Limits, Continuity & Derivatives · NDA previous year questions

2016 36 questions

1. Q532016 · 17 Apr 2016

   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.$\frac{1}{3}$
   Show answer

   Answer: D

2. Q542016 · 17 Apr 2016

   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
   Show answer

   Answer: B

3. Q552016 · 17 Apr 2016

   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
   Show answer

   Answer: C

4. Q562016 · 17 Apr 2016

   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
   Show answer

   Answer: C

5. Q572016 · 17 Apr 2016

   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
   Show answer

   Answer: D

6. Q582016 · 17 Apr 2016

   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.$-a$
   Show answer

   Answer: A

7. Q592016 · 17 Apr 2016

   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
   Show answer

   Answer: A

8. Q602016 · 17 Apr 2016

   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 $x=1$
   Show answer

   Answer: B

9. Q692016 · 17 Apr 2016

   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 $(1,\infty )$
   Show answer

   Answer: B

10. Q702016 · 17 Apr 2016

    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 $(-\infty ,\infty )$
    Show answer

    Answer: C

11. Q1102016 · 17 Apr 2016

    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
    Show answer

    Answer: A