Under certain conditions the constant quantity Q calories/sec of heat flowing through a wall is given by  where k is the conductivity of the material, A(cm³) is the area of the face of the wall perpendicular to the direction of flow and T is the temperature x cms from the face such that T decreases as x increases. Find the number of calories of heat per hour flowing through l square meters of the wall of a refrigerator room 125cm thick for which k = 0.0025 if the temperature of the inner face is –5°C and that of the outer face is 75 °.

A steam pipe 30cm long in diameter is protected with a covering 6cm thick for which k = 0.0003.
i) Find the heat loss per hour through a meter length of the pipe if the surface of the pipe is 200°C and that of the outer surface of the covering is 30°C.

ii) Find the temperature at a distance x > 10cm from the centre of the pipe.

A ship weighing 48,000 tons starts from rest under the force of a constant propeller thrust of 200,000 lb
i) Find the velocity as a function of time t, given that the resistance in pounds is 10,000V with velocity measured in ft/sec.

ii)Find the terminal velocity(i.e. v when t → ∞) in miles/hour (taking g = 32ft/sec²).

A boat is being towed at the rate of 12 miles per hour. At the instant (t = 0), that the towing line is cast off, a man in the boat begins to row in the direction of motion, exerting a force of 20 lb. If the combined weight of the man and boat is 480 lb and the resistance (lb) is equal to 1.75V where V is measured in ft/sec, find the speed of the boat after half a minute.

A mass is being pulled across the ice on a sled, the total weight including the sled being 80 lb. Under the assumption that the resistance offered by the ice to the runner is negligible and that the air offers a resistance in pounds equal to 5 times the velocity (V ft/sec) of the sled, find
i) The constant force (pounds) exerted on the sled which will give it a terminal velocity of 10 miles per hour
ii) The velocity and distance travelled at the end of 48 seconds.

A spring of negligible weight hangs vertically. A mass of m slugs is attached to the other end. If the mass is moving with velocity V₀ ft/sec when the spring is not stretched, find the velocity V as a function of the stretch x ft. Note: According to Hooke’s law, the spring force (force opposing the stretch) is proportional to the stretch.

A parachutist is falling at the speed 176 ft/sec when his parachute opens. The air resistance is wv²/256 lb where w is the total weight of the man and parachute. Find its speed as a function of time t after the parachute opens. Also find the terminal velocity.

A body of mass m slugs falls from rest in a medium for which the resistance (lb) is proportional to the square of the velocity (ft/sec). If the terminal velocity is 150 ft/sec find
i) the velocity at the end of two seconds
ii) the time required for the velocity to become 100 ft/sec.

A body of mass m falls from rest in a medium for which the resistance (lb) is proportional to the velocity (ft/sec). If the specific gravity of the medium is one fourth that of the body and if the terminal velocity is 24 ft/sec, find
i) the velocity at the end of three seconds
ii) the distance travelled in three seconds

The gravitational pull on a mass m at a distance s feet from the centre of the earth is proportional to m and inversely proportional to s²
i) Find the velocity attained by the mass in falling from rest a distance 5R from the centre to the earth’s surface, where R = 4000 miles is taken as the radius of the earth.
ii) What velocity would correspond to a fall from an infinite distance i.e. with what velocity must the mass be propelled vertically upwards to escape the gravitational pull? (All other forces, including friction are to be neglected)

One of the basic equations in electric circuits is
 

where L (henries) is called inductance, R(ohms) the resistance, I (amperes) the current and E (volts) the electromotive force or the emf.
i) Solve the equation where E(t) = E₀ and the initial current is I₀
ii) Solve the equation where L = 3 henries, R = 15 ohms, E (t) is the 60 cycles sine wave of amplitude 110 volts and I = 0 when t = 0

In an electric circuit containing the resistance R (ohms) and a condenser of capacitance C (farad) in series and an emf E (volts), the charge q (coulombs) on the condenser is given by  
 

If R = 10 ohms, C = 10⁻³ farad, E (t) = 100 sin (120 πt) volts
i) Find q assuming q = 0 when t = 0
ii) Use I =  to find I assuming I = 5 amperes when t = 0

A boy standing in corner A of a rectangular pool has a boat in the adjacent corner B on the end of a string 20 feet long. He walks along the side of the pool towards C keeping the string taut. Locate the boy and the boat when the latter is 12ft from AC.

A substance γ is being formed by the reaction of two substances α and β in which a grams of α and b grams of β form (a + b) grams of γ. If initially there are  grams of α,  grams of β and none of γ present, and if the rate of formation of γ is proportional to the product of the quantities of α and β uncombined, express the amount (z grams) of γ formed as a function of time t.

The solution of primitive equation
 is . If  and  then is

a)

b)

c)

d)

The differential equation  determines a family of circles with

a) Variable radii and a fixed centre ( 0, 1)

b) Variable radii and a fixed centre ( 0, -1)

c) Fixed radius and a variable centre along the X-axis

d) Fixed radius and a variable centre along the Y-axis

A curve  passes through  and the tangent at  cuts the X-axis and Y-axis at A and B respectively such that then

a) Equation of the curve is

b) Normal at  is

c) Curve passes through

d) Equation of the curve is

If  then prove that

If   and  , then show that
 

A normal is drawn at a point  of a curve meeting X-axis at Q. If PQ is of constant length k, then show that the differential equation of the curve is  

Let y = f (x) be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Find the differential equation and determine all such possible curves.

Let u (x) and v (x) satisfy the differential equations and  where p (x), f (x) and g (x) are continuous functions. If u (x₁) > v (x₁) for some x₁ and f (x) > g (x) for all x > x₁, prove that at any point (x, y) where x > x₁ does not satisfy the equations y = u (x) and y = v (x)

A curve passes through  and slope at the point  is

. Find the equation of the curve and the area between the

curve and the X-axis in the fourth quadrant.