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A boat is pulled into a dock by means of a rope attached to a pulley on the dock, Figure 2.1. The rope is attached to the bow of the boat at a point 1 m below the pulley. If the rope is pulled through the pulley at a rate of 1 m/sec, at what rate will the boat be approaching the dock when 10 m of rope is out
A hamburger wrapper is thrown from a third floor dorm window, and its motion can be modeled by the equation h(t) = -3t^2 + 6t + 30, where h is the height in feet and t is the time in seconds.
a. Find the vertex of the parabola, then describe what it tells us physically about the situation.
b. Find the x-intercept, then describe what it tells us physically about the situation.
Suppose f(x)={kx + 7 ; x>= 2 ,x^2+19 ; x < 2' , For what value of k is lim x→2 f(x) defined?
For any two subspace W1,W2 of R^3 of dimension 2, W1+ W2 is a direct sum . True or false with full explanation
If some eigenvalues of a matrix are repeated, the matrix is not diagonisable.true or false with full explanation
R^3 has infinitely many non zero, proper vector subspaces. True or false with full explanation
Let T: R^3 to R^3 be the linear transformation defined by
T(x,y,z)= (-x,x-y, 3x+2y+z)
Check whether T satifies the polynomial (x-1)(x+1)^2. Also the find of minimal polynomial of T.
Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.
Find the radius and centre of the circular section of sphere |r|= 4, cut off by the plane
r.(2i-j+4k)= 3.
Let P = [ -1 4 5] . Determine P^-1 using
[ 0 2 -3]
[ 0 0 8]
Cayley- Hamilton theorem. Further use P^-1 to express (x1, x2, x3) in terms of (-1,0,0), (4,2,0), ( 5,-3,8)
