2) if in the rectilinear motion of a particle s= at + 1/2 ft², when u and f are constants, then prove that velocity at time t is u + ft and acceleration is f.
3) Find the co-ordinates of the position of a particle moving along the paraboz y²= 4x at which the rate of increase of the abscissa is twice the rate of increase of the ordinate. (4,4)
4) Show that x/(1+ x) < log(1+ x) < x, when x> 0.
5) If f(x)= (x -1) eˣ +1, show that f(x) is positive for all positive values of x.
5) The time rate of change of radius of a sphere is 1/2π. When its radius is 5cm. Find the rate of change of the area of the surface of the sphere with time. 20cm²
6) Given y=(a sinx + b cosx)/(c sinx + d cosx), prove that
A) if a=1, b= 2, c= 3, d=4 then y decreases for all the value of x.
B) if a=2, b=1, c=3, d=4, then y increases for all values of x.
7) Find the range of value of x for which
A) the function x³ -9x²+ 24x - 16 increases as x increases. x<2 or x >4
B) the function 2x³ -9x²+ 12x - 3 decreases as x increases. 1< x < 2
8) A man 1.5 m tall walks away from a lampost 4.5 m at the rate of 4 km/hr.
A) how fast is shadow lenthening ?
B) how fast is the farther end of his shadow moving on the pavement ?
9) Water is poured into vertex into an inverted conical vessel of which the radius of the base is 2m and height 4 m at the rate of 77 litre/min . At what rate is the water-level rising at the instant when the depth is 70 cm ? 20cm/min
10) A spherical ice-ball is melting, the radius is decreasing at a constant rate of 0.1 cm per second. Find the amount of water formed in one second when the radius of the sphere is 7 cm. (Given π=22/7) sp. gr. of ice = 0.9). 55.44gm
11) the radius of a balloon is 7 cm. If an error of 0.1 cm is made as measuring the radius. Find the error in measuring the volume of the balloon. 6.16c.c
12) A spherical balloon is being inflated so that its volume increases uniformly at the rate of 40 cm³)min. How fast is its surface area increasing when the radius is 8 cm ? Find approximately, how much the radius will increase during the next 1/2 minute. 0.025. 0.025
13) Water is pumped out at a constant rate of 88 cubic metre per minute from a conical container held with its axis vertical. If the semivertical of this cone is 45°, find the rate of depression of the water level when the depth of the water level is 2 m. 7 m/min
13) find the set of all values 'a' for which the function f(x)= [{√(a+4)/(1- a)} -1] x⁵- 3x + log 5 decreases for all real x.
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