Left Hand Derivative

Q. The function $f\left(x\right)=|x^2-2x-3|\cdot e^{|9x^2-12x+4|}$ is not differentiable at exactly :

1. One point
2. Four points
3. Two points
4. Three points

Q. Find the derivative of $f\left(x\right)=3$ at $x=0$ and at $x=3$.

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Consider a differentiable function $f$ satisfying the relation $f(x-y+1)=\frac{f\left(x\right)}{f(y-1)}$ for all $x,y\in R$ and $f^{\prime }\left(0\right)=2,$ $f\left(0\right)=1.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\int f\left(x\right)dx=\frac{2f\left(x\right)}{p}+C\text{where}C\text{is constant of integration,}& \text{(P)}& 1\\ & \text{then the value of}p\text{is}& & \\ \text{(B)}& \text{The value of}\frac{d^{10}}{d{x}^{10}}\left(f\left(\frac{x}{2}\right)\right)\text{at}x=0\text{is}& \text{(Q)}& 2\\ \text{(C)}& \text{The number of solutions of the equation}f\left(x\right)=x^2\text{is}& \text{(R)}& 3\\ \text{(D)}& \text{The value of}\underset{x\to 0}{lim}\frac{f\left(x\right)-f\left(x^2\right)}{\sin x}\text{is}& \text{(S)}& 4\end{array}$

Which of the following is a CORRECT combination?

1. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(Q)}$
2. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(P)}$
3. $\text{(A)}\to \text{(P)},\text{(B)}\to \text{(S)}$
4. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(P)}$

Q.

Are Composite Functions Commutative?

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. If the function $f\left(x\right)=\{\begin{array}{ccc}2a^{2}x-1& ;& x\le 1\\ x^2+x+b& ;& x>1\end{array}$
is differentiable everywhere, where $a,b\in R$, then the value of $2a^2+b$ is

Q. Let $f\left(x\right)=e^{ax}+e^{bx},$ where $a\ne b$ and $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)-15f\left(x\right)=0$ for all $x\in R$. Then the product $ab$ is equal to

1. $25$
2. $9$
3. $-15$
4. $-9$

Q. If the function $f\left(x\right)=\{\begin{array}{ccc}2a^{2}x-1& ;& x\le 1\\ x^2+x+b& ;& x>1\end{array}$
is differentiable everywhere, where $a,b\in R$, then the value of $2a^2+b$ is

Q. Find the number of points of discontinuity for $f\left(x\right)=\left[6\sin x\right],0\le x\le \pi$ (Where $[.]$ denotes the greatest integer function)

Q. Assertion :If n is a positive integer then ${\int }_0^{n\pi }\left|\frac{\sin x}{x}\right|dx\ge \frac{2}{\pi }(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n})$ Reason: In the interval $(0,\frac{\pi }{2})$, $\frac{\sin x}{x}\ge \frac{2}{\pi }$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. Assertion (A): $f\left(x\right)=\frac{\sin \left\{\right[x\left]\pi \right\}}{1+x^2}$ is continuous on $R$ (where $\left[x\right]$ denotes greatest integer function of $x$).
Reason (R): Every constant function is continuous on $R$

1. Both A and R are true and R is the correct explanation of A
2. A is true but R is false
3. Both A and R are true and R is not the correct explanation of A
4. R is true but A is false

Q. The left-hand derivative of f(x) = [x] sin ($\pi$ x) at x = k, k is an integer and [x] = greatest integer $\le$ x, is

[IIT Screening 2001]

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. The left-hand derivative of f(x) = [x] sin ($\pi$ x) at x = k, k is an integer and [x] = greatest integer $\le$ x, is

[IIT Screening 2001]

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Consider a differentiable function $f$ satisfying the relation $f(x-y+1)=\frac{f\left(x\right)}{f(y-1)}$ for all $x,y\in R$ and $f^{\prime }\left(0\right)=2,$ $f\left(0\right)=1.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\int f\left(x\right)dx=\frac{2f\left(x\right)}{p}+C\text{where}C\text{is constant of integration,}& \text{(P)}& 1\\ & \text{then the value of}p\text{is}& & \\ \text{(B)}& \text{The value of}\frac{d^{10}}{d{x}^{10}}\left(f\left(\frac{x}{2}\right)\right)\text{at}x=0\text{is}& \text{(Q)}& 2\\ \text{(C)}& \text{The number of solutions of the equation}f\left(x\right)=x^2\text{is}& \text{(R)}& 3\\ \text{(D)}& \text{The value of}\underset{x\to 0}{lim}\frac{f\left(x\right)-f\left(x^2\right)}{\sin x}\text{is}& \text{(S)}& 4\end{array}$

Which of the following is a CORRECT combination?

1. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(Q)}$
2. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(S)}$
3. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(S)}$
4. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(Q)}$

Q. ${\int }_0^x\left[sint\right]dt$ where $x\in (2n\pi ,4n+1)\pi ,n\in N$ and $[.]$ denotes the greatest integer function is equal to .

Q. Assertion :Derivative of $si{n}^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to $co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is $1$ for $0<x<1.$ Reason: $si{n}^{-1}\left(\frac{2x}{1+x^2}\right)=co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ for $-1\le x\le 1$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. The integral ${\int }_0^{\pi }xf\left(\sin x\right)dx$ is equals to

1. $\frac{\pi }{2}{\int }_0^{\pi }f\left(\sin x\right)dx$
2. $\frac{\pi }{4}{\int }_0^{\pi }f\left(\sin x\right)dx$
3. $\pi {\int }_0^{\pi /2}f\left(\sin x\right)dx$
4. $\pi {\int }_0^{\pi /2}f\left(\cos x\right)dx$

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]sin\left(\pi x\right)$ at x= k, k is an integer and [x] = greatest integer, is

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}(k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. Evaluate: $\int \frac{{\left({\sin }^{-1}x\right)}^3}{\sqrt{(1-x^2)}}dx$

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

Q. Evaluate: $\int \left[\sin x\right]dx$ for $x\in (0,\frac{\pi }{2})$, where [.] represents greatest integer function.

1. $\cos x+c$, $c$ is a constant of integration
2. $c$, $c$ is a constant of integration
3. $0$
4. None of the above

Q. Find the values of $\sin (\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2}))$

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

Q. The value of ${\int }_{\pi }^{2\pi }\left[2\sin x\right]dx,$ where $\left[\right]$ represents the greatest integer function is

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

Q. $\underset{x\to \frac{\pi }{2}}{lim}\frac{1-\sin x}{(\pi -2x{)}^2}$=

1. $\frac{1}{4}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{8}$

Q. $\int \frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}dx$.

Q. Differentiate $\sin (x^2+1)$

1. $-2x\sin (x^2+1).$
2. $-2x\cos (x^2+1).$
3. $2x\cos (x^2+1).$
4. $2x\sin (x^2+1).$

Q. ${\int }_0^{\pi }xf\left(\sin x\right)dx$ is equal to

1. $\pi {\int }_0^{x}f\left(\cos x\right)dx$
2. $\pi {\int }_0^{x}f\left(\sin x\right)dx$
3. $\frac{\pi }{2}{\int }_0^{x/2}f\left(\sin x\right)dx$
4. $\pi {\int }_0^{\pi /2}f\left(\cos x\right)dx$

Q. The left-hand derivative of $f\left(x\right)=\left[x\right]\sin \pi x$ at $x=k$, where k is an integer and [k]=greatest integer not greater than x, is

1. $(-1{)}^k(k-1)\pi$
2. $(-1{)}^{k-1}\cdot (k-1)\pi$
3. $(-1{)}^{k}k\pi$
4. $(-1{)}^{k-1}k\pi$

Q. Find the values if ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$

Q. Assertion :${\int }_0^{2\pi }{\sin }^{3}xdx=0$ Reason: If $f\left(x\right)={\sin }^{3}x$ then $f(-x)=-f\left(x\right)$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect