Higher Order Derivatives

Q. If $f:R\to R,f(x^2+x+3)+2f(x^2-3x+5)=6x^2-10x+17\forall x\in R$
then which among the following options are correct

1. $f\left(x\right)$ is an odd function
2. $f\left(x\right)$ is invertible function.
3. $f\left(x\right)=0$ has a root in $(0,2)$
4. $f^{\prime }\left(x\right)$ is an even function

Q. $Ify=ta{n}^{-1}\left(\frac{lo{g}_e\left(e/x^2\right)}{lo{g}_e\left(e{x}^2\right)}\right)+ta{n}^{-1}\left(\frac{3+2lo{g}_{e}x}{1-6lo{g}_{e}x}\right),then\frac{d^{2}y}{d{x}^2}is$

1. 2
2. 1
3. \N
4. -1

Q. If $y^2=p\left(x\right)$ is a polynomial of degree 3, then $2\frac{d}{dx}$ $\left[y^3\frac{d^{2}y}{d{x}^2}\right]$ is equal to

1. p"' (x) + p' x
2. p''(x). p"'(x)
3. p (x). p"' (x)
4. None of these

Q. If $y=a\cos \left(\ln x\right)+b\sin \left(\ln x\right)$, then $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}$ is equal to

1. \N
2. $y$
3. $-y$
4. none of these

Q.

$\frac{d^{2}x}{d{y}^2}$ equals:

1. $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-3}$

2. $\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-2}$

3. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

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

Q. If $f\left(x\right)=(a{x}^2+b{)}^3,b\in R,a\in R-\{0\}$ and $g\left(x\right)$ is a function such that $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)=x,$ then $g\left(x\right)=$
(Given that $f$ and $g$ are bijective functions)

1. $\sqrt{\frac{x^{1/3}+b}{a}}$
2. $\sqrt{\frac{x^3-b}{a}}$
3. $\sqrt{\frac{x^{1/3}-b}{a}}$
4. $\sqrt{\frac{x^3+b}{a}}$

Q. $If\sqrt{(x+y)}+\sqrt{(y-x)}=a,then\frac{d^{2}y}{d{x}^2}equals$

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

Q. If $y=sinx,then\frac{d^{3}y}{d{x}^3}atx=\frac{\pi }{2}$ will be _____

1. \N
2. 1
3. -1
4. None of the above

Q. Let $f:R\to R$ be a function such that $f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+x{f}^{\prime \prime }\left(2\right)+f^{\prime \prime \prime }\left(3\right),x\in R$. Then $f\left(2\right)$ equals :

1. $-4$
2. $30$
3. $8$
4. $-2$

Q. If $y=ln{\left(\frac{x}{a+bx}\right)}^x$, then $x^3\frac{d^{2}y}{d{x}^2}$ is equal to

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

Q. Answer the following by appropriately matching the lists based on the information in Column I and Column II​​​​​​
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a}.y=f\left(x\right)\text{is given by}& \\ x=t^5-5t^3-20t+7& \\ \text{and}y=4t^3-3t^2-18t+3.& \\ \text{Then}-5\times \frac{dy}{dx}\text{at}t=1& \text{p.}0\\ \text{b}.\text{Let}P\left(x\right)\text{be a polynomial of degree}4,& \\ \text{with}P\left(2\right)=-1,P^{\prime }\left(2\right)=0,& \\ P^{\prime \prime }\left(2\right)=2,P^{\prime \prime \prime }\left(2\right)=-12\text{and}& \\ P^{iv}\left(2\right)=24,\text{then}{P}^{\prime \prime }\left(3\right)\text{is}& \text{q.}-2\\ \text{c}.y=\frac{1}{x},\text{then}\frac{\frac{dy}{\sqrt{1+y^4}}}{\frac{dx}{\sqrt{1+x^4}}}& r.2\\ \text{d}.f\left(\frac{2x+3y}{5}\right)=\frac{2f\left(x\right)+3f\left(y\right)}{5}\text{and}& \\ f^{\prime }\left(0\right)=p\text{and}f\left(0\right)=q.\text{Then},f^{\prime \prime }\left(0\right)\text{is}& \text{s.}-1\end{array}$

1. $a\to p,b\to r,c\to s,d\to q$
2. $a\to q,b\to r,c\to s,d\to p$
3. $a\to q,b\to s,c\to r,d\to p$
4. $a\to q,b\to r,c\to p,d\to s$

Q. If $y=f\left(x\right),then\frac{d^{2}y}{d{x}^2}=\left(f^{\prime }\right(x){)}^2.$

1. True
2. False

Q. If $y=sin\left(sinx\right),then\frac{d^{2}y}{d{x}^2}+tanx\frac{dy}{dx}+yco{s}^{2}x$ will be equal to

1. 1
2. -1
3. 2
4. \N

Q.

If $f\left(x\right)=2x^2+5\sqrt{x+2}$, complete the following statement $f\left(2\right)=\_\_\_$

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. If $f\left(x\right)=(1+x{)}^n$, then the value of $f\left(0\right)+f^{\prime }\left(0\right)+\frac{f^{\prime \prime }\left(0\right)}{2!}+\cdots +\frac{f^n\left(0\right)}{n!}$ is

1. $n$
2. $2^n$
3. $2^{n-1}$
4. None of these.

Q. If $e^y+xy=e,$ the ordered pair $(\frac{dy}{dx},\frac{d^{2}y}{d{x}^2})$ at $x=0$ is equal to :

1. $(-\frac{1}{e},\frac{1}{e^2})$
2. $(\frac{1}{e},-\frac{1}{e^2})$
   
3. $(\frac{1}{e},\frac{1}{e^2})$
4. $(-\frac{1}{e},-\frac{1}{e^2})$

Q. $\mathrm{If}y=\frac{{\sin }^{-1}x}{1-x^2},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\left(1-x^2\right)\frac{d^{2}y}{{\mathrm{dx}}^2}-3x\frac{\mathrm{dy}}{\mathrm{dx}}-y=$

Q.

$\frac{d^{2}x}{d{y}^2}$ equals:

1. $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-3}$

2. $\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-2}$

3. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

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

Q. Let $f:R\to R$ be a function such that $f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+x{f}^{\prime \prime }\left(2\right)+f^{\prime \prime \prime }\left(3\right),x\in R$. Then $f\left(2\right)$ equals :

1. $-4$
2. $30$
3. $8$
4. $-2$

Q. If $f:R\to R$ is a function such that $f\left(x\right)=x^3+x^2{f}^{\prime }\left(1\right)+x{f}^{\prime \prime }\left(2\right)+f^{\prime \prime \prime }\left(3\right)$ $\forall x\in R,$ then $f\left(2\right)-f\left(1\right)=$

1. $f\left(0\right)$
2. $-f\left(0\right)$
3. $f^{\prime }\left(0\right)$
4. $-f^{\prime }\left(0\right)$

Q. If $f:R\to R$ and $f\left(x\right)$ is a polynomial function of degree eleven and $f\left(x\right)=0$ has all real and distinct roots. Then the equation $\left(f^{\prime }\right(x){)}^2-f(x\left)f^{\prime \prime }\right(x)=0$ has

1. no real roots
2. 10 real roots
3. 11 real roots
4. 21 real roots

Q. A particle moves in a straight line according to the law $v^2=4a(x\sin x+\cos x)$ where $v$ is the velocity of a particle at a distance $x$ from the fixed point. Then the acceleration is

1. $2ax\cos x$
2. $ax\cos x$
3. $2ax\sin x$
4. $ax\sin x$

Q. If $\frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=k$, then k is equal to

1. \N
2. 1
3. 2
4. None of these