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12th Standard Maths English Medium Applications Of Differential Calculus Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  1. The volume of a sphere is increasing in volume at the rate of 3 πcm3 sec. The rate of change of its radius when radius is \(\cfrac { 1 }{ 2 } \) cm

    (a)

    3 cm/s

    (b)

    2 cm/s

    (c)

    1 cm/s

    (d)

    \(\cfrac { 1 }{ 2 } cm/s\)

  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. Angle between y2 = x and.x2= y at the origin is

    (a)

    \({ tan }^{ -1 }\cfrac { 3 }{ 4 } \)

    (b)

    \({ tan }^{ -1 }\left( \cfrac { 4 }{ 3 } \right) \)

    (c)

    \(\cfrac { \pi }{ 2 } \)

    (d)

    \(\cfrac { \pi }{ 4 } \)

  5. The value of the limit \(\\ \\ \\ \underset { x\rightarrow 0 }{ lim } \left( cotx-\cfrac { 1 }{ x } \right) \) 

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

  6. 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)

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

  7. 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

  8. Equation of the normal to the curve y=2x2+3 sin x at x=0 is

    (a)

    x + y = 0

    (b)

    3y = 0

    (c)

    x + 3y = 7

    (d)

    x + 3y = 0

  9. The least value of a when f f(x) =x2+ax+1 is increasing on (1, 2) is

    (a)

    -2

    (b)

    2

    (c)

    1

    (d)

    -1

  10. The angle made by any tangent to the curve y = x5 + 8x + 1 with the X-axis is a

    (a)

    obtuse

    (b)

    right angle

    (c)

    acute angle

    (d)

    no angle

  11. 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

  12. The equation of the tangent to the curve x = t cost, y = t sin t at the origin is

    (a)

    x = 0

    (b)

    y = 0

    (c)

    x +y = 0

    (d)

    x + y = 7

  13. The function -3x+12 is ________ function on R.

    (a)

    decreasing

    (b)

    strictly decreasing

    (c)

    increasing

    (d)

    strictly increasing

  14. \(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

  15. 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

    1. 2 Marks


    10 x 2 = 20
  16. Find the angle of intersection of the curve y = sin x with the positive x -axis.

  17. Find the slope of the tangent to the curves at the respective given points.
    y = x4 + 2x2 − x at x =1

  18. A truck travels on a toll road with a speed limit of 80 km/hr. The truck completes a 164 km journey in 2 hours. At the end of the toll road the trucker is issued with a speed violation ticket. Justify this using the Mean Value Theorem.

  19. The volume of a cylinder is given by the formula V = πr2 h. Find the greatest and least values of V if r + h = 6.

  20. Evaluate the limit \(\underset{x\rightarrow 0}{lim}(\frac{sin mx}{x})\)

  21. 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.

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

  23. Find the equation of the tangent to the curve y2=4x+5 and which is parallel to y=2x+7

  24. Evaluate the following limits, if necessary using L’Hopitalrule
    (i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
    (ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
    (iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
    (iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

  25. Find the absolute extreme of the function f(x)=x2-2x+2 on the closed interval [0,3]

    1. 3 Marks


    10 x 3 = 30
  26. For the function f(x) = x2 ∈ [0, 2] compute the average rate of changes in the subintervals [0,0.5], [0.5,1], [1,1.5], [1.5,2] and the instantaneous rate of changes at the points x = 0.5,1,1.5, 2

  27. The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.

  28. 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 \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

  29. Suppose that for a function f(x), f'(x) ≤ 1for all 1 ≤ x ≤ 4. Show that f(4) - f(1) ≤ 3.

  30. Evaluate the limit \(\underset{x\rightarrow 0^{+}}{lim} (\frac{sin \ x}{x^{2}})\)

  31. Evaluate: \(\underset{x\rightarrow 0^{+}}{lim}(\frac{1}{x}-\frac{1}{e^{x}-1})\).

  32. If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is \(A={ A }_{ 0 }{ \left( 1+\frac { r }{ n } \right) }^{ nt }\). If the interest is compounded continuously, (that is as n ➝∞), show that the amount after t years is A=Aoert.

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

  34. 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?

  35. Verify Rolle’s theorem for f(x)=exsinx,\(0\le x\le \pi \)

    1. 5 Marks


    7 x 5 = 35
  36. 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.

  37. 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?

  38. 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?

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

  40. Find the equations of the tangents to the curve y =1+ x3 for which the tangent is orthogonal with the line x +12y =12

  41. Find the intervals of mono tonicities and hence find the local extremum for the following function:
    f(x) = sin x cos x + 5, x ∈ (0,2π)

  42. Prove that among all the rectangles of the given area square has the least perimeter.

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