Limits, Continuity & Derivatives · NDA previous year questions

2017 49 questions

1. Q72017 · 23 Apr 2017

   If $|z+4|\le 3$, then the maximum value of $|z+1|$ is

   1. A.0
   2. B.4
   3. C.6
   4. D.10
   Show answer

   Answer: C

2. Q392017 · 23 Apr 2017

   The maximum value of $\sin \left(x+\frac{\pi }{6}\right)+\cos \left(x+\frac{\pi }{6}\right)$ in the interval $\left(0,\frac{\pi }{2}\right)$ is attained at

   1. A.$\frac{\pi }{12}$
   2. B.$\frac{\pi }{6}$
   3. C.$\frac{\pi }{3}$
   4. D.$\frac{\pi }{2}$
   Show answer

   Answer: A

3. Q402017 · 23 Apr 2017

   If $K=\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)$, then what is the value of $K$?

   1. A.$\frac{1}{2}$
   2. B.$\frac{1}{4}$
   3. C.$\frac{1}{8}$
   4. D.$\frac{1}{16}$
   Show answer

   Answer: C

4. Q412017 · 23 Apr 2017

   The expression $\frac{\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }$ is equal to

   1. A.$\tan \left(\frac{\alpha +\beta }{2}\right)$
   2. B.$\cot \left(\frac{\alpha +\beta }{2}\right)$
   3. C.$\sin \left(\frac{\alpha +\beta }{2}\right)$
   4. D.$\cos \left(\frac{\alpha +\beta }{2}\right)$
   Show answer

   Answer: A

5. Q422017 · 23 Apr 2017

   If $\sin \theta =3\sin (\theta +2\alpha )$, then the value of $\tan (\theta +\alpha )+2\tan \alpha$ is equal to

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

   Answer: B

6. Q432017 · 23 Apr 2017

   What is the value of $\tan 18^{\circ }$?

   1. A.$\frac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}}$
   2. B.$\frac{\sqrt{5}-1}{\sqrt{10+\sqrt{5}}}$
   3. C.$\frac{\sqrt{10+2\sqrt{5}}}{\sqrt{5}-1}$
   4. D.$\frac{\sqrt{10+\sqrt{5}}}{\sqrt{5}-1}$
   Show answer

   Answer: A

7. Q442017 · 23 Apr 2017

   If $\tan (\alpha +\beta )=2$ and $\tan (\alpha -\beta )=1$, then $\tan (2\alpha )$ is equal to

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

   Answer: A

8. Q452017 · 23 Apr 2017

   Consider the following for triangle $ABC$ : 1. $\sin \left(\frac{B+C}{2}\right)=\cos \left(\frac{A}{2}\right)$ 2. $\tan \left(\frac{B+C}{2}\right)=\cot \left(\frac{A}{2}\right)$ 3. $\sin (B+C)=\cos A$ 4. $\tan (B+C)=-\cot A$ Which of the above are correct?

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

   Answer: B

9. Q462017 · 23 Apr 2017

   If $\sec \theta -\mathrm{cosec}\theta =\frac{4}{3}$, then what is $(\sin \theta -\cos \theta )$ equal to?

   1. A.$-2$ only
   2. B.$\frac{1}{2}$ only
   3. C.Both $-2$ and $\frac{1}{2}$
   4. D.Neither $\frac{1}{2}$ nor $-2$
   Show answer

   Answer: B

10. Q472017 · 23 Apr 2017

    What is the maximum value of the function $f(x)=4{\sin }^{2}x+1$?

    1. A.5
    2. B.3
    3. C.2
    4. D.1
    Show answer

    Answer: A

11. Q732017 · 23 Apr 2017

    Let $f(x+y)=f(x)f(y)$ for all $x$ and $y$. Then what is $f^{\prime }(5)$ equal to [where $f^{\prime }(x)$ is the derivative of $f(x)$]?

    1. A.$f(5)f^{\prime }(0)$
    2. B.$f(5)-f^{\prime }(0)$
    3. C.$f(5)f(0)$
    4. D.$f(5)+f^{\prime }(0)$
    Show answer

    Answer: A

12. Q742017 · 23 Apr 2017

    Let $f(x)$ be defined as follows: $$f(x)=\{\begin{array}{ll}2x+1,& -3<x<-2\\ x-1,& -2\le x<0\\ x+2,& 0\le x<1\end{array}$$ Which one of the following statements is correct in respect of the above function?

    1. A.It is discontinuous at $x=-2$ but continuous at every other point.
    2. B.It is continuous only in the interval $(-3,-2)$.
    3. C.It is discontinuous at $x=0$ but continuous at every other point.
    4. D.It is discontinuous at every point.
    Show answer

    Answer: C

13. Q752017 · 23 Apr 2017

    Consider the following statements: 1. If ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist, then ${\lim }_{x\to a}\{f(x)g(x)\}$ exists. 2. If ${\lim }_{x\to a}\{f(x)g(x)\}$ exists, then both ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ must exist. 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
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    Answer: A

14. Q762017 · 23 Apr 2017

    What is $\frac{d^{2}x}{d{y}^2}$ equal to?

    1. A.$-{\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-3}$
    2. B.${\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-2}$
    3. C.$-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$
    4. D.${\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}$
    Show answer

    Answer: C

15. Q772017 · 23 Apr 2017

    What is the derivative of ${\log }_{10}(5x^2+3)$ with respect to $x$?

    1. A.$\frac{x{\log }_{10}e}{5x^2+3}$
    2. B.$\frac{2x{\log }_{10}e}{5x^2+3}$
    3. C.$\frac{10x{\log }_{10}e}{5x^2+3}$
    4. D.$\frac{10x{\log }_{e}10}{5x^2+3}$
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    Answer: C

16. Q782017 · 23 Apr 2017

    Suppose the function $f(x)=x^n$, $n\ne 0$ is differentiable for all $x$. Then $n$ can be any element of the interval

    1. A.$[1,\infty )$
    2. B.$(0,\infty )$
    3. C.$\left(\frac{1}{2},\infty \right)$
    4. D.None of the above
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    Answer: A

17. Q792017 · 23 Apr 2017

    What is the length of the longest interval in which the function $f(x)=3\sin x-4{\sin }^{3}x$ is increasing?

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

18. Q802017 · 23 Apr 2017

    Let $f(x)=x+\frac{1}{x}$, where $x\in (0,1)$. Then which one of the following is correct?

    1. A.$f(x)$ fluctuates in the interval
    2. B.$f(x)$ increases in the interval
    3. C.$f(x)$ decreases in the interval
    4. D.None of the above
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    Answer: C

19. Q812017 · 23 Apr 2017

    What is $\int \frac{dx}{x(x^7+1)}$ equal to?

    1. A.$\frac{1}{2}\ln \left|\frac{x^7-1}{x^7+1}\right|+c$
    2. B.$\frac{1}{7}\ln \left|\frac{x^7+1}{x^7}\right|+c$
    3. C.$\ln \left|\frac{x^7-1}{7x}\right|+c$
    4. D.$\frac{1}{7}\ln \left|\frac{x^7}{x^7+1}\right|+c$
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    Answer: D

20. Q822017 · 23 Apr 2017

    What is $\int \frac{(x^{e-1}+e^{x-1})dx}{x^e+e^x}$ equal to?

    1. A.$\frac{x^2}{2}+c$
    2. B.$\ln (x+e)+c$
    3. C.$\ln (x^e+e^x)+c$
    4. D.$\frac{1}{e}\ln (x^e+e^x)+c$
    Show answer

    Answer: D

21. Q852017 · 23 Apr 2017

    If $f(x)$ and $g(x)$ are continuous functions satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then what is ${\int }_0^{a}f(x)g(x)dx$ equal to?

    1. A.${\int }_0^{a}g(x)dx$
    2. B.${\int }_0^{a}f(x)dx$
    3. C.$2{\int }_0^{a}f(x)dx$
    4. D.0
    Show answer

    Answer: B

22. Q862017 · 23 Apr 2017

    What is ${\int }_{e^{-1}}^{e^2}\left|\frac{\ln x}{x}\right|dx$ equal to?

    1. A.$\frac{3}{2}$
    2. B.$\frac{5}{2}$
    3. C.3
    4. D.4
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    Answer: B

23. Q872017 · 23 Apr 2017

    If $xdy=y(dx+ydy);y(1)=1$ and $y(x)>0$, then what is $y(-3)$ equal to?

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

    Answer: A

24. Q1192017 · 23 Apr 2017

    What is the maximum area of a triangle that can be inscribed in a circle of radius $a$?

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

    Answer: C