InterviewSolution
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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. |
Let `A_n`be the area bounded by the curve `y=(tanx)^n`and the lines `x=0,y=0,`and `x=pi/4dot`Prove that for `n >2,A_n+A_(n-2)=1/(n-1)`and deduce `1/(2n+2)A. `A_(n)+A_(n-2)=(1)/(n-1)`B. `A_(n)+A_(n-2) lt (1)/(n-1)`C. `A_(n)-A_(n-2)=(1)/(n-1)`D. none of these |
| Answer» Correct Answer - A | |
| 2. |
The area bounded by `y=x^(2)+1` oand the tangents to it drawn from the origin, isA. `8//2` sq. unitsB. `1//3` sq. unitsC. `2//3` sq. unitsD. none of these |
| Answer» Correct Answer - C | |
| 3. |
The area enclosed by the curves`y= sinx+cosx and y = | cosx-sin x |` over the interval `[0,pi/2]`A. `4(sqrt(2)-1)`B. `2sqrt(2)(sqrt(2)-1)`C. `2(sqrt(2)+1)`D. `2sqrt(2)(sqrt(2)+1)` |
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Answer» Correct Answer - b Let A denote the area enclosed by the given curves. Then, `A=underset(0)overset(pi//2)int |(sinx + cos x )-|cos x - sin x ||dx` `because |cos x - sin x |={{:(cos x -sinx ","0 le x le (pi)/(4)),(-(cos x-sin x)","(pi)/(4) le x le (pi)/(2)):}` `therefore A=underset(0)overset(pi//4)int |(sin x + cosx )-(cosx-sinx)|dx+underset(pi//4)overset(pi//2)int |(sin x + cos x )+(cos x - sin x)|dx` `implies A =underset(0)overset(pi//4)int |2sin x|dx+ underset(pi//4)overset(pi//2)int |2cos x|dx` `implies A=2underset(0)overset(pi//4)int sin x dx + 2 underset(pi//4)overset(pi//2)int cos x dx` `implies A =[-2cos x]_(0)^(pi//4)+[2sinx ]_(pi//4)^(pi//2)` `implies A=(-2cos pi//4 + 2) + (2sin pi//2-2sin pi//4)` `implies A=(2-sqrt(2))+(2-sqrt(2))=2(2-sqrt(2))=2(2-sqrt(2))=2sqrt(2)(sqrt(2)-1)` |
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| 4. |
The area (in square units), bounded by `y=2-x^(2)` and `x+y=0` , isA. `(7)/(2)` sq. unitsB. `(9)/(2)` sq. unitsC. 9 sq. unitsD. none of these |
| Answer» Correct Answer - B | |
| 5. |
The area (in square units) bounded by the curve `y^(2)=8xand x^(2)=8y,` isA. `16/3`B. `3/16`C. `14/3`D. `3/14` |
| Answer» Correct Answer - A | |
| 6. |
The area of the figure bounded by `|y|=1-x^(2)` is in square units,A. `4//3`B. `8//3`C. `16//3`D. `5//3` |
| Answer» Correct Answer - B | |
| 7. |
The area (in square units) of the closed figure bounded by `x=-1,x=2and y={{:(-x^(2)+2","x le1),(2x-1","xgt1):}` and the abscissa axis, isA. `16//3`B. `13//3`C. `13//3`D. `7//3` |
| Answer» Correct Answer - A | |
| 8. |
Find the area bounded by the parabola `y=x^2+1`and the straight line `x+y=3.`A. `(45)/(7)`B. `(3)/(2)`C. `(32)/(3)`D. `(3)/(32)` |
| Answer» Correct Answer - D | |
| 9. |
Examples: Find the area bounded by the parabola `y^2 = 4ax` and its latus rectum. |
| Answer» Correct Answer - B | |
| 10. |
Area bounded by the parabola `y^2=x` and the line `2y=x` is (A) `4/3` (B) `1` (C) `2/3` (D) `1/3`A. `4//3`B. 1C. `2//3`D. `1//3` |
| Answer» Correct Answer - A | |
| 11. |
The area bounded by the curve `y=4x-x^2` and x-axis is (A) `30/7` sq. units (B) `31/7` sq. units (C) `32/3` sq. units (D) `34/3` sq. unitsA. `(30)/(7)`B. `(31)/(7)`C. `(32)/(3)`D. `(34)/(3)` |
| Answer» Correct Answer - C | |
| 12. |
Find the area bounded by the curve `x^2=4y`and the straight line `x=4y-2.`A. `3//8`B. `5//8`C. `7//8`D. `9//8` |
| Answer» Correct Answer - D | |
| 13. |
The area enclosed between the curve `y^2(2a-x)=x^3`and the line `x=2`above the `x-a xi s`is`pia^2s qdotu n i t s`(b) `(3pia^2)/2s qdotu n i t s``2pia^2s qdotu n i t s`(d) `3pia^2s qdotu n i t s`A. `pia^(2)`B. `3//2 pia^(2)`C. `2pia^(2)`D. `3pia^(2)` |
| Answer» Correct Answer - B | |
| 14. |
The area in square units of the region bounded by the curve `x^(2)=4y`, the line x=2 and the x-axis, isA. 1B. `2//3`C. `4//3`D. `8//3` |
| Answer» Correct Answer - B | |
| 15. |
The area of the region bounded by the curuse `y=|x-2|,x=1,x=3` and the x-axis isA. 4B. 2C. 3D. 1 |
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Answer» Correct Answer - d Let A be the required area. Then, `A=underset(1)overset(3)(int)|x-2|dx` `impliesA=underset(1)overset(2)(int)-(x-2)dx+underset(2)overset(3)(int)(x-2)dx` `impliesA=[2x-(x^(2))/(2)]_(1)^(2)+[(x^(2))/(2)-2x]_(2)^(3)=1/2+1/2=1` |
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| 16. |
The area of the region bounded by `y = |x - 1|` and `y = 1` isA. 1B. 2C. `1//2`D. `3//2` |
| Answer» Correct Answer - A | |
| 17. |
The area bounded by the curve `y=x|x|`, x-axis and the ordinates x=1,x=-1 is given by |
| Answer» Correct Answer - C | |
| 18. |
Area of the region bounded by the curves `y=2^(x),y=2x-x^(2),x=0" and "x=2` is given by :A. `(3)/(log 2)-(4)/(3)`B. `(3)/(log 2)+(4)/(3)`C. `2 "log" 2-(4)/(3)`D. `2 log^(2)-(4)/(3)` |
| Answer» Correct Answer - D | |
| 19. |
If `f(x) = max {sin x, cos x,1/2},` then the area of the region bounded by the curves `y =f(x),` x-axis, Y-axis and `x=(5pi)/3` isA. `sqrt2-sqrt3+(5pi)/(12)`B. `sqrt2+(sqrt3)/(2)+(5pi)/(12)`C. `sqrt2+sqrt3+(5pi)/(12)`D. none of these |
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Answer» Correct Answer - b `f(x)=max{sinx,cosx,1/2} ={{:(cos",",0lexle(pi)/(4)),(sinx",",(pi)/(4)lexle(5pi)/(6)),((1)/(2)",",(5pi)/(6)lexle(5pi)/(3)):}` Let A be the required area. Then, `A=underset(0)overset(5pi//3)(int)f(x)dx` `impliesA=underset(0)overset(pi//4)(int)cosx dx+underset(pi//4)overset(5pi//6)(int)sinx dx+underset(5pi//6)overset(5pi//3)(int)1/2dx` `impliesA=[sinx]_(0)^(pi//4)-[cosx]_(pi//4)^(5pi//6)+1/6((5pi)/(3)-(5pi)/(6))` `impliesA=(1)/(sqrt2)-(-(sqrt3)/(2)-(1)/(sqrt2))+(5pi)/(12)=(5pi)/(12)+(sqrt3)/(2)+sqrt2` |
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| 20. |
Area common to the circle `x^2+y^2=64` and the parabola `y^2=4x` isA. `(16)/(3)(4pi+sqrt(3))`B. `(16)/(3)(8pi-sqrt(3))`C. `(16)/(3)(4pi-sqrt(3))`D. none of these |
| Answer» Correct Answer - B | |
| 21. |
Find the area common to the circle `x^2+y^2=16a^2` and the parabola `y^2=6ax,agt0.`A. `(4a^(2))/(3)(4pi-sqrt(3))`B. `(4a^(2))/(3)(8pi-3)`C. `(4a^(2))/(3)(4pi+sqrt(3))`D. none of these |
| Answer» Correct Answer - C | |
| 22. |
The area bounded by `y = tan x, y = cot x,` X-axis in `0 lt=x lt= pi/2` isA. log 2B. `(1)/(2)log 2`C. `2 " log "((1)/(sqrt(2)))`D. `(3)/(2)log 2` |
| Answer» Correct Answer - A | |
| 23. |
The figure into which the curve `y^2 = 6x` divides the circle `x^2 + y^2 = 16` are in the ratioA. `2/3`B. `(4pi-sqrt(3))/(8pi+sqrt(3))`C. `(4pi+sqrt(3))/(8pi-sqrt(3))`D. none of these |
| Answer» Correct Answer - C | |
| 24. |
Area lying in the first quadrant and bounded by the circle `x^(2)+y^(2)=4` the line `x=sqrt(3)y` and x-axis , isA. `pi`B. `pi//2`C. `pi//3`D. `pi//4` |
| Answer» Correct Answer - C | |
| 25. |
The area of the region bounded by `x^(2)+y^(2)-2y-3=0 and y=|x|+1,` isA. `pi`B. `2pi`C. `4pi`D. `pi//2` |
| Answer» Correct Answer - A | |
| 26. |
Area common to the curves `y=sqrt(x)` and `x=sqrt(y)` is (A) `1` (B) `2/3` (C) `1/3` (D) none of theseA. 1B. `2//3`C. `1//3`D. `4//3` |
| Answer» Correct Answer - C | |
| 27. |
The area bounded by `y=x^2,y=[x+1],xle1` and the y axis isA. `1//3`B. `2//3`C. 1D. `7//3` |
| Answer» Correct Answer - B | |
| 28. |
The area of the region bounded by `y=|x-1|and y=3-|x|,` isA. 2B. 3C. 4D. 1 |
| Answer» Correct Answer - C | |
| 29. |
The area bounded by the curve `y^(2)=x` and the ordinate `x=36` is divided in the ratio `1:7` by the ordinate x=a. Then a=A. 8B. 9C. 7D. 0 |
| Answer» Correct Answer - B | |
| 30. |
Find the areaof the region bounded by the parabola `"x"^2=4"y "`and the line `"x"=4"y"-2`A. `9//8`B. `9//4`C. `9//2`D. `9//7` |
| Answer» Correct Answer - A | |
| 31. |
Find the area lying in the first quadrant and bounded by the curve `y=x^3`and the line `y=4xdot`A. 2B. 3C. 4D. 5 |
| Answer» Correct Answer - C | |
| 32. |
`A O B`is thepositive quadrant of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`in which `O A=a ,O B=b`. Then find the area between the arc `A B`and thechord `A B`of the ellipse.A. `(1)/(2)ab(pi+2)`B. `(1)/(4)ab(pi-4)`C. `(1)/(4)ab(pi-2)`D. none of these |
| Answer» Correct Answer - C | |
| 33. |
The area bounded by the curve `y^(2)(2a-x)=x^(3)` and the line x = 2a isA. `3 pia^(2)`B. `(3pia^(2))/(2)`C. `(3pia^(2))/(4)`D. `(pia^(2))/(4)` |
| Answer» Correct Answer - A | |
| 34. |
The area between `x^2/a^2+y^2/b^2=1` and the straight line `x/a+y/b=1` is (A) `1/2piab` (B) `1/2ab` (C) `(piab)/4-(ab)/2` (D) `1/4ab`A. `(1)/(2)ab`B. `(1)/(2) pi ab`C. `(1)/(4) ab`D. `(1)/(4)pi ab -(1)/(2)ab` |
| Answer» Correct Answer - D | |
| 35. |
If a curve `y=asqrtx+bx` passes through the point (1,2) and the area bounded by the curve, line x=4 and x-axis is 8 square units, thenA. `a=3,b=-1`B. `a=3,b=1`C. `a=-3,b=1`D. `a=-3,b=-1` |
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Answer» Correct Answer - a It is given that `y=asqrtx+bx` passes through the point (1,2). `therefore2=a+b" "...(i)` It is given that `underset(0)overset(4)(int)(asqrtx+bx)dx=8` `implies[(2)/(3)ax^(3//2)+b/2x^(2)]_(0)^(4)=8` `implies16/3a+8b=8implies2/3a+b=1" "...(ii)` Solving (i)and (ii), we get `a=3, b=-1.` |
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| 36. |
The area induced between the curves `y=(x^2)/(4a)` and `y=(8a^3)/(x^2+4a^2)` is given byA. `a^(2)(2pi -(4)/(3))`B. `a^(2)(pi -(4)/(3))`C. `a^(2)(2pi +(1)/(3))`D. `a^(2)(pi +(4)/(3))` |
| Answer» Correct Answer - A | |
| 37. |
Area bounded by the curve `y = sin x` between .`x = 0` and `x = 2pi` isA. `2pi`B. `2pi`C. `4pi`D. `pi` |
| Answer» Correct Answer - C | |
| 38. |
The area bounded by the x-axis and the curve `y = 4x - x^2 - 3` isA. `4//3`B. `3//4`C. 7D. `3//2` |
| Answer» Correct Answer - A | |
| 39. |
The area of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` isA. `pi ab`B. `(pi)/(4)(a^(2)+b^(2))`C. `pi(a+b)`D. `pia^(2)b^(2)` |
| Answer» Correct Answer - A | |
| 40. |
The line `y=m x`bisects the area enclosed by the curve `y=1+4x-x^2`and the lines `x=0,x=3/2a n dy=0.`Then the value of `m`is`(13)/6`b. `6/(13)`c. `3/2`d. `4`A. `13//8`B. `13//32`C. `13//16`D. `13//14` |
| Answer» Correct Answer - C | |
| 41. |
Smaller area enclosed by the circle `x^2+y^2=4` and line x+y=2 isA. `2(pi-2)`B. `pi-2`C. `2pi-1`D. `pi-1` |
| Answer» Correct Answer - B | |
| 42. |
If `A_(1)` is the area enclosed by the curve `xy=1,` x-axis and the ordinates `x=1,x=2,and A_(2)` is the area enclosed by the curve `xy=1,` x-axis and the ordinates `x=2, x=4,` thenA. `A_(1)=2A_(2)`B. `A_(2)=2A_(1)`C. `A_(2)=3A_(1)`D. `A_(1)=A_(2)` |
| Answer» Correct Answer - D | |
| 43. |
Find the area enclosed between the parabola `y^2=4a x`and the line `y = m x`.A. `(5a^(2))/(3)`B. `(8a^(2))/(3m^(3))`C. `(7a^(2))/(4m^(2))`D. `(3a^(2))/(5m)` |
| Answer» Correct Answer - B | |
| 44. |
The area between the curve `x=-2y^(2)and x=1-3y^(2),` isA. `4//3`B. `3//4`C. `3//2`D. `2//3` |
| Answer» Correct Answer - A | |
| 45. |
The value of a for which the area between the curves `y^(2) = 4ax` and `x^(2) = 4ay` is 1 unit isA. `sqrt3`B. 4C. `4sqrt3`D. `sqrt3//4` |
| Answer» Correct Answer - D | |
| 46. |
The area bounded by the curve `y = sin2x,` axis and `y=1,` isA. 1B. `1//4`C. `pi//4`D. `pi//4-1//2` |
| Answer» Correct Answer - D | |
| 47. |
The area of the figure bounded by the curve `|y|=1-x^(2),` isA. `2//3`B. `4//3`C. `8//3`D. `-5//3` |
| Answer» Correct Answer - C | |
| 48. |
The area bounded by `y = x |sinx|` and x - axis between `x = 0, x = 2pi` isA. `2pi`B. `3pi`C. `4 pi`D. `5 pi` |
| Answer» Correct Answer - C | |
| 49. |
Find the area of the figure bounded by the parabolas `x=-2y^2, x=1-3y^2dot`A. `8//3`B. `6//3`C. `4//3`D. `2//3` |
| Answer» Correct Answer - C | |
| 50. |
The equation of the common tangent to the parabolas `y^2= 4ax `and `x^2= 4by` is given byA. (8/3) abB. (16/3) abC. (4/3) abD. (5/3) ab |
| Answer» Correct Answer - B | |