6^{\left|x\right|}$ is

1. One-One and Onto

2. Many-One and Onto

3. One-One and Into

4. Many-One and Into

Q.

Part of the domain of the function$f\left(x\right)=\sqrt{\frac{\cos x-\left({\displaystyle \frac{1}{2}}\right)}{6+35x-6x^2}}$ lying in the interval $\left[-1,6\right]$is

1. $\left[\frac{-1}{6},\frac{\pi }{3}\right]$

2. $\left[\frac{5\pi }{3},6\right]$

3. $\left[\frac{-1}{5},\frac{\pi }{3}\right]\cup \left[\frac{5\pi }{3},6\right]$

4. $(\frac{-1}{6},\frac{\pi }{3}]\cup \left[\frac{5\pi }{3},6\right]$

Q.

The minimum value of $2a^2-3a+10$ is

1. $\frac{15}{2}$

2. $\frac{11}{2}$

3. $-\frac{13}{2}$

4. $\frac{71}{8}$

Q.

If $u=\log \left(\tan \left[\frac{\mathrm{\pi }}{4}+\frac{x}{2}\right]\right)$, then $\cos hu=$

1. $secx$

2. $cosecx$

3. $\tan x$

4. $\sin x$

Q. If $x^y.y^x=16,then\frac{dy}{dx}at(2,2)$ is

1. - 1
2. 0
3. 1
4. None of these

Q.

Let $y=y\left(x\right)$ be the solution of the differential equation $\cos x\frac{dy}{dx}+2y\sin x=\sin 2x,x\in (0,\pi /2).$If $y\left(\frac{\mathrm{\pi }}{3}\right)=0$, then $y\left(\frac{\mathrm{\pi }}{4}\right)$ is equal to:

1. $2+\sqrt{2}$

2. $\sqrt{2}-2$

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

4. $2-\sqrt{2}$

Q.

The greatest term in the expansion of ${(1+3\mathrm{x})}^{54}$, where $x=\frac{1}{3}$ is

1. ${\mathrm{T}}_{28}$

2. ${\mathrm{T}}_{25}$

3. ${\mathrm{T}}_{26}$

4. ${\mathrm{T}}_{24}$

Q.

Evaluate:$\frac{\tan {26}^0}{\cot {64}^0}$

Q.

If $y^2=a{x}^2+bx+c$, then $y^3\frac{d^{2}y}{d{x}^2}$ is

1. a constant

2. a function of $x$ only

3. a function of $y$ only

4. a function of $x$ and $y$

Q.

If $cot\theta +\tan \theta =2\cos ec\theta$, then the general value of $\theta$ is

1. $n\pi \pm \frac{\pi }{3}$

2. $n\pi \pm \frac{\pi }{6}$

3. $2n\pi \pm \frac{\pi }{3}$

4. $2n\pi \pm \frac{\pi }{6}$

Q.

If $y=\frac{1-x}{x^2}$, then $\frac{dy}{dx}$is

1. $\frac{2}{x}+\frac{2}{x^3}$

2. $-\frac{2}{x^3}+\frac{1}{x^2}$

3. $-\frac{2}{x^3}+\frac{2}{x^2}$

4. None of these

Q.

If $x{e}^{xy}=y+{\sin }^{2}x$, then at $x=0$, $\frac{dy}{dx}=$

1. $-1$

2. $-2$

3. $1$

4. $2$

Q.

Factorize the following expression: $x^4-{(x-z)}^4$.

Q.

If in a $∆ABC,2b^2=a^2+c^2$, then $\frac{\sin 3B}{\sin B}$ is equal to

1. $\frac{(c^2–a^2)}{2ac}$

2. $\frac{(c^2–a^2)}{ac}$

3. ${\left(\frac{c^2–a^2}{ac}\right)}^2$

4. ${\left[\frac{(c^2–a^2)}{2ac}\right]}^2$