Higher Order Derivatives

Q.

write the formula for $\cos 3A$.

Q. Let $f\left(x\right)$ be a polynomial of degree $3$ such that $f\left(k\right)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10f\left(10\right)$ is equal to

Q.

If $z_1,z_2,$ and $z_3$ are complex number such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1$ then $\left|z_1+z_2+z_3\right|$ equal

1. equals $1$

2. less than $1$

3. greater than $1$

4. equals $3$

Q.

Solution set of the inequality $5^{x+2}>{\left(\frac{1}{25}\right)}^{\frac{1}{x}}$ is

1. $(-2,0)$

2. $(-2,2)$

3. $(-5,5)$

4. $(0,\infty )$

Q.

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

1. $-4x\sin 2x^2$

2. $-x\sin x^2$

3. $-2x\sin 2x^2$

4. $-x\cos 2x^2$

Q.

Evaluate the derivative of $\frac{8^x}{x^8}$

Q.

The greatest and least values of ${\left[{\sin }^{-1}\left(x\right)\right]}^2+{\left[{\cos }^{-1}\left(x\right)\right]}^2$ are respectively

1. $\frac{{\mathrm{\pi }}^4}{4}$ and $0$

2. $\frac{\mathrm{\pi }}{2}$ and $-\frac{\mathrm{\pi }}{2}$

3. $\frac{5{\mathrm{\pi }}^4}{4}$ and $\frac{{\mathrm{\pi }}^2}{8}$

4. $\frac{{\mathrm{\pi }}^2}{4}$ and $-\frac{{\mathrm{\pi }}^2}{4}$

Q.

If $y=2^{\log x}$, then$\frac{dy}{dx}$ is equal to

1. $\frac{2^{\log x}}{\log 2}$

2. $2^{\log x}\times \log 2$

3. $\frac{2^{\log x}}{x}$

4. $\frac{2^{\log x}\times \log 2}{x}$

Q.

$\frac{d\left({\tan }^{-1}\left({\displaystyle \frac{a-x}{1+ax}}\right)\right)}{dx}=$

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

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

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

4. none of these

Q.

Find two positive numbers x and y such that their sum is 35 and the product $x^2{y}^5$ is maximum.

Q.

If $f\left(x\right)=\frac{x}{\sqrt{1+x^2}}$, then $fofof\left(x\right)=$

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

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

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

4. None of these

Q.

Let $f:\left[-1,1\right]\to R$ be defined as $f\left(x\right)=a{x}^2+bx+c$ for all $x\in \left[-1,1\right]$, where $a,b,c\in R$ such that$f\left(-1\right)=2$, $f\left(-1\right)=1$ and for $x\in \left[-1,1\right]$ the maximum value of $f\left(x\right)$ is $\frac{1}{2}$. If $f\left(x\right)\le a$, $x\in \left[-1,1\right]$, then the least value of $\alpha$ is equal to

Q.

If $a$and $b$are unit vectors$\left|a+b\right|=1$, then $\left|a-b\right|$is equal to?

1. $\sqrt{2}$

2. 1

3. $\sqrt{5}$

4. $\sqrt{3}$

Q. If f(x) = cos (log x), then $f\left(x^2\right)f\left(y^2\right)-\frac{1}{2}[f\left(\frac{x^2}{y^2}\right)+f(x^2{y}^2\left)\right]$ has the value

1. -2
2. -1
3. None of these
4. $\frac{1}{2}$

Q.

If $f\left(x\right)$ and $g\left(x\right)$ are two polynomials such that the polynomial $P\left(x\right)=f\left(x^3\right)+xg\left(x^3\right)$ is divisible by $x^2+x+1$, then $P\left(1\right)$ is equal to:

Q. The value of integral $\int {\text{cosec}}^{3}xdx$ is equal to
$($where $C$ is integration constant$)$

1. $-\text{cosec}x\cot x+\ln |\text{cosec}x-\cot x|+C$
2. $\frac{-\text{cosec}x\cot x+\ln |\text{cosec}x-\cot x|}{2}+C$
3. $\frac{-\text{cosec}x\cot x+\ln |\text{cosec}x-\cot x|}{3}+C$
4. $\frac{\text{cosec}x\cot x-\ln |\text{cosec}x-\cot x|}{2}+C$

Q.

Give five examples for linear polynomial, quadratic, cubic polynomial.

Q.

The maximum value of $e^{\left(2+\sqrt{3}\cos \left(x\right)+\sin \left(x\right)\right)}$ is

1. $e^2$

2. $e^{\left(2-\sqrt{3}\right)}$

3. $e^4$

4. $1$

Q.

If ${\log }_{\frac{1}{2}}\left(x^2-5x+7\right)>0$, then an exhaustive range of values of $x$ is

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

2. $(2,3)$

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

4. None of these

Q.

If ${\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{2}{3}={\sin }^{-1}x$, then $x$ is equal to

1. $0$

2. $\frac{(\sqrt{5}-4\sqrt{2})}{9}$

3. $\frac{(\sqrt{5}+4\sqrt{2})}{9}$

4. $\frac{\mathrm{\pi }}{2}$

Q.

The value of $cot\left[\cos e{c}^{-1}\left(\frac{5}{3}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]$ is

1. $\frac{3}{17}$

2. $\frac{4}{17}$

3. $\frac{5}{17}$

4. $\frac{6}{17}$

Q. Find the second order derivative of the function $y=x^3\log x$

Q. Differential equation representing the family of curves $y=e^x(Acosx+Bsinx)$ is $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$

Q.

If $n\left(A\right)=4$ and $n\left(B\right)=6$. then, the number of one-one function from $A$ to $B$ is.

1. $24$

2. $60$

3. $120$

4. $360$

Q. Find d²y/dx²: y=acost x=bsint

Q. The number of real roots of the polynomial equation $x^4-x^2+2x-1=0$ is

1. 0
2. 2
3. 3
4. 4

Q.

If $I={\int }_0^1\frac{dx}{1+x^{{\displaystyle \frac{\mathrm{\pi }}{2}}}}$, then

1. ${\log }_{e}2<I<\frac{\pi }{4}$

2. ${\log }_{e}2>I$

3. $I=\frac{\mathrm{\pi }}{4}$

4. $I={\log }_{e}2$

Q.

$\frac{(1+7i)}{(2-i{)}^2}$ is equal to

1. $\sqrt{2}[\cos (3\pi /4)+i\sin (3\pi /4\left)\right]$

2. $\sqrt{2}[\cos (\pi /4)+i\sin (\pi /4\left)\right]$

3. $\cos (3\pi /4)+i\sin (3\pi /4)$

4. $Noneofthese$

Q. If $y=x^5\left(\cos \right(\ln x)+\sin (\ln x\left)\right)$, then the value of $(a+b)$ in the relation $x^2{y}_2+ax{y}_1+by=0$ is
$(y_1$ and $y_2$ denote the first and second derivative of $y$ with respect to $x$, respectively.$)$

Q.

The expression $\frac{\left(\cos 6x+6\cos 4x+15\cos 2x+10\right)}{\left(\cos 5x+5\cos 3x+10\cos x\right)}$ equal to

1. $\cos 2x$

2. $2\cos x$

3. ${\cos }^{2}x$

4. $1+\cos x$