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Application of Differential Calculus Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t2. The stone reaches the maximum height in time t seconds is given by

    (a)

    2

    (b)

    2.5

    (c)

    3

    (d)

    3.5

  2. The tangent to the curve y2 - xy + 9 = 0 is vertical when 

    (a)

    y = 0

    (b)

    \(\\ \\ y=\pm \sqrt { 3 } \)

    (c)

    \(y=\cfrac { 1 }{ 2 } \)

    (d)

    \(y=\pm 3\)

  3. The number given by the Mean value theorem for the function \(\cfrac { 1 }{ x } \),x∈[1,9] is

    (a)

    2

    (b)

    2.5

    (c)

    3

    (d)

    3.5

  4. The point on the curve y=x2 is the tangent parallel to X-axis is

    (a)

    (1,1)

    (b)

    (2,2)

    (c)

    (4,4)

    (d)

    (0,0)

  5. The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is

    (a)

    0

    (b)

    (c)

    e

    (d)

    \(\frac{1}{e}\)

  6. 5 x 2 = 10
  7. A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres
    Find the average velocity of the points between t = 3 and t = 6 seconds.

  8. A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres
    Find the instantaneous velocities at t = 3 and t = 6 seconds.

  9. Find the values in the interval \((\frac{1}{2},2)\) satisfied by the Rolle's theorem for the function \(f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]\)

  10. A particle moves in a line so that x=\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

  11. Find the intervals of increasing and decreasing function for f(x) =x3 + 2x2 - 1.

  12. 5 x 3 = 15
  13. A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

  14. Find the point on the curve y = x2 − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

  15. Prove that \(\frac { x }{ 1+x } \) < < log (1+ x) for x > 0.

  16. Find the equation of normal to the cure y = sin2x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).

  17. 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?

  18. 4 x 5 = 20
  19. A particle moves along a line according to the law s(t) = 2t3 − 9t2 +12t − 4, where t ≥ 0.
    Find the total distance travelled by the particle in the first 4 seconds.

  20. A particle moves along a line according to the law s(t) = 2t3 − 9t2 +12t − 4, where t ≥ 0.
    Find the particle’s acceleration each time the velocity is zero.

  21. If f(x) = a log x + bx2 +x has entreme values at x = - 1 and x = 2, then find a and b.

  22. 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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