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Application of Differential Calculus Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 40
    5 x 1 = 5
  1. Find the point on the curve 6y = .x3 + 2 at which y-coordinate changes 8 times as fast as x-coordinate is

    (a)

    (4,11)

    (b)

    (4,-11)

    (c)

    (-4,11)

    (d)

    (-4,-11)

  2. The function sin4 x + cos4X is increasing in the interval

    (a)

    \(\left[ \cfrac { 5\pi }{ 8 } ,\cfrac { 3\pi }{ 4 } \right] \)

    (b)

    \(\left[ \cfrac { \pi }{ 2 } ,\cfrac { 5\pi }{ 8 } \right] \)

    (c)

    \(\left[ \cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 2 } \right] \)

    (d)

    \(\left[ 0,\cfrac { \pi }{ 4 } \right] \)

  3. One of the closest points on the curve x2 - y2.= 4 to the point (6, 0) is

    (a)

    (2,0)

    (b)

    \(\left( \sqrt { 5 } ,1 \right) \)

    (c)

    \(\left( 3,\sqrt { 5 } \right) \)

    (d)

    \(\left( \sqrt { 13 } ,-\sqrt { 3 } \right) \)

  4. The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are

    (a)

    -1, 2

    (b)

    1, \(\frac { 1 }{ 2 } \)

    (c)

    1, 2

    (d)

    none

  5. The statement " If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

    (a)

    the extreme value theorem

    (b)

    Fermats' theorem

    (c)

    Law of mean

    (d)

    Rolle's theorem

  6. 1 x 2 = 2
  7. Which of the following statement is incorrect?
    (1) Initial velocity means velocity at t = 0.
    (2) Initial acceleration means, acceleration at t=0.
    (3) If the motion is upward, at the maximum height the velocity is not zero.
    (4) If the motion is horizontal, u = 0 when the particle comes to rest.

  8. 3 x 2 = 6
  9. The temperature in celsius in a long rod of length 10 m, insulated at both ends, is a function of
    length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the
    rod is zero.

  10. A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what
    time the velocity and acceleration are zero respectively?

  11. Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).

  12. 4 x 3 = 12
  13. A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t3 − t2 + t + 6 9 1, where s is measured in metres and t in seconds?
    (1) At what time the particle is at rest?
    (2) At what time the particle changes direction?
    (3) Find the total distance travelled by the particle in the first 2 seconds.

  14. Find the tangent and normal to the following curves at the given points on the curve
    y = x sin x at \(\left( \frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right) \)

  15. Find the absolute maximum and absolute minimum values of the function f (x) = 2x3 + 3x2 −12x on [−3, 2]

  16. The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?

  17. 3 x 5 = 15
  18. A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the easst of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

  19. Find the acute angle between the curves y = x2 and x = y2 at their points of intersection (0,0), (1,1).

  20. Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan-1(\(\sqrt { 2 } \)).

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