Integral Calculus & Area · NDA previous year questions

2010 20 questions

1. Q632010 · 22 Aug 2010

   What is the area of the region bounded by the curve $f(x)=1-\frac{x^2}{4},x\in [-2,2]$ and the $x$-axis?

   1. A.8/3 square units
   2. B.4/3 square units
   3. C.2/3 square unit
   4. D.1/3 square unit
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   Answer: A

2. Q642010 · 22 Aug 2010

   What is ${\lim }_{x\to 0}\frac{a^x-b^x}{x}$ equal to?

   1. A.$\ln (ab)$
   2. B.$\frac{\ln a}{\ln b}$
   3. C.$\ln \left(\frac{a}{b}\right)$
   4. D.$\ln \left(\frac{b}{a}\right)$
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   Answer: C

3. Q652010 · 22 Aug 2010

   If $\sqrt{x}+\sqrt{y}=2$, then what is $\frac{dy}{dx}$ at $y=1$ equal to?

   1. A.5
   2. B.2
   3. C.4
   4. D.$-1$
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   Answer: D

4. Q662010 · 22 Aug 2010

   For a point of inflection of $y=f(x)$, which one of the following is correct?

   1. A.$\frac{dy}{dx}$ must be zero
   2. B.$\frac{d^{2}y}{d{x}^2}$ must be zero
   3. C.$\frac{dy}{dx}$ must be non-zero
   4. D.$\frac{d^{2}y}{d{x}^2}$ must be non-zero
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   Answer: B

5. Q702010 · 22 Aug 2010

   What is the value of the integral ${\int }_{-1}^1|x|dx$ ?

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

6. Q732010 · 22 Aug 2010

   What is $\int \frac{e^x(1+x)}{{\cos }^2(x{e}^x)}dx$ equal to? (where $c$ is a constant of integration.)

   1. A.$x{e}^x+c$
   2. B.$\cos (x{e}^x)+c$
   3. C.$\tan (x{e}^x)+c$
   4. D.$x\mathrm{cosec}(x{e}^x)+c$
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   Answer: C

7. Q772010 · 22 Aug 2010

   What is $\int \sqrt{x}{e}^{\sqrt{x}}dx$ equal to? (where $c$ is a constant of integration.)

   1. A.$2e^{\sqrt{x}}\left(x-2\sqrt{x}+2\right)+c$
   2. B.$2e^{\sqrt{x}}\left(x+2\sqrt{x}+2\right)+c$
   3. C.$2e^{\sqrt{x}}\left(x+2\sqrt{x}-2\right)+c$
   4. D.$2e^{\sqrt{x}}\left(x-2\sqrt{x}-2\right)+c$
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   Answer: A

8. Q792010 · 22 Aug 2010

   What is the general solution of $(1+e^x)ydy=e^{x}dx$ ? (where $c$ is a constant of integration.)

   1. A.$y^2=\ln \left[c^2(e^x+1{)}^2\right]$
   2. B.$y=\ln \left[c(e^x+1)\right]$
   3. C.$y^2=\ln \left[c(e^x+1)\right]$
   4. D.None of the above
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   Answer: A

9. Q802010 · 22 Aug 2010

   What is the value of ${\int }_{\pi /6}^{\pi /4}\frac{dx}{\sin x\cos x}$ ?

   1. A.$2\ln \sqrt{3}$
   2. B.$\ln \sqrt{3}$
   3. C.$2\ln 3$
   4. D.$4\ln 3$
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   Answer: B

10. Q942010 · 22 Aug 2010

    What is $\int {\sec }^{n}x\tan xdx$ equal to? (where $c$ is a constant of integration.)

    1. A.$\frac{{\sec }^{n}x}{n}+c$
    2. B.$\frac{{\sec }^{n-1}x}{n-1}+c$
    3. C.$\frac{{\tan }^{n}x}{n}+c$
    4. D.$\frac{{\tan }^{n-1}x}{n-1}+c$
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    Answer: A

11. Q952010 · 22 Aug 2010

    What is the area bounded by the curve $y=x^2$ and the line $y=16$ ?

    1. A.$32/3$
    2. B.$64/3$
    3. C.$256/3$
    4. D.$128/3$
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    Answer: C

12. Q322010 · 18 Apr 2010

    Consider the equation $(x-p)(x-6)+1=0$ having integral coefficients. If the equation has integral roots, then what values can p have?

    1. A.4 or 8
    2. B.5 or 10
    3. C.6 or 12
    4. D.3 or 6
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    Answer: A

13. Q662010 · 18 Apr 2010

    What is $\int e^{\ln x}\sin xdx$ equal to? where $c$ is a constant of integration.

    1. A.$e^{\ln x}(\sin x-\cos x)+c$
    2. B.$(\sin x-x\cos x)+c$
    3. C.$(x\sin x+\cos x)+c$
    4. D.$(\sin x+x\cos x)+c$
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    Answer: B

14. Q682010 · 18 Apr 2010

    What is $\int \frac{x^4+1}{x^2+1}dx$ equal to? where $c$ is a constant of integration.

    1. A.$\frac{x^3}{3}-x+4{\tan }^{-1}x+c$
    2. B.$\frac{x^3}{3}+x+4{\tan }^{-1}x+c$
    3. C.$\frac{x^3}{3}-x+2{\tan }^{-1}x+c$
    4. D.$\frac{x^3}{3}-x-4{\tan }^{-1}x+c$
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    Answer: C

15. Q692010 · 18 Apr 2010

    What is the value of ${\lim }_{x\to \infty }{\left(\frac{x+6}{x+1}\right)}^{x+4}$?

    1. A.$e$
    2. B.$e^2$
    3. C.$e^4$
    4. D.$e^5$
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    Answer: D

16. Q792010 · 18 Apr 2010

    What is the area enclosed between the curves $y^2=12x$ and the lines $x=0$ and $y=6$?

    1. A.2 square units
    2. B.4 square units
    3. C.6 square units
    4. D.8 square units
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    Answer: C

17. Q832010 · 18 Apr 2010

    What is $\int \frac{1}{1+e^x}dx$ equal to? where c is a constant of integration.

    1. A.$x-\ln x+c$
    2. B.$x-\ln (\tan x)+c$
    3. C.$x-\ln (1+e^x)+c$
    4. D.$\ln (1+e^x)+c$
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    Answer: C

18. Q972010 · 18 Apr 2010

    If $\int x^2\ln xdx=\frac{x^3}{m}\ln x+\frac{x^3}{n}+c$, then what are the values of m and n respectively? where c is a constant of integration.

    1. A.1/3, -1/9
    2. B.3, -9
    3. C.3, 9
    4. D.3, 3
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    Answer: B

19. Q982010 · 18 Apr 2010

    What is the principal value of ${\mathrm{cosec}}^{-1}(-\sqrt{2})$?

    1. A.$\frac{\pi }{4}$
    2. B.$\frac{\pi }{2}$
    3. C.$-\frac{\pi }{4}$
    4. D.0
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    Answer: C

20. Q992010 · 18 Apr 2010

    If $f:R\to R,g:R\to R$ and $g(x)=x+3$ and $(fog)(x)=(x+3{)}^2$, then what is the value of $f(-3)$?

    1. A.-9
    2. B.0
    3. C.9
    4. D.3
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    Answer: C