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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Find the area enclosed by circle `x^(2) + y^(2) = 4` ,parabola `y = x^(2) + x + 1` , the curve `y = [|sin^2 ""(x)/(4) + cos""(x)/(4)|]` and x-axis |
| Answer» `((2pi)/(3) + sqrt3 - (1)/(6))` sq. unit | |
| 2. |
Find the ratio in which the area bounded by the curves `y^2=12 xa n dx^2=12 y`is divided by the line `x=3.`A. `15 : 16`B. `15 : 49`C. `1 : 2`D. `15 : 29` |
| Answer» Correct Answer - B | |
| 3. |
Find the area of the region bounded by the curve `y^2= x`and the lines `x = 1, x = 4`and the x-axis. |
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Answer» I=`int_1^4 yd(x)` =`int_1^4sqrtxdx` =`2/3(x^(3/2))^4` =`2/3(4^(3/2)-1)` =`2/3(4*2-1)` =`2/3*7` =`14/3 unit^2` |
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| 4. |
Let `f(x)=` maximum `{x^2, (1-x)^2, 2x(1 - x)}` where `x in [0, 1].` Determine the area of the region bounded by the curve `y=f(x)` and the lines `y = 0,x=0, x=1.` |
| Answer» `(17)/(27)` sq. units | |
| 5. |
The area of the region in first quadrant bounded by the curves `y = x^(3)` and `y = sqrtx` is equal toA. `(12)/(5)` sq. unitsB. `(5)/(12)` sq. unitsC. `(5)/(3)` sq. unitsD. `(3)/(5)` units |
| Answer» Correct Answer - B | |
| 6. |
The area of the region bounded by the curve `y = x^(2) - 2` and line y = 2 , is equal toA. `(32)/(3)` sq. unitB. `(19)/(2)` sq. unitC. `(21)/(5)` unitD. `(16)/(3)` sq. unit |
| Answer» Correct Answer - A | |
| 7. |
The area of figure bounded by the curves `y=a^x(a gt 1) and y=b^-x (b gt 1)` and the straight line `x = 1` is (1) `1/(loga) (a-1)+1/(logb) (1/b-1)` (2) `loga(a-1)+logb(1/b-1)` (3) `1/(loga)(a-1)+1/(log b)(b-1)` (4) `loga(a-1)+logb.(b-1)`A. `(1)/(log a)""(a-1) + (1)/(log b) ((1)/(b) - 1)`B. log a (a - 1) + log `((1)/(b) - 1)`C. `(1)/(log a) (a-1)+ (1)/(log b) (b-1)`D. log (a-1) + log (b-1) |
| Answer» Correct Answer - A | |
| 8. |
The area bounded between curves `y^2 = x and y= |x|`A. `(1)/(3)`B. `(2)/(3)`C. 1D. `(1)/(6)` |
| Answer» Correct Answer - D | |
| 9. |
The area between the curves `y= x^(2)` and `y = (2)/(1 + x^(2))` is equal toA. `pi - (1)/(3)` sq. unitsB. `pi - 2` sq. unitsC. `pi - (2)/(3)` sq. unitsD. `pi + (2)/(3)` sq. units |
| Answer» Correct Answer - C | |
| 10. |
STATEMENT-1 : The area bounded by the region `{(x,y) : 0 le y le x^(2) + 1 , 0 le y le x + 1 , 0 le x le 2}` is `(23)/(9)` sq. units and STATEMENT-2 : Required Area is ` int_(a)^(b) (y_(2) - y_(1)) `dxA. Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement-1B. Statement-1 is True , Statement-2 is True , Statement-2 is NOT a correct explanation for Statement-1C. Statement-1 is True , Statement-2 is FalseD. Statement-1 is False , Statement-2 is True |
| Answer» Correct Answer - A | |
| 11. |
The area of the region bounded by the curves `y = xe^x, y = e^x` and the lines `x = +-1,` is equal toA. e sq. unitsB. `e + (2)/(e)` sq. unitC. `e - (3)/(e)` sq. unitD. `e + (3)/(e)` sq. unit |
| Answer» Correct Answer - C | |
| 12. |
The area of the region bounded by the curve `y = x^(2)` and y `= |x|` is equal toA. `(5)/(3)` sq. unitsB. `(1)/(3)` sq. unitsC. `(5)/(3)` sq. unitsD. `(1)/(6)` sq. unit |
| Answer» Correct Answer - B | |
| 13. |
The slope of the tangent to a curve `y=f(x)` at `(x,f(x))` is `2x+1.` If the curve passes through the point `(1,2)` then the area of the region bounded by the curve, the x-axis and the line `x=1` is (A) `5/6` (B) `6/5` (C) `1/6` (D) `1`A. `(5)/(3)` sq. unitsB. `(5)/(6)` sq. unitsC. `(6)/(5)` sq. unitsD. 6 sq. units |
| Answer» Correct Answer - B | |
| 14. |
Using integration find the area of the region `{(x,y):x^2+y^2=ax,x,y>=0}` |
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Answer» `x^2+y^2<=2ax` `x^2-2ax+y^2+a^2-a^2<=0` `(x-a)^2+y^2<=a^2` `y^2>=ax` `(x-a)^2+ax=a^2` `x^2+a^2-2ax+ax=a^2` `x(x-a)=0` `x=0,a` `y^2=ax` `y^2=a*a=a^2` `y=a` Area of region`=int_0^(2a)ydx` `int_0^asqrt(ax)dx+int_a^(2a)sqrt(2ax-x^2)dx` `sqrtaint_0^asqrtxdx+int_a^(2a)sqrt(a^2-(x-a))dx` `2/3a^2+[0+(a^2pi)/4-0-0]` `=2/3a^2+pi/4a^2` `=a^2(2/3+pi/4)`. |
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| 15. |
The area of the region bounded by the x-axis , the function `y =-x^(2) + 4x - 8` and x = -1 and x = 4 is equal toA. `31""(2)/(3)` sq. unitsB. `31` sq unitsC. `32""(2)/(3)` sq. unitsD. 32 sq. units |
| Answer» Correct Answer - A | |
| 16. |
The area of the region bounded by the function `f(x) = x^(3)` , the x-axis and the lines `x = -1` and x = 1 is equal toA. `(1)/(4)` sq. unitB. `(1)/(2)` sq.unitC. `1` sq. unitsD. `(1)/(8)` sq. unit |
| Answer» Correct Answer - B | |
| 17. |
The area bounded by `y = x^(2) , x + y = 2` isA. `(9)/(2)`B. `(15)/(2)`C. `9`D. `(15` |
| Answer» Correct Answer - A | |
| 18. |
The area bounded by `y = -x^(2) + 1` and the x-axis isA. `(1)/(3)`B. `(2)/(3)`C. `(4)/(3)`D. `(8)/(3)` |
| Answer» Correct Answer - C | |
| 19. |
The value of a for which the area between the curves `y^(2) = 4ax` and `x^(2) = 4ay` is 1 unit isA. `sqrt3`B. `4`C. `4sqrt3`D. `(sqrt3)/(4)` |
| Answer» Correct Answer - D | |
| 20. |
The area of the region bounded by `y=x^2`, `y=[x+1]`, `x |
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Answer» `int_0^1(1-x^2)dx` `|x-x^3/3|_0^1` `1-1/3=2/3`. |
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| 21. |
Find the area bounded by the curve `y = sin x` between `x = 0` and `x=2pi`. |
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Answer» area=2 area of` 1^(st)` part =`2int_0^pi f(x) dx` =`2int_0^pi sinx dx` =`2(-cosx)_0^pi` =`2(-cospi+cos0)` =`2(1+1)` =`4 unit^2` |
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| 22. |
Find the area of the smaller part of the circle `x^2+y^2=a^2`cut off by the line `x=a/(sqrt(2))` |
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Answer» require Area= `int ydx` `= int_(a/sqrt2)^a 2 sqrt(a^2 - x^2) dx` `=> x = aSin theta` `dx= a cos theta d theta` when `x=a/sqrt2, sin theta= 1/sqrt2` `theta= pi/4` when `x=a, sin theta = 1` `theta= pi/2` `= 2 int _(pi/4)^ (pi/2) (a cos theta)(a cos theta) d theta` `= 2a^2 int_ (pi/4)^(pi/2) (cos 2 theta +1)d theta` `= a^2[ int_(pi/2)^(pi/4)Cos 2 theta d theta + int 1 d theta]` `= a^2[[sin 2 theta/2] + [ theta]]` `= a^2[[-1/2] +[pi/4]]` `= a^2[pi/4 - 1/2]` answer |
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| 23. |
Find the area of the region bounded by the two parabolas `y=x^2`and `y^2=x`. |
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Answer» `y_1=f(x)=x^2` `y_2=g(x)=sqrtx` area=`int_0^1g(x)-f(x)dx` =`int_0^1(x^2-sqrtx)dx` =`(-x^3/3+x^(3/2)/(3/2))_0^1` =`2/3-1/3` =`1/3 unit^2` |
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| 24. |
Statement - I : The value of the integral `int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))`is equal to `pi/6`.Statement - II : `int_a^bf(x)dx=int_a^bf(a+b-x)dxdot`(1)Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I(2)Statement -I is True; Statement -II is False.(3)Statement -I is False; Statement -II is True(4)Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I |
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Answer» `i = int_(pi/6)^(pi/3) dx/(1 + sqrt tan x) ` `= int_(pi/6)^(pi/3) dx/(1 + sqrt(tan(pi/3 + pi/6 - x))` `= int_(pi/6)^(pi/3) dx/(1 + sqrt cot x) ` `= int_(pi/6)^(pi/3) (sqrt tan x)/(sqrt tan x + 1) dx` equation `1+ 2` `2i = int_(pi/6)^(pi/3) (1 + sqrt tan x)/(1 + sqrt tan x) dx` `2i = x` `2i = pi/3 - pi/6` `2i = pi/6` `i=pi/12` option 3 is correct |
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| 25. |
The area of `int_(-10)^(10) ` f(x) dx , where f(x) = min {x -[x] , -x-[-x]} , isA. 20B. 40C. 5D. 30 |
| Answer» Correct Answer - C | |
| 26. |
The area bounded by the curve `y = cos^-1(cos x)` and `y=|x-pi|` isA. `pi^(2)` sq. unitB. `2pi^(2)` sq. unitC. `(pi^(2))/(2)` sq. unitD. `(pi^(2))/(4)` sq. unit |
| Answer» Correct Answer - C | |
| 27. |
The area bounded by curve `|x/9|+|y/9|=log_b axxlog_a b` where `a,b > 0 a !=b != 1` isA. 81B. 27C. 162D. 36 |
| Answer» Correct Answer - C | |
| 28. |
The area bounded by the curve `f(x)=||tanx+cotx|-|tanx-cotx||` between the lines `x=0,x=pi/2` and the X-axis isA. log 4B. log `sqrt2`C. 2 log 4D. `sqrt2` log 2 |
| Answer» Correct Answer - A | |
| 29. |
If the area bounded by x-axis, the curve `y = f(x)` and the ordinates `x = c and x = d` is independent of `d,AA d > c` (c is contant), then f isA. A non-zero constant functionB. Identify functionC. Zero functionD. Parabolic function |
| Answer» Correct Answer - C | |
| 30. |
If area bounded by the curve x = `ay^(2)` and `y = 1` is equal toA. `(1)/(sqrt3)`B. `(1)/(3)`C. `(1)/(2)`D. 3 |
| Answer» Correct Answer - A | |
| 31. |
The area of the region bounded by `y = |x - 1|` and `y = 1` isA. 1 sq. unitsB. 2 sq. unitsC. `(1)/(2)` sq. unitD. `3` sq. units |
| Answer» Correct Answer - A | |
| 32. |
The area between `x=y^2`and `x = 4`is divided into two equal parts by the line`x = a`, find the value of a. |
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Answer» area of first portion=area of second portion `int_0^a(2sqrtx)(dx)=int_a^4(2sqrtx dx` `2(x^(3/2)/(3/2))_0^a=2(x^(3/2)/(3/2))_a^4` `a^(3/2)=4^(3/2)-a^(3/2)` `2a^(3/2)=2^(2*(2/3))` `a=4^(2/3)` |
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| 33. |
The area bounded by the curve `y = x | x |`, x-axis and the ordinates `x = - 1`and `x = 1`is given by(A) 0 (B) `1/3` (C) `2/3` (D) `4/3`[Hint : `y=x^2`if `x > 0`and `y=-x^2`if `x < 0`]. |
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Answer» area=`int_-1^0ydx+int_0^1ydx` =`int_-1^0-x^2dx+int_0^1x^2dx` =`(-x^3/3)_-1^0+(x^3/3)_0^1` =`|-1/3|+1/3` =`2/3 unit^2` |
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| 34. |
Area bounded by the curve `y=x^3`, the x-axis and the ordinates `x = 2`and `x = 1`is(A) `-9` (B) `(-15)/4` (C) `(15)/4` (D) `(17)/4` |
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Answer» From, the given details, we can draw the diagram. Please refer video for the diagram. As per diagram, required area will be, `int_-2^0x^3dx+int_0^1x^3dx` `=|x^4/4|_-2^0+|x^4/4|_0^1` `=16/4+1/4 = 17/4`, which is the required area. |
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| 35. |
The area bounded by curve `|x|+|y| >= 1 and x^2+y^2 = 0` isA. 2 sq. unitB. `(pi)/(2)` sq. unitC. `((pi-2))/(2)` sq. unitD. `(pi - 2)` sq. unit |
| Answer» Correct Answer - C | |
| 36. |
The area bounded by the curve `y = (x - 1)^(2) , y = (x + 1)^(2)` and the x-axis isA. `(1)/(3)`B. `(2)/(3)`C. `(4)/(3)`D. `(8)/(3)` |
| Answer» Correct Answer - B | |
| 37. |
The area bounded by the curve `y=|x|-1 and y=-|x|+1` isA. 1 sq. unitB. 2 sq. unitC. `2sqrt2` sq. unitD. 4 sq. unit |
| Answer» Correct Answer - B | |
| 38. |
Smaller area enclosed by the circle `x^2+y^2=4`and the line `x + y = 2`is(A) `2(pi-2)` (B) `pi-2` (C) `2pi-1` (D) `2(pi+2)` |
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Answer» required area=area of triangle OAB -(1/4) area of circle =`|1/2*2*2-pi*4/4|` =`|2-pi|` =`pi-2 unit^2` |
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| 39. |
The area bounded by `y^(2) = 4x ` and `x^(2) = 4y` is `A_(1)` and the area bounded by `x^(2) = 4y , x = 4` and x-axis is `A_(2)` . If `A_(1) : A_(2) = K : 1` then K is _______ |
| Answer» Correct Answer - 1 | |
| 40. |
The smaller area bounded by `x^2/16+y^2/9=1` and the line `3x+4y=12` isA. `3pi` sq. unitsB. `(3pi - 6)` sq. unitsC. `(3pi - 2)` sq. unitsD. `(3pi - 4)` sq. units |
| Answer» Correct Answer - B | |
| 41. |
The area bounded between the parabolas `x^2=y/4"and"x^2=9y`and thestraight line `y""=""2`is(1) `20sqrt(2)`(2) `(10sqrt(2))/3`(3) `(20sqrt(2))/3`(4) `10sqrt(2)` |
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Answer» area of region 1 = area of region 2 area = `int x dy = int y dx` `| int_0^2 (3 sqrt y - sqrty/2) dy |` `= 2[ (3y^(3/2))/(3/2) - (y^(3/2))/(3/2 xx2) ]` `= [(3 xx 2^(3/2) xx 2)/3 - 2^(3/2)/3] = 2 xx 2^(3/2) [2 -1/3]` `= 2 xx 5/3 xx 2^(3/2) ` `= 10/3xx 2^(3/2) = 20/3sqrt2` option2 is correct |
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| 42. |
The maximum area of a rectangle whose two vertices lie on the x-axis and two on the curve `y=3-|x| , -3 |
| Answer» Correct Answer - B::D | |
| 43. |
For which of the following values of `m`is the area of the regions bounded by the curve `y=x-x^2`and the line `y=m x`equal `9/2?``-4`(b) `-2`(c) 2(d) 4A. 3B. 1C. 2D. 4 |
| Answer» Correct Answer - D | |
| 44. |
The area of the region enclosed by the curves `y""=""x ,""x""=""e ,""y""=1/x`and thepositive x-axis is(1) `1/2`squareunits(2) 1 square units(3) `3/2`squareunits(4) `5/2`square units |
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Answer» area of 1 = `1/2 xx 1 xx 1 = 1/2` area of 2 = `int x dx = int y dx` `int_1^e 1/x dx` `= ln x` `= ln e = 1` total area =`1/2 + 1 = 3/2 unit^2` option 3 is correct |
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| 45. |
Consider a square with vertices at `(1,1),(-1,1),(-1,-1),a n d(1,-1)dot`Set `S`be the region consisting of all points inside the square which are nearerto the origin than to any edge. Sketch the region `S`and find its area. |
| Answer» `(4)/(3) (4sqrt2 - 5)` sq. units | |
| 46. |
The sine and cosine curves intersect infinitely many times , bounding regions of equal areas . Sketch one of these regions and find its area . |
| Answer» Correct Answer - `2sqrt2` sq. units | |
| 47. |
The area of the region included between the curves `x^2+y^2=a^2 and sqrt|x|+sqrt|y|=sqrta(a > 0)` is |
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Answer» Given curves are ` sqrt(|x|) + sqrt(|y|) = sqrta " " … (i)` and `x^(2) + y^(2) = a^(2) " "…. (ii)` Curve (ii) is a circle . Now we have to draw the graph of curve (i) (1) Curve (i) cuts x-axis at (-a,0) and (a,0) and y-axis at (0,-a) and (0,a) (2) Curve (i) is symmetrical about x-axis as well as y-axis (3) In first quadrant from (i) we have `sqrtx + sqrty = sqrta implies (dy)/(dx) = - (sqrty)/(sqrtx) lt 0` `implies ` y is decreasing (4) `(dy)/(dx) = - ((sqrta - sqrtx)/(sqrtx)) = 1 - (sqrta)/(sqrtx)` `therefore (d^(2)y)/(dx^(2)) = (1)/(2) sqrta . x^(-(3)/(2))= (sqrta)/(2.x^((3)/(2))) gt 0` `therefore` Curve is convex downward . Now , required area = 4[shaded area in the first quadrant] =` 4[(pi a^(2))/(4) - underset(0)overset(a)(int) (sqrta - sqrtx)^(2) dx ] = 4[(pi a^(2))/(4) - underset(0)overset(a)(int)(a + x - 2sqrta sqrtx) dx`] `= 4 {(pia^(2))/(4) - [ax + (x^(2))/(2) - (4)/(3) sqrta x^((3)/(2)) ]_(0)^(a)}` `= (pi - (2)/(3)) a^(2)` sq. units |
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| 48. |
Find area of region bounded by `y = x^2 – 3x + 2, x = 1, x = 2` and `y = 0`. |
| Answer» `(1)/(6)` sq. units | |
| 49. |
Find the area bounded by the curves y = x and y = x^3 |
| Answer» `(1)/(2)` sq. units | |
| 50. |
Find the area of the region bounded by the ellipse `(x^2)/(16)+(y^2)/9=1`. |
| Answer» Correct Answer - `12 pi` sq. units | |