Integral Calculus & Area · NDA previous year questions

Passage: Consider the curves $y=x^2$ and $y=2|x|$. Question: What is the area bounded by the curves, the lines $x=0$ and $x=1$ ?

1. A.1 square unit
2. B.$\frac{2}{3}$ square unit
3. C.$\frac{1}{2}$ square unit
4. D.$\frac{1}{3}$ square unit

For the following three (03) items: Consider the function f(x) = x|x| What is the area bounded by the curve f(x) the x-axis and the lines x = - 2 and x = 1

1. A.1/3
2. B.2/3
3. C.5/2
4. D.3

What is the area of the region bounded by |x| <= 2k and |y| <= k where k is a positive real number?

1. A.$2k^2$
2. B.$4k^2$
3. C.$5k^2$
4. D.$8k^2$

. What is the area bounded by the curve, the x-axis and the line x = 4?

1. A.8 square units
2. B.16 square units
3. C.32 square units
4. D.64 square units

Consider the following for the two (02) items that follow: Let the function $y=(1-\cos x{)}^{-1}$ where $x\ne 2n\pi$ and n is an integer What is $\int ydx$ equal to? where c is the constant of integration.

1. A.$-\tan (x/2)+c$
2. B.$-\cot (x/2)+c$
3. C.$\tan (x/2)+c$
4. D.$\cot (x/2)+c$

What is the area bounded by the curve f(x) and y = 3?

1. A.3 square units
2. B.4.5 square units
3. C.7.5 square units
4. D.9 square units

If f(x) = px + q then what is the value of (p + q) ?

1. A.-1
2. B.0
3. C.1
4. D.5

Consider the following for the two (02) items that follow : Let the function $f(x)=x^2-1$ What is the area bounded by the function $f(x)$ and the x-axis?

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

Direction : Consider the following for the items that follow : The area bounded by the parabola $y^2=kx$ and the line $x=k$, where $k>0$, is 4/3 square units. What is the value of k?

1. A.1/2
2. B.1
3. C.$\sqrt{2}$
4. D.2

If $x^2+mx+n$ is an integer for all integral values of $x$, then which of the following is/are correct? 1. $m$ must be an integer 2. $n$ must be an integer 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

For how many integral values of k, the equation $x^2-4x+k=0$, where k is an integer has real roots and both of them lie in the interval $(0,5)$ ?

1. A.3
2. B.4
3. C.5
4. D.6

For what value of m with m < 0, is the area bounded by the lines y = x, y = mx and x = 2 equal to 3 ?

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

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.$\frac{\pi }{12}$

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

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.$f(\alpha \beta )$

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.$f(x)=2f(-x)$

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.$\frac{{\pi }^2}{4}-2$

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

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.$\frac{3}{2}$

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.$\sqrt{2\tan x}+c$

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.$\ln (e^x-e^{-x})+c$

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

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

What is the integral of $f(x)=1+x^2+x^4$ with respect to $x^2$?

1. A.$x+\frac{x^3}{3}+\frac{x^5}{5}+C$
2. B.$\frac{x^3}{3}+\frac{x^5}{5}+C$
3. C.$x^2+\frac{x^4}{4}+\frac{x^6}{6}+C$
4. D.$x^2+\frac{x^4}{2}+\frac{x^6}{3}+C$

What is the area bounded by y = [x], where [⋅] is the greatest integer function, the x-axis and the lines x = -1.5 and x = -1.8?

1. A.0.3 square unit
2. B.0.4 square unit
3. C.0.6 square unit
4. D.0.8 square unit