AREA – CARTESIAN COORDINATES – MCQ- LEVEL

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Question 1 of 10

1. Question
The area bounded by $y = 1 + 2 \sin^2x$ , x – axis, $x=0$ and $x=\pi$ is :
- $2\pi$ sq. units
- $4\pi$ sq. units
- $6 \pi$ sq. units
- $8\pi$ sq. units
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Question 2 of 10

2. Question
If A₁ : Area of the region bounded by $\cos^2x$ , y–axis and $x= \pi$

A₂ : Area of the region bounded by $\sin^2x$ , y–axis and $x=\pi$ . Then,2
- $A_{1} > A_{2}$
- $A_{2} > A_{1}$
- $A_{1} = A_{2}$
- $A_{1} = -A_{2}$
Correct

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Question 3 of 10

3. Question
The area of the region bounded by the curves, $xy \leq 1$ ; $y^{2} = 8x$ ; $x^{2} = y$ ;
$x = 0$ ; $y = 0$ will be :
- $\frac{1}{3}-\log (1/2) \ \mathrm{sq. units}$
- $\frac{2}{3} \ \mathrm{sq. units}$
- $\frac{1}{3}+\log (1/2) \ \mathrm{sq. units}$
- $\frac{2}{3} - \log (1/2) \ \mathrm{sq. units}$
Correct

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Question 4 of 10

4. Question
Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$ , x – axis and the ordinates $x = \frac{\pi}{4}$ and [/latex] x = \beta > \frac{\pi}{4} [/latex] is $\beta \sin \beta +\frac{\pi}{4}\cos \beta + \sqrt{2} \beta [latex]. Then, [latex] f\left ( \frac{\pi}{2} \right )$ is :
- $\frac{\pi}{4} + \sqrt{2} - 1$
- $1-\frac{\pi}{4} - \sqrt{2}$
- $\frac{\pi}{4} - \sqrt{2} + 1$
- $1- \frac{\pi}{4} + \sqrt{2}$
Correct

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Question 5 of 10

5. Question
The area of the figure bounded by the straight lines $x=0$ , $x=2$ and the curves $y = 2^x$ and $y = 2x - x^{2}$ is :
- $\left ( \frac{4}{\log 2 } + \frac{8}{3} \right )\ \mathrm{sq. units}$
- $\left ( \frac{4}{\log 2 } - \frac{8}{3} \right )\ \mathrm{sq. units}$
- $\left ( \frac{8}{\log 2 } - \frac{4}{3} \right )\ \mathrm{sq. units}$
- $\left ( \frac{3}{\log 2 } - \frac{4}{3} \right )\ \mathrm{sq. units}$
Correct

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Question 6 of 10

6. Question
The area bounded by the curve $y = \log x$ and $y = (\log x)^2$ is :
- $(e - 3)\ \mathrm{sq. units}$
- $(3-e)\ \mathrm{sq. units}$
- $\frac{1}{2}(e - 3)\ \mathrm{sq. units}$
- $\frac{1}{2}(3 - e)\ \mathrm{sq. units}$
Correct

Incorrect
Question 7 of 10

7. Question
The area contained between the curve $y = \frac{a^3}{x^2 + a^2}$ and x – axis is :
- $\pi a^2 \ \mathrm{sq. units}$
- $3 \pi a^{2} \mathrm{sq. units}$
- $2 \pi a^2 \ \mathrm{sq. units}$
- None of these
Correct

Incorrect
Question 8 of 10

8. Question
The area bounded by $y = 4 - x^2$ and $y = \left [ 3 + \frac{x^2}{4} \right ]$ . Where [.] denotes the greatest integer function, is :
- $1 \ \mathrm{sq. unit}$
- $\frac{4}{3} \ \mathrm{sq. unit}$
- $\frac{1}{3} \ \mathrm{sq. unit}$
- None of these
Correct

Incorrect
Question 9 of 10

9. Question
If $a > 0,$ $b > 0$ , the maximum area of the triangle formed by the points $O(0, 0) ,$ $A \left ( a\cos \theta, b \sin \theta \right ) [latex] and [latex] B\left ( a\cos \theta - b\sin \theta \right )$ is (in sq. units) :
- $\frac{ab}{2}\ \mathrm{when} \ \theta = \frac{\pi}{2}$
- $\frac{ab}{2}\ \mathrm{when} \ \theta = \frac{\pi}{4}$
- $\frac{3ab}{4}\ \mathrm{when} \ \theta = \frac{\pi}{4}$
- None of these
Correct

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Question 10 of 10

10. Question
The area bounded by the curves $y^{2} = x - 1,$ $2y = x,$ x – axis and y – axis will be equal to :
- $\frac{1}{3} \ \mathrm{sq. units}$
- $2 \ \mathrm{sq. units}$
- $\frac{9}{2} \ \mathrm{sq. units}$
- $\frac{4}{5} \ \mathrm{sq. units}$
Correct

Incorrect