Limits, Continuity & Derivatives · NDA previous year questions

2016 36 questions

1. Q12016 · 18 Sept 2016

   In an examination, 70% students passed in Physics, 80% students passed in Chemistry, 75% students passed in Mathematics and 85% students passed in Biology, and x% students failed in all the four subjects. What is the minimum value of x ?

   1. A.10
   2. B.12
   3. C.15
   4. D.None of the above
   Show answer

   Answer: D

2. Q642016 · 18 Sept 2016

   Consider the following in respect of the function $f(x)=\{\begin{array}{ll}2+x,& x\ge 0\\ 2-x,& x<0\end{array}$ 1. ${\lim }_{x\to 1}f(x)$ does not exist. 2. $f(x)$ is differentiable at $x=0$. 3. $f(x)$ is continuous at $x=0$. Which of the above statements is/are correct?

   1. A.1 only
   2. B.3 only
   3. C.2 and 3 only
   4. D.1 and 3 only
   Show answer

   Answer: B

3. Q652016 · 18 Sept 2016

   Let $f:A\to \mathbf{R}$ where $A=\mathbf{R}\setminus \{0\}$ is such that $f(x)=\frac{x+|x|}{x}$. On which one of the following sets is $f(x)$ continuous?

   1. A.$A$
   2. B.$B=\{x\in \mathbf{R}:x\ge 0\}$
   3. C.$C=\{x\in \mathbf{R}:x\le 0\}$
   4. D.$D=\mathbf{R}$
   Show answer

   Answer: A

4. Q662016 · 18 Sept 2016

   Consider the following for the next three (03) items that follow: Let $f(x)=[x]$, where $[.]$ is the greatest integer function and $g(x)=\sin x$ be two real valued functions over $\mathbf{R}$. Which of the following statements is correct?

   1. A.Both $f(x)$ and $g(x)$ are continuous at $x=0$.
   2. B.$f(x)$ is continuous at $x=0$, but $g(x)$ is not continuous at $x=0$.
   3. C.$g(x)$ is continuous at $x=0$, but $f(x)$ is not continuous at $x=0$.
   4. D.Both $f(x)$ and $g(x)$ are discontinuous at $x=0$.
   Show answer

   Answer: C

5. Q692016 · 18 Sept 2016

   Let $f(x)=\{\begin{array}{ll}-2,& -3\le x\le 0\\ x-2,& 0<x\le 3\end{array}$ and $g(x)=f(|x|)+|f(x)|$ Which of the following statements is/are correct? 1. $g(x)$ is differentiable at $x=0$. 2. $g(x)$ is differentiable at $x=2$. Select the correct answer using the code given below:

   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. Q712016 · 18 Sept 2016

   Consider the following for the next three (03) items that follow: Let $f(x)=\{\begin{array}{ll}-2,& -3\le x\le 0\\ x-2,& 0<x\le 3\end{array}$ and $g(x)=f(|x|)+|f(x)|$ Which of the following statements are correct? 1. $g(x)$ is continuous at $x=0$. 2. $g(x)$ is continuous at $x=2$. 3. $g(x)$ is continuous at $x=-1$. Select the correct answer using the code given below:

   1. A.1 and 2 only
   2. B.2 and 3 only
   3. C.1 and 3 only
   4. D.1, 2 and 3
   Show answer

   Answer: D

7. Q722016 · 18 Sept 2016

   Consider the following for the next two (02) items that follow: Let $f(x)=\{\begin{array}{ll}\frac{e^x-1}{x},& x>0\\ 0,& x=0\end{array}$ be a real valued function. Which of the following statements is/are correct? 1. $f(x)$ is right continuous at $x=0$. 2. $f(x)$ is discontinuous at $x=1$. Select the correct answer using the code given below:

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

   Answer: D

8. Q732016 · 18 Sept 2016

   Consider the following for the next three (03) items that follow: Let $f(x)=(|x|-|x-1|{)}^2$. What is $f^{\prime }(x)$ equal to when $x>1$?

   1. A.0
   2. B.$2x-1$
   3. C.$4x-2$
   4. D.$8x-4$
   Show answer

   Answer: A

9. Q742016 · 18 Sept 2016

   Consider the following for the next three (03) items that follow: Let $f(x)=(|x|-|x-1|{)}^2$. What is $f^{\prime }(x)$ equal to when $0<x<1$?

   1. A.0
   2. B.$2x-1$
   3. C.$4x-2$
   4. D.$8x-4$
   Show answer

   Answer: D

10. Q752016 · 18 Sept 2016

    Consider the following for the next three (03) items that follow: Let $f(x)=(|x|-|x-1|{)}^2$. Which of the following equations is/are correct? 1. $f(-2)=f(5)$ 2. $f^{\prime \prime }(-2)+f^{\prime \prime }(0.5)+f^{\prime \prime }(3)=4$ Select the correct answer using the code given below:

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

    Answer: A

11. Q762016 · 18 Sept 2016

    Which one of the following statements is correct in respect of the function $f(x)=x^3\sin x$?

    1. A.It has local maximum at $x=0$.
    2. B.It has local minimum at $x=0$.
    3. C.It has neither maximum nor minimum at $x=0$.
    4. D.It has maximum value 1.
    Show answer

    Answer: B

12. Q772016 · 18 Sept 2016

    Consider the following function for the next two (02) items that follow: $f(x)=\{\begin{array}{ll}3x^2+12x-1,& -1\le x\le 2\\ 37-x,& 2<x\le 3\end{array}$ Which of the following statements is/are correct? 1. $f(x)$ is increasing in the interval $[-1,2]$. 2. $f(x)$ is decreasing in the interval $(2,3]$. Select the correct answer using the code given below:

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

    Answer: C

13. Q782016 · 18 Sept 2016

    Consider the following function for the next two (02) items that follow: $f(x)=\{\begin{array}{ll}3x^2+12x-1,& -1\le x\le 2\\ 37-x,& 2<x\le 3\end{array}$ Which of the following statements are correct? 1. $f(x)$ is continuous at $x=2$. 2. $f(x)$ attains greatest value at $x=2$. 3. $f(x)$ is differentiable at $x=2$. Select the correct answer using the code given below:

    1. A.1 and 2 only
    2. B.2 and 3 only
    3. C.1 and 3 only
    4. D.1, 2 and 3
    Show answer

    Answer: A

14. Q792016 · 18 Sept 2016

    Consider the following for the next two (02) items that follow: Let $f(x)=\{\begin{array}{ll}\frac{e^x-1}{x},& x>0\\ 0,& x=0\end{array}$ be a real valued function. Which one of the following statements is correct?

    1. A.$f(x)$ is a strictly decreasing function in $(0,x)$.
    2. B.$f(x)$ is a strictly increasing function in $(0,x)$.
    3. C.$f(x)$ is neither increasing nor decreasing in $(0,x)$.
    4. D.$f(x)$ is not decreasing in $(0,x)$.
    Show answer

    Answer: B

15. Q802016 · 18 Sept 2016

    What is $\int \frac{x^4-1}{x^2\sqrt{x^4+x^2+1}}dx$ equal to?

    1. A.$\sqrt{\frac{x^4+x^2+1}{x}}+c$
    2. B.$\sqrt{x^4+2-\frac{1}{x^2}}+c$
    3. C.$\sqrt{x^2+\frac{1}{x^2}+1}+c$
    4. D.$\frac{\sqrt{x^4-x^2+1}}{x}+c$
    Show answer

    Answer: C

16. Q812016 · 18 Sept 2016

    What is $\int e^{\sin x}\frac{x{\cos }^{3}x-\sin x}{{\cos }^{2}x}dx$ equal to?

    1. A.$(x+\sec x)e^{\sin x}+c$
    2. B.$(x-\sec x)e^{\sin x}+c$
    3. C.$(x+\tan x)e^{\sin x}+c$
    4. D.$(x-\tan x)e^{\sin x}+c$
    Show answer

    Answer: B

17. Q202016 · 17 Apr 2016
    Passage

    For the next three (3) items that follow : Consider the function

    For the next three (3) items that follow : Consider the function $f(\theta )=4({\sin }^2\theta +{\cos }^4\theta )$. What is the maximum value of the function $f(\theta )$?

    1. A.1
    2. B.2
    3. C.3
    4. D.4
    Show answer

    Answer: D

18. Q212016 · 17 Apr 2016
    Passage

    For the next three (3) items that follow : Consider the function

    For the next three (3) items that follow : Consider the function $f(\theta )=4({\sin }^2\theta +{\cos }^4\theta )$. What is the minimum value of the function $f(\theta )$?

    1. A.0
    2. B.1
    3. C.2
    4. D.3
    Show answer

    Answer: D

19. Q272016 · 17 Apr 2016
    Passage

    For the next two (2) items that follow : Consider the equation

    For the next two (2) items that follow : Consider the equation $k\sin x+\cos 2x=2k-7$. If the equation possesses solution, then what is the minimum value of $k$?

    1. A.1
    2. B.2
    3. C.4
    4. D.6
    Show answer

    Answer: B

20. Q282016 · 17 Apr 2016
    Passage

    For the next two (2) items that follow : Consider the equation

    For the next two (2) items that follow : Consider the equation $k\sin x+\cos 2x=2k-7$. If the equation possesses solution, then what is the maximum value of $k$?

    1. A.1
    2. B.2
    3. C.4
    4. D.6
    Show answer

    Answer: D

21. Q292016 · 17 Apr 2016

    Consider the following statements: 1. If $ABC$ is an equilateral triangle, then $3\tan (A+B)\tan C=1$. 2. If $ABC$ is a triangle in which $A=78^{\circ },B=66^{\circ }$, then $\tan \left(\frac{A}{2}+C\right)<\tan A$. 3. If $ABC$ is any triangle, then $\tan \left(\frac{A+B}{2}\right)\sin \left(\frac{C}{2}\right)<\cos \left(\frac{C}{2}\right)$. Which of the above statements is/are correct?

    1. A.1 only
    2. B.2 only
    3. C.1 and 2
    4. D.2 and 3
    Show answer

    Answer: B

22. Q302016 · 17 Apr 2016

    If $A=(\cos 12^{\circ }-\cos 36^{\circ })(\sin 96^{\circ }+\sin 24^{\circ })$ and $B=(\sin 60^{\circ }-\sin 12^{\circ })(\cos 48^{\circ }-\cos 72^{\circ })$, then what is $\frac{A}{B}$ equal to?

    1. A.-1
    2. B.0
    3. C.1
    4. D.2
    Show answer

    Answer: C

23. Q312016 · 17 Apr 2016

    Consider the following statements: 1. There exists $\theta \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ for which ${\tan }^{-1}(\tan \theta )\ne \theta$. 2. ${\sin }^{-1}\left(\frac{1}{3}\right)-{\sin }^{-1}\left(\frac{1}{5}\right)={\sin }^{-1}\left(\frac{2\sqrt{2}(\sqrt{3}-1)}{15}\right)$ 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: B

24. Q322016 · 17 Apr 2016

    Consider the following statements: 1. ${\tan }^{-1}x+{\tan }^{-1}\left(\frac{1}{x}\right)=\pi$ 2. There exist $x,y\in [-1,1]$, where $x\ne y$ such that ${\sin }^{-1}x+{\cos }^{-1}y=\frac{\pi }{2}$. 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

25. Q522016 · 17 Apr 2016

    For the next two (2) items that follow : Consider the equation $x+|y|=2y$. Which of the following statements are *not* correct? 1. $y$ as a function of $x$ is not defined for all real $x$. 2. $y$ as a function of $x$ is not continuous at $x=0$. 3. $y$ as a function of $x$ is differentiable for all $x$. Select the correct answer using the code given below.

    1. A.1 and 2 only
    2. B.2 and 3 only
    3. C.1 and 3 only
    4. D.1, 2 and 3
    Show answer

    Answer: D