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1. Water is being drained at a constant rate from an inverted conical tank into a tank in the shape of an inverted square pyramid. The concial tank has a semivertical angle of 30 degrees. The dimensions of the square pyramid tank are 2x2 for the base and the height is 4. The dept of the water in...

1. A particle is in SHM where its maximum speed is Vm/s^1 and its period is Ts. Find an expression for the particle's (a) amplitude (b) maximum acceleration 2. P, Q, R are three distinct points on the positive x axis, so that PQ = QR = 2 metres. A particle performs SHM along the x axis and...

1. Find thé maximum area of a rectangle that can be inscribed in an equilateral triangle of side-length 1 unit. 2. Find the dimension of the right square priem with maximum volume, which can be inscribed in a sphere of radius 10cm. 3. U is at (-2,-6) and V is at (2,6). Point A divides UV in...

The distinct points P, Q correspond respectively to the values t=t1, t=t2 on the parabool x = 2t, y = t^2 (a) (i) write down the equation of the tangent to the parabola at P. (ii) show that the equation of the chord PQ is 2y- (t1+t2)x+2t1t2=0 (iii) show that M, the point of intersection of...

1. A particle is projected with a speed V, from a height h, above a horiztonal plane, at an angle of θ to the horizontaal. (a) if its range on the horizontal plane is R, show that: R^2sec^2θ -[2V^2R/g](tanθ)- [2hV^2/g] = 0 (b) without solving this equation, show that it has one negative...

1. Prove that for any positive integer n, the largest value of 2nCr for 0< r < 2n (r integers) is 2nCn and that it occurs only when r = n. 2. By considering the value (1+x)^2n when x = 1, prove that ∑_(r=0)^n▒(2n¦r) = 2^(2n-1)+(2n)!/(2〖(n!)〗^2 ) 3. By integrating both sides of the expansion...

A pool of volume 5 x 10^4 litres is initiale full of water containing chlorine at a concentration 5g/L. The contents of the pool are drained at 50L/min while the pool is kept full from the reservoir containing 2g/L of chlorine. (a) Assuming the pool is thoroughly stirred at all times, show...

An object falling directly on the earth from space moves according to the equation (d^2 x)/(dt^2 )= (-k)/x^2 , where x is the distance of the object from the centre of the earth at time t. The constant k is related to g, the value of gravity at the earth's surface, and the radius of the...