Limits, Continuity & Derivatives · NDA previous year questions

Consider the following statements : 1. $f(x)=[x]$, where $[.]$ is the greatest integer function, is discontinuous at $x=n$, where $n\in Z$. 2. $f(x)=\cot x$ is discontinuous at $x=n\pi$, where $n\in Z$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Consider the function $f(x)=\{\begin{array}{ll}-1& \text{if }x\le 0\\ ax+b& \text{if }0<x<1\\ 1& \text{if }x\ge 1\end{array}$ where $a,b$ are constants. The function is continuous everywhere. What is the value of $b$?

1. A.$-1$
2. B.$1$
3. C.$0$
4. D.$2$

Consider the function $f(x)=\{\begin{array}{ll}\frac{\alpha \cos x}{\pi -2x}& \text{if }x\ne \frac{\pi }{2}\\ 3& \text{if }x=\frac{\pi }{2}\end{array}$ which is continuous at $x=\frac{\pi }{2}$, where $\alpha$ is a constant. What is the value of $\alpha$?

1. A.$6$
2. B.$3$
3. C.$2$
4. D.$1$

What is the derivative of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to ${\tan }^{-1}x$?

1. A.$0$
2. B.$1/2$
3. C.$1$
4. D.$x$

Consider the following statements: 1. $y=\frac{e^x+e^{-x}}{2}$ is an increasing function on $[0,\infty )$. 2. $y=\frac{e^x-e^{-x}}{2}$ is an increasing function on $(-\infty ,\infty )$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next two (2) items that follow : Consider the function $f(x)=\frac{x^2-1}{x^2+1}$, where $x\in \mathbb{R}$. At what value of $x$ does $f(x)$ attain minimum value?

1. A.$-1$
2. B.$0$
3. C.$1$
4. D.$2$

Consider the function $f(x)=\frac{x^2-1}{x^2+1}$, where $x\in \mathbb{R}$. What is the minimum value of $f(x)$?

1. A.$0$
2. B.$1/2$
3. C.$-1$
4. D.$2$

Consider the function $f(x)=0.75x^4-x^3-9x^2+7$ What is the maximum value of the function?

1. A.1
2. B.3
3. C.7
4. D.9

For the next three (03) items that follow : Consider the function $f(x)=\{\begin{array}{ll}{x}^2-5,& x\le 3\\ \sqrt{x+13},& x>3\end{array}$ Consider the following statements : 1. The function is discontinuous at $x=3$. 2. The function is not differentiable at $x=0$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Consider the function $f(x)=\{\begin{array}{ll}\frac{\tan kx}{x},& x<0\\ 3x+2k^2,& x\ge 0\end{array}$ What is the non-zero value of k for which the function is continuous at $x=0$?

1. A.1/4
2. B.1/2
3. C.1
4. D.2

Consider the following statements: 1. The function $f(x)=[x]$, where $[.]$ is the greatest integer function defined on $\mathbb{R}$, is continuous at all points except at $x=0$. 2. The function $f(x)=\sin |x|$ is continuous for all $x\in \mathbb{R}$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is ${\lim }_{x\to 0}\frac{{\log }_5(1+x)}{x}$ equal to?

1. A.1
2. B.${\log }_{5}e$
3. C.${\log }_{e}5$
4. D.5

What is ${\lim }_{x\to 0}\frac{5^x-1}{x}$ equal to?

1. A.${\log }_{e}5$
2. B.${\log }_{5}e$
3. C.5
4. D.1

What is ${\lim }_{n\to \infty }\frac{1+2+3+\cdots +n}{1^2+2^2+3^2+\cdots +n^2}$ equal to?

1. A.5
2. B.2
3. C.1
4. D.0

For the next two (02) items that follow: Consider the curve $x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$. What is $\frac{dy}{dx}$ equal to?

1. A.$\tan \theta$
2. B.$\cot \theta$
3. C.$\sin 2\theta$
4. D.$\cos 2\theta$

For the next two (02) items that follow: Consider the curve $x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$. What is $\frac{d^{2}y}{d{x}^2}$ equal to?

1. A.${\sec }^2\theta$
2. B.$-{\mathrm{cosec}}^2\theta$
3. C.$\frac{{\sec }^3\theta }{a\theta }$
4. D.None of the above

If $y=x\ln x+x{e}^x$, then what is the value of $\frac{dy}{dx}$ at $x=1$?

1. A.$1+e$
2. B.$1-e$
3. C.$1+2e$
4. D.None of the above

What is the height of the cylinder of maximum volume?

1. A.$\frac{2r}{\sqrt{3}}$
2. B.$\frac{r}{\sqrt{3}}$
3. C.$2r$
4. D.$\sqrt{3}r$

What is the radius of the cylinder of maximum volume?

1. A.$\frac{2r}{\sqrt{3}}$
2. B.$\frac{\sqrt{2}r}{\sqrt{3}}$
3. C.$r$
4. D.$\sqrt{3}r$

A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length $x$ from the four corners and turning up the sides. What is the value of $x$ for which the volume is maximum?

1. A.1 inch
2. B.1.5 inch
3. C.2 inch
4. D.2.5 inch

A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length $x$ from the four corners and turning up the sides. What is the maximum volume of the box?

1. A.200 cubic inch
2. B.400 cubic inch
3. C.100 cubic inch
4. D.None of the above

Consider the following statements: 1. The function $f(x)=\sqrt[3]{x}$ is continuous at all x except at $x=0$. 2. The function $f(x)=[x]$ is continuous at $x=2.99$ where [.] is the bracket function. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Consider the following statements: 1. The function $f(x)=|x|$ is not differentiable at $x=1$. 2. The function $f(x)=e^x$ is differentiable at $x=0$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next three (03) items that follow: Let $f(x)$ be a function defined in $1\le x<\infty$ by $$f(x)=\{\begin{array}{ll}2-x& \text{for }1\le x\le 2\\ 3x-x^2& \text{for }x>2.\end{array}$$ Consider the following statements: 1. The function is continuous at every point in the interval $[1,\infty )$. 2. The function is differentiable at $x=1.5$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next three (03) items that follow: Let $f(x)$ be a function defined in $1\le x<\infty$ by $$f(x)=\{\begin{array}{ll}2-x& \text{for }1\le x\le 2\\ 3x-x^2& \text{for }x>2.\end{array}$$ Consider the following statements: 1. The function is continuous at every point in the interval $[1,\infty )$. 2. The function is differentiable at $x=1.5$. What is the differentiable coefficient of $f(x)$ at $x=3$?

1. A.1
2. B.2
3. C.$-1$
4. D.$-3$