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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. |
Evaluate `(1001)^(1/3)` upto six places of decimal. |
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Answer» `(100)^(1//3) = (1000 +1)^(1//3) = [1000(1+(1)/(1000))]^(1//3) = 10 (1+001)^(1//3)` `= 10 [ 1+(1)/(3)(0.001)+((1)/(3)((1)/(3) - 1))/(2!) (.001)^2 +….] = 10 [ 1 + 0.0003333 - (1)/(9)(0.000001)+…..` `=10[1+ 0.00003333- 0.0000001]` ` = 10.003332` |
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| 2. |
A particle is at rest, It starts rotating about a fixed point. Its angle of rotation `(theta)` with time (t) is given by the relation : `theta = (6t^3)/(15) - (t^2)/(2)` where `theta` is in radian and t is seconds. Find the angular velocity and angular acceleration of a particle at the end of 6 second. |
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Answer» Here, `theta = (6t^3)/(15) - (t^2)/(2)` Angular velocity, `Omega = (d theta)/(dt) = (d)/(dt) [(6t^3)/(15) - (t^2)/(2)] = (6)/(5)t^2 - t` Angular acceleration, `alpha = (domega)/(dt) = (d)/(dt) ((6)/(5)t^2- t) = (12)/(5) t -1` When t =6s, `omega = (6)/(5)xx6^2 -6 =43.2-6 = 37.2 rad//s` `alpha = (12)/(5)x6-1 =13.4 rad//s^2` |
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| 3. |
A particle starts from rest and its angular displacement (in red) is given by `theta = (t^2)/(20) +(t)/(5),` calculate the angular velocity at the end of t =4 second. |
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Answer» Correct Answer - ` 0.6 rad//s` Angular velocity `omega = (d theta)/(dt) = (d)/(dt)((t^2)/(20) + (t)/(5)) = (2t)/(20)+(1)/(5)` when t =4 s, `omega = (2xx4)/(20) +(1)/(5) = 0.6 rad//s` |
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| 4. |
An inclined plane rises 1 in 10. if the length of the inclined plane is 5 cm.. Calculate the hight of the reised end above the horizontal. |
| Answer» Correct Answer - 0.5 m | |
| 5. |
Find the value of the following : (a) `sin 210^(@)` (b) `cos 220^(@)` (c ) `tan 245^(@)` |
| Answer» Correct Answer - (a) `-1//2` (b) `-0.7660` (c ) 2.1445` | |
| 6. |
If `sin theta = 3//5,` find the values of cosine `theta` and tangent `theta.` |
| Answer» Correct Answer - `4//5 ; 3//4` | |
| 7. |
If `x = a cos theta` and `y = b sin theta`, find `(dy)/(dx)` |
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Answer» Correct Answer - ` - (b)/(alpha) cot theta` `(dx)/(d theta) =-a sin theta , (dy)/(d theta) = b cos theta :. (dy)/(dx) = (dy)/(d theta)/(dx)/(d theta) = (b cos theta)/(-a sin theta) =-(b)/(a) cot theta` |
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| 8. |
`Lt sin theta = ….. ("in radian") theta rarr 0` |
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Answer» Correct Answer - `theta` |
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| 9. |
Find the (i) surface area (ii) volume of the cylinder of length 10 cm and radius 2cm. |
| Answer» Correct Answer - (i)`48 pi cm^2` (ii) `40 pi cm^3` | |
| 10. |
Simplify the following : (i) `(0.05246)^(1//8) - 2.6055` (ii) `(3.142xx(80.2)^(1//2))/((9.8)^(1//2)` |
| Answer» Correct Answer - (i) -1.9137 (ii) 8.989 | |
| 11. |
Simplify the following. Using Binomial theorem`E=K[(1+(Deltatheta)/theta_(0))-1]` |
| Answer» Correct Answer - `(4K Delta theta)/(theta_0)` | |
| 12. |
Simplify the following : (i) `(0.0036)^(1//5)` (ii) `(0.056)^(2//3)` (iii) `10^(-1//5)` |
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Answer» Correct Answer - (i) 0.3245 (ii) 0.1463 (iii) 0.6310 (iii) `x= 10^(-1//5)` `log x =-(1)/(5) log 10 = - (1)/(5)xx1 = - 0.2 = bar1 + 1 - 0.2 = bar1.8` Taking antilog on both the sides, we get x = 0.6310 |
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| 13. |
The value of `int_0^(10) (x +(1)/(x))` dx isA. 51B. 52.3C. 101D. 103.3 |
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Answer» Correct Answer - B |
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| 14. |
Differentiate the following with respect to x. (i) `(4 x +2)(5x^2 +4)` (ii) `(2x^3 _3) (2x^(-3) +1)` |
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Answer» (i) `y = (4x +2)(5x^2+4)` `(dy)/(dx) = (4x +2) (5x^2+4)` `(dy)/(dt) = (4x+2) (d)/(dx) (5x^2+4)+(5x^2+4) (d)/(dx) (4x+2)` `=(4x+2)xx(5xx2x^(2-1) + 0) + (5x^2 +4) (4xx1x^(1-1) +0)` `=(4x+2)10x+(5x^2+4) 4 = 40x^2 +20x +20 x^2 +16` `= 60x^2 + 20x+16` (ii) `y = (2x^3+3)(2x^(-3) +1)` `(dy)/(dx) = (2x^3 +3) (d)/(dx)(2x^(-3) +1) (2x^(-3) +1) (d)/(dx) (2x^3+3)` `=(2x^3+3)[2xx(-3)x^(-3-1) +0] + (2x^(-3) +1) (2xx3x^(3-1) +0)` `=(2x^3 + 3)((-6)/(x^4)) +((2)/(x^3) +1) 6x^2 = (-12)/(x) - (18)/(x^4) +(12)/(x) + 6x^2` `=6x^2 - (18)/(x^4)` |
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| 15. |
Differentiate the following w.r.t., x (i) `tan^3 x` (ii) `(cos x)/((1 - sin x))` (iii) `sin (ax + b)^2` |
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Answer» Correct Answer - (i) `3 tan ^2 x sec^2 x` (ii) `(1)/((1 - sin x))` (iii) `2a (ax + b) cos (ax +b)^2` (i)`y= tan^3x = (tan x)^3, (dy)/(dx) = (d)/(dx)(tanx)^3 = 3(tan x)^2(d)/(dx) (tan x) = 3 tan^2 x sec^2x` (ii) `y= (cos x)/(1 -sin x) : (dy)/(dx) = ((1-sin s)(d)/(dx) cos x - cos x(d)/(dx) (1 - sinx))/((1 - sinx)^2) = (-sin x +1)/((1 - sinx)^2) = (1)/((1-sinx))` (iii) `y = sin (ax +b)^2, (dy)/(dx) = (d)/(dx) [sin (ax +b)^2] = cos (ax +b)^2 (d)/(dx) (ax +b)^2` = `cos (ax +b)^2 xx2(ax +b) (d)/(dx) (ax +b) = cos (ax +b)^2 xx2(ax +b)xxa` `=2a(ax +b )cos (ax +b)^2` |
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| 16. |
Differentiate the following with respect to x. (i) `4x^3 - 3x^2 +(4)/(x^2) -8` (ii) `5x^4 +4x^(3//4) - 3x^2+2x` |
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Answer» (i) `y = 4x^3 - 3x^2 +(4)/(x^2) - 8` `(dy)/(dt) = (d)/(dx) (4x^3 - 3x^2+(4)/(x^2) - 8) = (d)/(dx) (4x^3) - (d)/(dx) (3x^2) +(d)/(dx) ((4)/(x^2)) - (d)/(dx)(8)` ` = 4xx3x^(3-1) - 3xx2x^(2-1) +4(-2)x^(-2-1) - 0 =12 x^2 - 6x -8x^(-3) = 12x^2 -6x (8)/(x^3)` (ii) ` y = 5x^4 +4x^(3//4) -3x^2+2x` `(dy)/(dx) = (d)/(dx) (5x^4 + 4x^(3//4) - 3x^2 +2x) = (d)/(dx) (5x^4) +(d)/(dx)(4x^(3//4)) - (d)/(dx)(3x^2) + (d)/(dx)(2x)` ` = 5xx4 x^(4-1)+4xx(3)/(4)x ^((3)/(4)x-1) - 3xx2x^2 +2xx1 = 20x^3 +3x^(-1//4) - 6x +2` |
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| 17. |
Find the value of the following : (a) `sin 120^(@)` (b) `cos 135^(@)` (c ) `tan 150^(@)` |
| Answer» Correct Answer - (a) `sqrt 3//2` (b) ` -1//sqrt2` (c ) `-1 // sqrt3` | |
| 18. |
Find the value of the following : (a) `sin ( - 30^(@))` (b) `cos (-60^(@))` (c ) `tan (- 45^(@))` |
| Answer» Correct Answer - (a) `-1//2,` (b) `1//2` (c ) `-1` | |
| 19. |
`x^2 +x - 2 = 0.` |
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Answer» Here, a = 1, b = 1, c = -2 `:. X = (-b+-sqrt(b^2 - 4ac))/(2a) = (-1+-sqrt(1)^(2)-4xx1xx(-2))/(2xx1) = (-1+- 3)/(2) = 1 or -2 :. X =1or -2` |
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| 20. |
Identify , whether the following equation represent a straight line , parabola or circle:`x^(2) + y^(2) =16` |
| Answer» Correct Answer - Circle | |
| 21. |
Identify , whether the following equation represent a straight line , parabola or circle: `y = 7` |
| Answer» Correct Answer - St. line | |
| 22. |
Solve the following : `3 x^2 - 8 x +5 = 0` |
| Answer» Correct Answer - `1 or 5//3` | |
| 23. |
Solve the followings : ` 9 x^2 +15x +4 = 0` |
| Answer» Correct Answer - `- 1//3, - 4//3` | |
| 24. |
Identify , whether the following equation represent a straight line , parabola or circle: `Y^2 -6x +3y +5 = 0` |
| Answer» Correct Answer - Parabola | |
| 25. |
`6x^2-13+6 = 0` |
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Answer» Here `a = 6,b = -13` and `c=6` `: . x = (-b +-sqrt b^2 -4ac)/(2a) = (+12+-sqrt(-13)^(2) - 4xx6xx6)/(2xx6)` or `x= (13+-sqrt169 -144)/(12) = (13+-5)/(12) = (18)/(12), (8)/(12)`, i.e., `x= (3)/(2), (3)/(2)` These are the roots of the given equation. |
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| 26. |
Solve the followings : `x^2 +4x - 5 = 0` |
| Answer» Correct Answer - 1 or -5 | |
| 27. |
Solve the following equation , `3x^2 +6x -9 =0` |
| Answer» Correct Answer - 1 or -3 | |
| 28. |
Solve the following : `5 x^2 + 4x -7 = 0` |
| Answer» Correct Answer - 0.894 or - 1.649 | |
| 29. |
Solve the equation for x : ` 4x^2 - 4ax + (a^2 -b^2) = 0.` |
| Answer» Correct Answer - `((a+b))/(2), ((a - b))/(2)` | |
| 30. |
Expand using binomial `(1+3 x)^2` upto the term having `x^3`. |
| Answer» Correct Answer - `1 + 15 x + 90x^2 +270x^3` | |
| 31. |
Evaluate `int x cos x dx.` |
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Answer» `int x cos x dx =x int cos x dx - int[((d)/(dx) (x) int cos x dx)] dx = x sin x = - int 1. sin x dx` `= x sin x + cos x +c` |
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| 32. |
`int cos x dx = …….` |
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Answer» Correct Answer - `sin x + C` |
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| 33. |
Evaluate w.r.t.x , (i) `"in"t x log x dx` (ii) `ing(x^2 - cos x +(1)/(x))dx` |
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Answer» Correct Answer - (i) `(x^2)/(2) (log x- (1)/(2)) + C` (ii) `(x^3)/(3) - sin x + log_e x +C` (i) `int x log xdx = int log x.xdx = log x intxdx -int[(d)/(dx)(logx) int xdx]dx` `=log x xx(x^2)/(2) - int(1)/(x)xx(x^2)/(2) dx = (x^2)/(2) log x =(x^2)/(4)+C` `=(x^2)/(2)(log x - (1)/(2))+C` (ii) `int(x^2 - cos x +(1)/(x)) dx = int x^2 dx -int cos x dx + int (1)/(x) dx= (x^3)/(3) - sin x + log_e x +c` |
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| 34. |
Evaluate : `overset2underset0int` |
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Answer» `int_0^2 (1)/((1 +3 x))dx = [log(1+3x)/(3)]_0^2 = (1)/(3)[ log(1 +3xx2) - log (1+3xx0)]` `= (1)/(3) [log 7-log 1] = (1)/(3) (log 7- 0) = (1)/(3) log7` |
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| 35. |
Evaluate : `int_(1)^(4) x^3 dx ` |
| Answer» `overset4underset1int x^3 dx = ((x^3 +1)/(3 +1 ))_1^4 = ((4^4)/(4) - (1^4)/(4)) = 64 -0.25 = 63.75` x^3 dx = ((x^3 +1)/(3 +1 ))_1^4 = ((4^4)/(4) - (1^4)/(4)) = 64 -0.25 = 63.75` | |
| 36. |
Evaluate (i) `int_0^(pi//4) sin x cos x dx` (ii) `int_0^(pi//2) (1 + cos x)^(1//2) dx` (iii) `int_0^(pi//2) (1 + sin x)^(1//2) dx` (iv) `int_0^(pi//4) (1 -cos 2x)^(1//2)dx` |
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Answer» Correct Answer - (i)`(1)/(4)` (ii) 2 (iii) 2 (iv) 0.414 (i) `int_0^(pi//4) sin x cos x dx = (1)/(2) int_0^(pi//4) sin 2x dx = (1)/(2)[(-cos 2x)/(2)]_0^(pi//4) = (1)/(4)` (ii) `overset(pi//2)underset0int(1+cosx^(1/2))dxoverset(pi//2)underset0intsqrt2cosx/2dx=sqrt2((sin(x)/(2))/((1)/(2)))=2sqrt2[sin(pi//2)/2-(sin0)/2]` `=2sqrt2[sinpi/4]=2sqrt2xx1/sqrt2=2` ` =2sqrt2[sin(pi)/(4)] = 2sqrt2xx(1)/(sqrt2) =2` (iii) `overset(pi//2)underset0int(1 + sinx )^(1//2) dx = overset(pi//2)underset0int(sin^2(x)/(2) +cos^2(x)/(2) +2sin (x)/(2) cos (x)/(2))^(1//2) dx = overset(pi//2)underset0int(sin (x)/(2) + cos (x)/(2)) dx = [-2 cos (x)/(2) + 2 sin(x)/(2)]_0^(pi//2) = 2[ sin(x)/(2) - cos (x)/(2)]_0^(pi//2)` `=2 [(sin pi//4 - cos pi//4) - (sin 0 -cos 0)] = 2[((1)/(sqrt2) - (1)/(sqrt2)) - (0-1)] =2` (iv) `overset(pi//4)underset0int (1 -cos 2x)^(1//2) dx overset(pi//4)underset0int (2sin^2 x)^(1//2) dx = sqrt2 overset(pi//4)underset0int sin x dx = sqrt2 (-cos x)_0^(pi//4)` ` = - sqrt2 [ cos pi//4 - cos 0]= - sqrt2[(1)/(sqrt2) -1]` `=-1 +sqrt2 =0.414` |
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| 37. |
Evaluate `int_(R)^(oo)(GMm)/x^(2)dx` |
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Answer» Correct Answer - `(GMm)/(R )` `overset(oo)undersetRint(GMm)/x^(2)dx (GM m )/(x^2)dx =GM m overset(00)undersetRint(GMm)/x^(2)dx x^(-2) dx = GM m [x^(-2+1)/( - 2+1)]_R^(oo) = (GM m)/(R )` |
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| 38. |
An inclined plane rises 1 in 10. if the length of the inclined plane is 5 cm.. Calculate the hight of the reised end above the horizontal.A. 0.1 mB. 0.25 mC. 0.5 mD. 1.0 m |
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Answer» Correct Answer - C |
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| 39. |
Differentiate the following w.r.t.x. (i) 2002 (ii) `e^((-1))` (iii) `pi^(2)` |
| Answer» Since these are constant quantities, so (i) `(d)/(dx) (2002) = 0` (ii) `(d)/(dx)(e^(-1) = 0` (iii) `(d)/(dx) (pi^2) = 0` | |
| 40. |
Differentiate the following w.r.t.x (i) `pi^2` (ii) `e^2` (iii) -14 |
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Answer» Correct Answer - (i) 0 (ii) 0 (iii) 0 All are constant quantities, hence, their differentiation is zero. |
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| 41. |
Differentiate the following with respect to x. (i) `x^6` (ii) `(1)/(x^3)` (iii) `sqrtx` |
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Answer» (i) `y = x^6 , (dy)/(dx) = 6x^(6-1) = 6x^5` (ii) `y = (1)/(x^3) = x^(-3) , (dy)/(dx) = (-3)x^(-3-1) = -3x^(-4) = (-3)/(x^4)` (iii) `y = sqrtx = x^(1/2) , (dy)/(dx) = (1)/(2) x ^(((1)/(2)-1)) = (1)/(2)x ^(-(1)/(2)) = (1)/(2sqrtx)` |
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| 42. |
Differentiate the following w.r.t.x (1) `(3x +5)` (ii)` x^(-2)` (iii) `x^(3//2)` (iv) `sqrtx - (1)/(sqrtx)` (v) `(1)/((x +2))` |
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Answer» Correct Answer - `[(i) 3` (ii) `(-2)/(x^3)` (iii) `(3)/(2) s^(1//2)` (iv) `(1)/(2sqrtx) ((x +1)/(x))` (v) `-(1)/((x +2)^2)]` (iv) `(d)/(dx) (sqrtx) - (d)/(dx)((1)/(sqrtx)) = (d)/(dx)(x^(1//2)) - (d)/(dx)(x^(-1//2)) = (1)/(2)x^(-1//2) = (1)/(2)x^(-3//2) = (1)/(2sqrtx)(1+(1)/(x)) = (1)/(2sqrtx) ((x +1)/(x))` (v) `(d)/(dx)((1)/(x+2)) =(d)/(dx)(x + 2)^(-1) = (-1)(x+2)^(-2)(d)/(dx)(x+2) = -(1)/((x+2)^2) [1+0] = -(1)/((x+2)^2)` |
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| 43. |
When x =4, the derivative of `sqrtx` isA. `(1)/(2)`B. `(1)/(4)`C. 2D. 4 |
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Answer» Correct Answer - B |
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| 44. |
Evaluate : `intsec^4 x tan x dx.` |
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Answer» Put sec x =z so, `(d)/(dx) (sec x) = (dz)/(dx) or sec x tan x dx =dx` `:. Int sec^4 x tan x dx = intsec^3 x (sec x tan x ) dx` `int x^3 dz = (z^4)/(4) +C = (sec^4 x)/(4) +C = (1)/(4) sec^4 x +c` |
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| 45. |
Differentiate the following w.r.t.x (i)` (2)/((3x +1)` (ii) `(3x +5)/(4x +6)` |
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Answer» Correct Answer - (i) `(-6)/((3x +1)^2)` (ii) `(-2)/((4x +6)^2)` (i) `(dy)/(dx) = ((3x+1)(d)/(dx)(2) - 2(d)/(dx)(3x+1))/((3x+1)^2) =((3x+1)xx0-2(3+0))/((3x+1)^(2))=(-6)/((3x+1)^(2))` `(ii)(dy)/(dx)=((4x+6)d/(dx)(3x+5)-(3x+5)d/(dx)(4x+6))/((4x+6)^(2))=((4+6)(3+0)-(3x+5)(4+0))/((4x+6)^2)` `=(12x +18 - 12x - 20)/((4x+6)^2) = (-2)/((4x+6)^2)` |
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| 46. |
Differentiate the following w.r.t.x sin 3x (ii) `cos^2x` (iii) `sin x^3` |
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Answer» Correct Answer - (i) `3cos 3x` (ii) - sin 2 x (iii) `3x^2 cos x^3` (i) `y = sin 3 x,` Let `u = 3x , (du)/(dx) = 3` Then, `y = sin u, (dy)/(du) = cos u = cos 3x :. (dy)/(dx) = (dy)/(du)xx(du)/(dx) =(cos 3x)xx3 = 3 cos 3x` (ii) Let `u = cos x,(du)/(dx) = - sin x, y= cos^2 x = u^2 , (dy)/(du) =2u = 2 cos x` `(dy)/(dx) = (dy)/(du)xx(du)/(dx) =2cos x (-sin x)=-sin 2x` (iii) Let `u =x^3 , (du)/(dx)=3x^2, y= sinu, (dy)/(dx) = cos u =cos x^3` `(dy)/(dx) = (dy)/(du)xx(du)/(dx) = 3x^2 cos x^3` |
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| 47. |
Differentiate the following w.r.t.x and find the value when x =9. (i) sin 5x (ii) tan 5x (iii) cos 5x |
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Answer» Correct Answer - (i) `5//sqrt2` (ii) 10 (iii) `-5 // sqrt2` (i) Let `u =5x, (du)/(dx) = 5 so, y =sin u and (dy)/(du) = cos u = cos 5 x` `:. (dy)/(dx) =(dy)/(du)xx(du)/(dx) = cos 5x xx5 = 5 cos 5x when x =9, (dy)/(dx) = 5cos 5xx9 = 5 cos 45^@ =5xx(1)/(sqrt2)` (ii) `(dy)/(dx) = (d)/(dx)(tan 5x) = sec^25x xx(d)/(dx)(5x) = sec^25x xx5` when `x =9, (dy)/(dx) = 5 sec^2 5xx9 = 5xx(sqrt2)^2 =10` (iii) `(dy)/(dx) = (d)/(dx)(cos 5 x) = - 5 sin 5x = - 5 sin 5xx9 = -(5)/(sqrt2)` |
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| 48. |
The mass of a body is 2.5 Kg. it is in motion and its velocity `upsilon`after time t is ` upsilon = (t^3)/(3) +(t^2)/(2) +1` Calculate the force acting on the body at the time t =3 s. |
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Answer» Acceleration, `a = (dupsilon)/(dt) = (d)/(dt)((t^3)/(3) +(t^2)/(2) +1) = (d)/(dt)((t^3)/(3)) +(d)/(dt) ((t^2)/(2)) +(d)/(dt) (1)` `=(1)/(3)xx3t^(3-1) +(1)/(2)2t^(2-1) +0 =t^2 +t` When t =3 s, `a = 3^2 +3 = 12 m//s^2` As, force = mass xx acceleration = `2.5xx12 = 30N` |
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| 49. |
The air if filled in a balloon and the volume of balloon increases gradually. Find the rate of increase of volume of balloon with radius when of balloon becomes 30 cm. |
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Answer» Here radius of balloon r = 30 cm Volume of balloon, `V =(4)/(3)pi r^3` Rate of increase of volume of balloom w.r.t radius `(dV)/(dt) = (d)/(dr) ((4)/(3) pi r^3) = (4)/(3) pi xx3r^2 = 4pi r^2` When r =30 cm = 0.30m `(dV)/(dr) = 4xx(22)/(7)xx(0.30)^2 = 1.13 m^2` |
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| 50. |
Find the (i) surface area (ii) volume of the rectangular body of dimension 30 cm xx10 cm xx5 cm. |
| Answer» Correct Answer - (i) `1000 cm^2` (ii) `1500 cm^3` | |