Limits, Continuity & Derivatives · NDA previous year questions

For the next two (2) items that follow : Consider the equation $x+|y|=2y$. What is the derivative of $y$ as a function of $x$ with respect to $x$ for $x<0$?

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

For the next two (2) items that follow : Consider the function $$f(x)=\frac{a^{[x]+x}-1}{[x]+x}$$ where $[\cdot ]$ denotes the greatest integer function. What is ${\lim }_{x\to 0^{+}}f(x)$ equal to?

1. A.1
2. B.$\ln a$
3. C.$1-a^{-1}$
4. D.Limit does not exist

For the next two (2) items that follow : Consider the function $$f(x)=\frac{a^{[x]+x}-1}{[x]+x}$$ where $[\cdot ]$ denotes the greatest integer function. What is ${\lim }_{x\to 0^{-}}f(x)$ equal to?

1. A.0
2. B.$\ln a$
3. C.$1-a^{-1}$
4. D.Limit does not exist

For the next two (2) items that follow : A function $f(x)$ is defined as follows : $$f(x)=\{\begin{array}{ll}x+\pi & \text{for }x\in [-\pi ,0)\\ \pi \cos x& \text{for }x\in [0,\frac{\pi }{2}]\\ {\left(x-\frac{\pi }{2}\right)}^2& \text{for }x\in (\frac{\pi }{2},\pi ]\end{array}$$ Consider the following statements : 1. The function $f(x)$ is continuous at $x=0$. 2. The function $f(x)$ is continuous at $x=\frac{\pi }{2}$. 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 : A function $f(x)$ is defined as follows : $$f(x)=\{\begin{array}{ll}x+\pi & \text{for }x\in [-\pi ,0)\\ \pi \cos x& \text{for }x\in [0,\frac{\pi }{2}]\\ {\left(x-\frac{\pi }{2}\right)}^2& \text{for }x\in (\frac{\pi }{2},\pi ]\end{array}$$ Consider the following statements : 1. The function $f(x)$ is differentiable at $x=0$. 2. The function $f(x)$ is differentiable at $x=\frac{\pi }{2}$. 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

If ${\lim }_{x\to 0}\varphi (x)=a^2$, where $a\ne 0$, then what is ${\lim }_{x\to 0}\varphi \left(\frac{x}{a}\right)$ equal to?

1. A.$a^2$
2. B.$a^{-2}$
3. C.$-a^2$
4. D.$-a$

What is ${\lim }_{x\to 0}{e}^{-\frac{1}{x^2}}$ equal to?

1. A.0
2. B.1
3. C.$-1$
4. D.Limit does not exist

For the next five (5) items that follow : Consider the function $$f(x)=|x-1|+x^2$$ where $x\in \mathbb{R}$. Which one of the following statements is correct?

1. A.$f(x)$ is continuous but not differentiable at $x=0$
2. B.$f(x)$ is continuous but not differentiable at $x=1$
3. C.$f(x)$ is differentiable at $x=1$
4. D.$f(x)$ is not differentiable at $x=0$ and $x=1$

For the next five (5) items that follow : Consider the function $$f(x)=|x-1|+x^2$$ where $x\in \mathbb{R}$. Which one of the following statements is correct?

1. A.$f(x)$ is increasing in $(-\infty ,\frac{1}{2})$ and decreasing in $(\frac{1}{2},\infty )$
2. B.$f(x)$ is decreasing in $(-\infty ,\frac{1}{2})$ and increasing in $(\frac{1}{2},\infty )$
3. C.$f(x)$ is increasing in $(-\infty ,1)$ and decreasing in $(1,\infty )$
4. D.$f(x)$ is decreasing in $(-\infty ,1)$ and increasing in $(1,\infty )$

For the next five (5) items that follow : Consider the function $$f(x)=|x-1|+x^2$$ where $x\in \mathbb{R}$. Which one of the following statements is correct?

1. A.$f(x)$ has local minima at more than one point in $(-\infty ,\infty )$
2. B.$f(x)$ has local maxima at more than one point in $(-\infty ,\infty )$
3. C.$f(x)$ has local minimum at one point only in $(-\infty ,\infty )$
4. D.$f(x)$ has neither maxima nor minima in $(-\infty ,\infty )$

Let $f(x)$ and $g(x)$ be twice differentiable functions on $[0,2]$ satisfying $f^{\prime \prime }(x)=g^{\prime \prime }(x)$, $f^{\prime }(1)=4$, $g^{\prime }(1)=6$, $f(2)=3$ and $g(2)=9$. Then what is $f(x)-g(x)$ at $x=4$ equal to?

1. A.$-10$
2. B.$-6$
3. C.$-4$
4. D.2

Consider the function $$f(x)=\{\begin{array}{ll}ax-2& \text{for }-2<x<-1\\ -1& \text{for }-1\le x\le 1\\ a+2(x-1{)}^2& \text{for }1<x<2\end{array}$$ What is the value of $a$ for which $f(x)$ is continuous at $x=-1$ and $x=1$?

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

The function $f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at $x=\pi$. The value of $f(\pi )$ so that $f(x)$ is continuous at $x=\pi$ is

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

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

The derivative of $\ln (x+\sin x)$ with respect to $(x+\cos x)$ is

1. A.$\frac{1+\cos x}{(x+\sin x)(1-\sin x)}$
2. B.$\frac{1-\cos x}{(x+\sin x)(1+\sin x)}$
3. C.$\frac{1-\cos x}{(x-\sin x)(1+\cos x)}$
4. D.$\frac{1+\cos x}{(x-\sin x)(1-\cos x)}$

If $y={\cot }^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$, where $0<x<\frac{\pi }{2}$, then $\frac{dy}{dx}$ is equal to

1. A.$\frac{1}{2}$
2. B.2
3. C.$\sin x+\cos x$
4. D.$\sin x-\cos x$

The function $f(x)=\frac{x^2}{e^x}$ is monotonically increasing if

1. A.$x<0\text{ only}$
2. B.$x>2\text{ only}$
3. C.$0<x<2$
4. D.$x\in (-\infty ,0)\cup (2,\infty )$

If $x^a{y}^b=(x-y{)}^{a+b}$, then the value of $$\frac{dy}{dx}-\frac{y}{x}$$ is equal to

1. A.$\frac{a}{b}$
2. B.$\frac{b}{a}$
3. C.1
4. D.0

If $s=\sqrt{t^2+1}$, then $\frac{d^{2}s}{d{t}^2}$ is equal to

1. A.$\frac{1}{s}$
2. B.$\frac{1}{s^2}$
3. C.$\frac{1}{s^3}$
4. D.$\frac{1}{s^4}$

Consider the following statements: 1. The function $f(x)=x^2+2\cos x$ is increasing in the interval $(0,\pi )$ 2. The function $f(x)=\ln \left(\sqrt{1+x^2}-x\right)$ is decreasing in the interval $(-\infty ,\infty )$ 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

Consider the following statements: 1. $f(x)=\ln x$ is an increasing function on $(0,\infty )$. 2. $f(x)=e^x-x(\ln x)$ is an increasing function on $(1,\infty )$. 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 (02) items that follow: Consider the function $$f(x)={\left(\frac{1}{x}\right)}^{2x^2}\text{, where }x>0$$ At what value of $x$ does the function attain maximum value?

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

For the next two (02) items that follow: Consider the function $$f(x)={\left(\frac{1}{x}\right)}^{2x^2}\text{, where }x>0$$ The maximum value of the function is

1. A.$e$
2. B.$e^{\frac{2}{e}}$
3. C.$e^{\frac{1}{e}}$
4. D.$\frac{1}{e}$

Consider the function $$f(x)=-2x^3-9x^2-12x+1$$ The function $f(x)$ is an increasing function in the interval

1. A.$(-2,-1)$
2. B.$(-\infty ,-2)$
3. C.$(-1,2)$
4. D.$(-1,\infty )$

Consider the function $$f(x)=-2x^3-9x^2-12x+1$$ The function $f(x)$ is a decreasing function in the interval

1. A.$(-2,-1)$
2. B.$(-\infty ,-2)\text{ only}$
3. C.$(-1,\infty )\text{ only}$
4. D.$(-\infty ,-2)\cup (-1,\infty )$