Limits, Continuity & Derivatives · NDA previous year questions

2018 35 questions

1. Q592018 · 9 Sept 2018

   Consider the function $$f(x)=\{\begin{array}{ll}\frac{\sin 2x}{5x}& \text{if }x\ne 0\\ \frac{2}{15}& \text{if }x=0\end{array}$$ Which one of the following is correct in respect of the function?

   1. A.It is not continuous at $x=0$
   2. B.It is continuous at every $x$
   3. C.It is not continuous at $x=\pi$
   4. D.It is continuous at $x=0$
   Show answer

   Answer: A

2. Q602018 · 9 Sept 2018

   For the function $f(x)=|x-3|$, which one of the following is not correct?

   1. A.The function is not continuous at $x=-3$
   2. B.The function is continuous at $x=3$
   3. C.The function is differentiable at $x=0$
   4. D.The function is differentiable at $x=-3$
   Show answer

   Answer: A

3. Q612018 · 9 Sept 2018

   If the function $f(x)=\frac{2x-{\sin }^{-1}x}{2x+{\tan }^{-1}x}$ is continuous at each point in its domain, then what is the value of $f(0)$?

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

   Answer: B

4. Q622018 · 9 Sept 2018

   If $f(x)=\sqrt{25-x^2}$, then what is ${\lim }_{x\to 1}\frac{f(x)-f(1)}{x-1}$ equal to?

   1. A.$-\frac{1}{\sqrt{24}}$
   2. B.$\frac{1}{\sqrt{24}}$
   3. C.$-\frac{1}{4\sqrt{3}}$
   4. D.$\frac{1}{4\sqrt{3}}$
   Show answer

   Answer: A

5. Q632018 · 9 Sept 2018

   What is ${\lim }_{\theta \to 0}\frac{\sqrt{1-\cos \theta }}{\theta }$ equal to?

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

   Answer: Dropped in the final answer key (full marks to everyone)

6. Q642018 · 9 Sept 2018

   Let $f(x+y)=f(x)f(y)$ and $f(x)=1+xg(x)\varphi (x)$, where ${\lim }_{x\to 0}g(x)=a$ and ${\lim }_{x\to 0}\varphi (x)=b$. Then what is $f^{\prime }(x)$ equal to?

   1. A.$1+abf(x)$
   2. B.$1+ab$
   3. C.$ab$
   4. D.$abf(x)$
   Show answer

   Answer: D

7. Q652018 · 9 Sept 2018

   What is ${\lim }_{x\to \frac{\pi }{6}}\frac{2{\sin }^{2}x+\sin x-1}{2{\sin }^{2}x-3\sin x+1}$ equal to?

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

   Answer: D

8. Q662018 · 9 Sept 2018

   If $y={\tan }^{-1}\left(\frac{5-2\tan \sqrt{x}}{2+5\tan \sqrt{x}}\right)$, then what is $\frac{dy}{dx}$ equal to?

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

   Answer: A

9. Q672018 · 9 Sept 2018

   If $u=e^{ax}\sin bx$ and $v=e^{ax}\cos bx$, then what is $u\frac{du}{dx}+v\frac{dv}{dx}$ equal to?

   1. A.$a{e}^{2ax}$
   2. B.$(a^2+b^2)e^{ax}$
   3. C.$ab{e}^{2ax}$
   4. D.$(a+b)e^{ax}$
   Show answer

   Answer: A

10. Q682018 · 9 Sept 2018

    If $y=\sin (\ln x)$, then which one of the following is correct?

    1. A.$\frac{d^{2}y}{d{x}^2}+y=0$
    2. B.$\frac{d^{2}y}{d{x}^2}=0$
    3. C.$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$
    4. D.$x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0$
    Show answer

    Answer: C

11. Q692018 · 9 Sept 2018

    If $y=|\sin x{|}^{|x|}$, then what is the value of $\frac{dy}{dx}$ at $x=-\frac{\pi }{6}$?

    1. A.$\frac{2^{-\frac{\pi }{6}}(6\ln 2-\sqrt{3}\pi )}{6}$
    2. B.$\frac{2^{\frac{\pi }{6}}(6\ln 2+\sqrt{3}\pi )}{6}$
    3. C.$\frac{2^{-\frac{\pi }{6}}(6\ln 2+\sqrt{3}\pi )}{6}$
    4. D.$\frac{2^{\frac{\pi }{6}}(6\ln 2-\sqrt{3}\pi )}{6}$
    Show answer

    Answer: A

12. Q702018 · 9 Sept 2018

    What is $\frac{d\sqrt{1-\sin 2x}}{dx}$ equal to, where $\frac{\pi }{4}<x<\frac{\pi }{2}$?

    1. A.$\cos x+\sin x$
    2. B.$-(\cos x+\sin x)$
    3. C.$\pm (\cos x+\sin x)$
    4. D.None of the above
    Show answer

    Answer: A

13. Q722018 · 9 Sept 2018

    Which one of the following is correct in respect of the function $f(x)=x\sin x+\cos x+\frac{1}{2}{\cos }^{2}x$?

    1. A.It is increasing in the interval $\left(0,\frac{\pi }{2}\right)$
    2. B.It remains constant in the interval $\left(0,\frac{\pi }{2}\right)$
    3. C.It is decreasing in the interval $\left(0,\frac{\pi }{2}\right)$
    4. D.It is decreasing in the interval $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$
    Show answer

    Answer: A

14. Q732018 · 9 Sept 2018

    In which one of the following intervals is the function $f(x)=x^2-5x+6$ decreasing?

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

    Answer: A

15. Q742018 · 9 Sept 2018

    A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector?

    1. A.25 m
    2. B.20 m
    3. C.10 m
    4. D.5 m
    Show answer

    Answer: C

16. Q752018 · 9 Sept 2018

    What is the minimum value of $[x(x-1)+1{]}^{\frac{1}{3}}$, where $0\le x\le 1$?

    1. A.${\left(\frac{3}{4}\right)}^{\frac{1}{3}}$
    2. B.$1$
    3. C.$\frac{1}{2}$
    4. D.${\left(\frac{3}{8}\right)}^{\frac{1}{3}}$
    Show answer

    Answer: A

17. Q392018 · 22 Apr 2018

    The maximum value of $\sin \left(x+\frac{\pi }{5}\right)+\cos \left(x+\frac{\pi }{5}\right)$, where $x\in \left(0,\frac{\pi }{2}\right)$, is attained at

    1. A.$\frac{\pi }{20}$
    2. B.$\frac{\pi }{15}$
    3. C.$\frac{\pi }{10}$
    4. D.$\frac{\pi }{2}$
    Show answer

    Answer: A

18. Q402018 · 22 Apr 2018

    What is the maximum value of $16\sin \theta -12{\sin }^2\theta$?

    1. A.$3/4$
    2. B.$4/3$
    3. C.$16/3$
    4. D.$4$
    Show answer

    Answer: C

19. Q412018 · 22 Apr 2018

    If $\sin x=\frac{1}{\sqrt{5}},\sin y=\frac{1}{\sqrt{10}}$, where $0<x<\frac{\pi }{2},0<y<\frac{\pi }{2}$, then what is $(x+y)$ equal to?

    1. A.$\pi$
    2. B.$\pi /2$
    3. C.$\pi /4$
    4. D.$0$
    Show answer

    Answer: C

20. Q422018 · 22 Apr 2018

    What is the principal value of ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$?

    1. A.$\pi /4$
    2. B.$\pi /2$
    3. C.$\pi /3$
    4. D.$2\pi /3$
    Show answer

    Answer: C

21. Q432018 · 22 Apr 2018

    What is ${\tan }^{-1}\left(\frac{1}{4}\right)+{\tan }^{-1}\left(\frac{3}{5}\right)$ equal to?

    1. A.$0$
    2. B.$\pi /4$
    3. C.$\pi /3$
    4. D.$\pi /2$
    Show answer

    Answer: B

22. Q442018 · 22 Apr 2018

    If a flag-staff of 6 m height placed on the top of a tower throws a shadow of $2\sqrt{3}$ m along the ground, then what is the angle that the sun makes with the ground?

    1. A.$60^{\circ }$
    2. B.$45^{\circ }$
    3. C.$30^{\circ }$
    4. D.$15^{\circ }$
    Show answer

    Answer: A

23. Q682018 · 22 Apr 2018

    If $f(x)=\frac{x^2-9}{x^2-2x-3}$, $x\ne 3$ is continuous at $x=3$, then which one of the following is correct?

    1. A.$f(3)=0$
    2. B.$f(3)=1.5$
    3. C.$f(3)=3$
    4. D.$f(3)=-1.5$
    Show answer

    Answer: B

24. Q692018 · 22 Apr 2018

    What is ${\lim }_{x\to 0}\frac{\tan x}{\sin 2x}$ equal to?

    1. A.1/2
    2. B.1
    3. C.2
    4. D.Limit does not exist
    Show answer

    Answer: A

25. Q702018 · 22 Apr 2018

    What is ${\lim }_{h\to 0}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h}$ equal to?

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

    Answer: D