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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($\sqrt { 2 }$).