A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is .
Show that, for a positive integer n, the coefficient of xk, () in the expansion of 1 + (1 + x) + (1 + x) + ... + (1 + x)n is
Find the values of 'a' for which the expression x2 - (3a - 1)x + 2a2 + 2a - 11 is always positive
Find the angle subtended by the double ordinate of length 2a of the parabola y2 = ax at its vertex.
The equation to the pairs of opposite sides of a parallelogram are x2 - 5x + 6 = 0 and y2 - 6y + 5. Find the equations its diagonals.
Opposite sides of a parallelogram are,
x2 - 5x + 6 = 0 ie, (x - 3) (r - 2) = 0
or x = 2, x = 3 ...(i)
and y2 - 6y + 5 = 0
i.e, (y - 1)(y - 5) = 0
or y = 1, y = 5 ...(ii)
The intersection lines of Eqs. (i) and (ii) are A(2, 1), B(3, 1), C(3, 5), D(2, 5).
Equation of diagonal AC is,
And equation of diagonal BD is,