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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 abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

    (a)

    -8

    (b)

    -4

    (c)

    -2

    (d)

    0

  3. The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \)

    (a)

    \(-4\sqrt { 3 } \)

    (b)

    -4

    (c)

    \(\cfrac { \sqrt { 3 } }{ 12 } \)

    (d)

    \(4\sqrt { 3 } \)

  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 equation of the tangent to the curve y=x2-4x+2 at (4,2) is

    (a)

    x + 4y + 12 = 0

    (b)

    4x + y + 12 = 0

    (c)

    4x - y - 14 = 0

    (d)

    x + 4y - 12 = 0

  6. 5 x 2 = 10
  7. A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t2 in t seconds
    What is the instantaneous velocity of the camera when it hits the ground?

  8. A particle moves along a line according to the law s(t) = 2t3 − 9t2 +12t − 4, where t ≥ 0.
    At what times the particle changes direction?

  9. Find the local extrema for the following function using second derivative test:
    f(x) = x2 e-2x

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

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

  12. 5 x 3 = 15
  13. A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) =128t −16t2.
    (1) Compute the maximum height of the particle reached.
    (2) What is the velocity when the particle hits the ground?

  14. If we blow air into a balloon of spherical shape at a rate of 10003 cm per second. At what rate the radius of the baloon changes when the radius is 7cm? Also compute the rate at which the surface area changes.

  15. Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\) there is a zero of the polynomial \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

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

  17. Verify LMV theorem for f (x) = x3 - 2x2 - x + 3 in [0, 1].

  18. 2 x 5 = 10
  19. Find the equation of the tangent and normal to the Lissajous curve given by x = 2cos3t and y = 3sin 2t, t ∈ R

  20. If the curves ax2+by2=1 and cx2+dy2=1 intersect each other orthogonally then,  \(\frac{1}{a}-\frac{1}{b}=\frac{1}{c}-\frac{1}{d}\)

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