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#### Application of Differential Calculus Model Question Paper 1

12th Standard EM

Reg.No. :
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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($\sqrt { 2 }$).