Let a be an integer such that all the real roots of the polynomial
2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

Let L be a common tangent line to the curves

4x² + 9y² = 36 and (2x)² + (2y)² = 31. Then the

square of the slope of the line L is __________.

Draw the output signal Y in the given combination of gates.

An aeroplane, with its wings spread 10 m, is flying at a speed of 180 km/h in a horizontal direction. The total intensity of earth's field at that part is 2.5 $$\times$$ 10^$$-$$4 Wb/m² and the angle of dip is 60$$^\circ$$. The emf induced between the tips of the plane wings will be __________.