Part A: JEE Main Level

Transcript

0:00Introduction

Hello everyone! Okay, let us start with a quick puzzle. Add up cos one degree, cos two degrees, cos three degrees, and so on, all the way to cos one hundred and eighty degrees. That is one hundred and eighty cosines, and no calculator. One student says the answer is zero. Another student says it is minus one. So, which one is right? Keep your answer ready; by the end of this class, you will see exactly who slipped, and where. This is Part A of our unit problem bank, at JEE Main level, and quick questions like this follow the JEE pattern. So, let's begin.

0:39Part A: a 30-minute test

So, here is what Part A holds. Sixteen solved problems, all at JEE Main level. Short, quick questions, okay? And they are mixed: values, identities, equations, inverse functions, triangles and heights, all shuffled together. Nobody tells you which chapter a question comes from, just like the real exam. Right? The plan, from the chapter overview: finish Chapters one to six, then attempt Part A as a single test of thirty minutes. Today, we sharpen two tools, and then we solve the first two problems of Part A together. With me?

1:18Tool 1: allied angles

Tool number one comes from Chapter one, and it is allied angles. These are angles like n pi over two, plus or minus theta, where n is an integer. Pi and theta are Greek letters, and the sign with a plus over a minus means plus or minus. Okay? First, the function. If n is even, the function stays the same. If n is odd, it switches: sine becomes cos, tan becomes cot, and sec becomes cosec, and the other way round too. That is what the double-headed arrows mean. Clear? Then the sign. Treat theta as acute, find the quadrant where the angle lies, and attach the sign that the original function has in that quadrant. Right? Now, let me ask you this. What is cos of one hundred and eighty degrees minus theta? The small circle up top means degrees. One hundred and eighty degrees is pi, which is two times pi over two, so n is two, an even number, and the function stays cos. And one hundred and eighty degrees minus theta lies in the second quadrant, where cos is negative. So cos of one hundred and eighty degrees minus theta equals minus cos theta. Got it?

2:28Tool 2: general solution of tan theta = tan alpha

Tool number two comes from Chapter three: the general solution of tan theta equals tan alpha, where alpha is another Greek letter. Tan theta equals tan alpha implies theta equals n pi plus alpha; the double arrow means implies. Here n belongs to the double-lined capital Z, the set of integers; the curved E sign means belongs to. Keep an eye on the graphs on the right; we will use them in a moment. Okay? Now think about this. For tan, a single n pi gives every solution. Why is that enough? Yes, exactly, tan repeats every pi! In the figure, the left half shows sine and cos, and the right half shows the green curve of tan x, with period pi. But see the dashed vertical lines there. At odd multiples of pi over two, that is, two n plus one, in round brackets, times pi over two, the curve shoots off, and tan theta is not defined at all. Clear? So here is the warning from Chapter three. After you solve, reject every value at which a term of the original equation is undefined. Skip this check, and you lose marks. We will need it very soon. With me?

3:42Example 1: cos 1 + cos 2 + ... + cos 180 degrees by pairing

Now, remember our puzzle from the start? It is problem one of Part A, so let's settle it. Find cos one degree, plus cos two degrees, and so on, up to cos one hundred and eighty degrees; the three dots mean and so on. Pause the video and try it first. Done? Okay, let's check. Use tool one. Cos of one hundred and eighty minus k, in brackets, degrees, is minus cos k degrees. So cos k degrees plus cos of one hundred and eighty minus k degrees is zero, for every k from one to eighty nine. Okay? So pair them up: cos one degree with cos one hundred and seventy nine degrees, and so on, up to cos eighty nine degrees with cos ninety one degrees. That makes eighty nine pairs, and every pair is zero. Right? So the whole sum is zero, isn't it? That was the first student's answer. Is it correct? No, no, wait, count the terms! Eighty nine pairs use only one hundred and seventy eight terms, but there are one hundred and eighty. Two terms have no partner: cos ninety degrees, which is zero, and cos one hundred and eighty degrees, which is minus one. Clear? So the pairs give zero, cos ninety degrees adds zero, and cos one hundred and eighty degrees adds minus one. The sum is minus one. The second student was right; the first one forgot that cos one hundred and eighty degrees has no partner in the sum. So the puzzle is solved, perfect!

5:11Example 2: solve tan 5x = tan 3x, reject undefined values

Now the harder one, problem two of Part A, okay? Solve tan five x equals tan three x. Here is a hint: use tool two, and remember its warning. Pause the video and try it first. Done? Okay, let's check. Tan of five x equals tan of three x, so by tool two, five x equals n pi plus three x, with n an integer. Okay? Take three x across, and two x equals n pi. So x equals n pi over two. Right? So the answer is n pi over two, and we are done, right? Hmm, let me ask you this. Put n equal to one, so x is pi over two. Is that really a solution? No, it is not, and this is the trap! At x equals pi over two, five x is five pi over two, and three x is three pi over two. Both are odd multiples of pi over two, so neither tan is even defined. The same happens for every odd n, so we reject every odd n. Clear? Now the even values of n. Then x is a whole multiple of pi, so five x and three x are multiples of pi too. Both tans are zero, and the equation holds. Got it? So the answer is x equals n pi, where n is any integer, and not n pi over two. Half of those values were never solutions at all, and the warning from tool two caught them. That is a perfect finish!

6:42Quick revision: allied angles, tan equation, undefined check

So, quick revision. First, allied angles: cos of one hundred and eighty degrees minus theta is minus cos theta, so pair the terms, and then count the leftovers. Second, tan theta equals tan alpha gives theta equals n pi plus alpha, with n an integer. Third, always reject values where a term of the equation is undefined. Now attempt the other problems of Part A on this page as a thirty minute test. You can do this. All the best!