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Examine the discriminant d for this equation: d = 22 4 1 ( 15) = 4 ( 60) = 4 + 60 = 64 How can we change the stand-alone constant, which is 15 right now, to bring the discriminant down to 0 We can find out by substituting 0 for d, and by substituting a letter constant for 15 in the above equation If use k as the letter constant, we get 0 = 22 4 1 k which simplifies to 0 = 4 4k and further to 4k = 4 This resolves to k = 1 In the quadratic we got by mixing, let s change the stand-alone constant k from 15 to 1 That gives us x 2 + 2x + 1 = 0 If we increase the constant any more, then d becomes negative, and the system has no real solutions Now we know that if we increase the stand-alone constant by 16 in the quadratic represented by the solid parabola in Fig 28-2, the two solution points will merge in the graph So let s increase that constant by 17! When we do that, the function becomes y = 2x 2 3x + 13 The graph of this function is a parabola with the same contour as the original one, but displaced by 17 units upward in the coordinate plane For extra credit, try solving the system y = 2x 2 3x + 13 and y = 3x 2 5x + 11 and see for yourself that it has no real solutions

At frequencies above about 30 MHz, the lower atmosphere bends radio waves towards the surface (Fig 27-3) Tropospheric banding or tropo occurs because the index of refraction of air, with respect to EM waves, decreases with altitude The effect is similar to sound waves hugging the surface of a lake in the early morning, letting you hear a conversation a mile away Tropo makes it possible to communicate for hundreds of miles when the ionosphere will not return waves to the earth

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(c) +5 V (d) +50 V 13 Fill in the blank to make the following sentence true: bits sent along a single line, one bit after another (a) Serial (b) Synchronous (c) Parallel (d) Analog 14 The inputs of an R-S flip-flop are known as (a) low and high (b) octal and binary (c) synchronous and asynchronous (d) reset and set 15 When both inputs of an R-S flip-flop are 0, (a) the outputs stay as they are (b) Q = 0 and Q = 1 (c) Q = 1 and Q = 0 (d) the resulting outputs can be absurd 16 The fifth digit from the right in a binary number carries which decimal value (a) 64 (b) 32 (c) 24 (d) 16 17 What is the octal equivalent of binary 1010 (a) 4 (b) 10 (c) 12 (d) There is no way to tell without more information 18 What is the largest possible radix 10 number that can be represented as a six-digit binary number (a) Decimal 256 (b) Decimal 128 (c) Decimal 64 (d) Decimal 63 19 If one of the inputs to a two-input NOR gate is high, what is the output state (a) There is not enough information to tell (b) Low data transmission refers to digital

Here s another challenge!

Another type of tropospheric propagation is called ducting It takes place when EM waves are trapped in a layer of cool, dense air sandwiched between two layers of warmer air Like bending, ducting occurs almost entirely at frequencies above 30 MHz Still another tropospheric-propagation mode is troposcatter This takes place because air molecules, dust grains, and water droplets scatter some of the EM field at very high and ultra-high frequencies (above 30 MHz)

(shown by the dashed curve) can be modified to produce a system with no real solutions, assuming that the other quadratic function (shown by the solid curve) stays the same, and also assuming that the contour of the graph for the modified function stays the same

(c) High (d) It depends on the state of the other input 20 Suppose a logic circuit has four inputs W, X, Y, and Z How many possible input combinations are there (a) 4 (b) 8 (c) 16 (d) 32

To cause the two solution points to merge, we must move the dashed parabola downward to the same extent we moved the solid parabola upward in the previous example That means we must reduce the constant in the function for the dashed parabola by 16, giving us y = 3x 2 5x 5 If we reduce the constant to anything smaller than 5, the real solutions of the two-by-two system vanish, and the parabolas no longer intersect

Radio waves can bounce off the aurora (northern and southern lights) This is auroral propagation, and it occurs at frequencies from roughly 15 to 250 MHz It can take place between stations separated by up to about 2000 miles Meteors entering the upper atmosphere produce ionized trails that persist for several seconds up to about a minute; these ions reflect EM waves and cause meteor-scatter propagation This mode allows communication for hundreds of miles at frequencies from 20 to 150 MHz The moon, like the earth, reflects EM fields This makes it possible to communicate via earth-moon-earth (EME), also called moonbounce High-powered transmitters, sophisticated antenna systems, and sensitive receivers are needed for EME Most EME is done by radio hams at frequencies from 50 MHz to over 2 GHz

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