Integral Calculus & Area · NDA previous year questions

2022 11 questions

1. Q802022 · 4 Sept 2022

   What is the area of the region (in the first quadrant) bounded by $y=\sqrt{1-x^2}$, $y=x$ and $y=0$ ?

   1. A.$\frac{\pi }{4}$
   2. B.$\frac{\pi }{6}$
   3. C.$\frac{\pi }{8}$
   4. D.$\frac{\pi }{12}$
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   Answer: C

2. Q812022 · 4 Sept 2022

   What is the area of the region bounded by x − |y| = 0 and x − 2 = 0 ?

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

3. Q822022 · 4 Sept 2022

   If $f(\alpha )=\sqrt{{\sec }^2\alpha -1}$, then what is $\frac{f(\alpha )+f(\beta )}{1-f(\alpha )f(\beta )}$ equal to ?

   1. A.$f(\alpha -\beta )$
   2. B.$f(\alpha +\beta )$
   3. C.$f(\alpha )f(\beta )$
   4. D.$f(\alpha \beta )$
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   Answer: B

4. Q832022 · 4 Sept 2022

   If $f(x)=\ln (x+\sqrt{1+x^2})$, then which one of the following is correct ?

   1. A.$f(x)+f(-x)=0$
   2. B.$f(x)-f(-x)=0$
   3. C.$2f(x)=f(-x)$
   4. D.$f(x)=2f(-x)$
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   Answer: A

5. Q972022 · 4 Sept 2022

   What is the area of the region enclosed in the first quadrant by $x^2+y^2={\pi }^2$, $y=\sin x$ and $x=0$ ?

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

6. Q602022 · 10 Apr 2022

   The area of the region bounded by the parabola $y^2=4kx$, where $k>0$ and its latus rectum is 24 square units. What is the value of k?

   1. A.1
   2. B.2
   3. C.3
   4. D.4
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   Answer: C

7. Q612022 · 10 Apr 2022

   What is ${\int }_0^{\frac{\pi }{4}}\frac{\mathrm{d}x}{(\sin x+\cos x{)}^2}$ equal to?

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

8. Q622022 · 10 Apr 2022

   What is $\int (\sin x{)}^{-1/2}(\cos x{)}^{-3/2}\mathrm{d}x$ equal to?

   1. A.$\sqrt{\tan x}+c$
   2. B.$2\sqrt{\tan x}+c$
   3. C.$\sqrt{\cot x}+c$
   4. D.$\sqrt{2\tan x}+c$
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   Answer: B

9. Q632022 · 10 Apr 2022

   If $I_1=\int \frac{e^x\mathrm{d}x}{e^x+e^{-x}}$ and $I_2=\int \frac{\mathrm{d}x}{e^{2x}+1}$, then what is $I_1+I_2$ equal to?

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

10. Q642022 · 10 Apr 2022

    What is ${\int }_{-2}^{-1}\frac{x}{|x|}\mathrm{d}x$ equal to?

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

11. Q652022 · 10 Apr 2022

    How many extreme values does $\sin 4x+2x$, where $0<x<\frac{\pi }{2}$ have ?

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