Analysis only. All three worked equations are the guide's own examples, which makes this the examined shape rather than a selection.
Put the three equations on the board before the figure and ask which looks hardest. Classes pick the quartic. It has integer roots. They pick ex = sin x as the easy one, and it has no solution you can write down at all.
Then step the figure. View 2 shows the quartic crossing twice at whole numbers; view 3 shows the two curves crossing again and again as they go left, with the exponential shrinking towards the axis. The ranking they just gave is reversed by the pictures, which is a better argument for reading structure than any instruction to do so.
Worth pointing out on view 1: the two roots are x = 0 and x = 1.386, and the one at the origin is the one students lose. It looks like the graph passing through a landmark rather than a solution.
| Question | Answer |
|---|---|
| 1. Smaller root of e2x − 5ex + 4 = 0 | u = 1, so x = 0. |
| 2. The larger root | u = 4, so x = ln 4 = 1.386. |
| 3. Real roots of x⁴ + 5x − 6 | 2: x = 1 and x = −2. |
| 4. Solutions of ex = sin x with x ≥ 0 | 0. |
| 5. What signals the substitution | B. e2x is the square of ex. |
Question 4 is a "zero is an answer" question, and students hesitate to write it. Say beforehand that a count can be zero and that "none, because ex ≥ 1 and sin x ≤ 1 here" is the full answer.
1 markThe substitution or factorisation, stated.
1 markThe values of the new variable.
1 markUndoing the substitution, both ways.
1 markRejecting any impossible value, with a reason.
Marks three and four are the ones this sub-topic turns on. A student who reaches u = 1 and u = 4 has done the thinking and can still lose half the question by reporting one root and not mentioning the other.
| They wrote | What happened |
|---|---|
| 1 or 4 on question 1 | Stopped at u. They have solved the quadratic and not come back. This is the single most common shape of answer here. |
| 1.386 on question 1 | Found both and reported only the one that looks like a real answer. Ask what ex = 1 gives. |
| 0.602 | log104 rather than ln 4. The base is e, so it is the ln key. |
| 0.693 | ln 2. Half of ln 4, probably from ln√4 or from halving by habit. |
| 2.773 | 2 ln 4. They have remembered that ln 4 = 2 ln 2 and applied the 2 twice. |
| 4 on question 3 | A quartic has four roots, so four were reported. Two are complex, because x² − x + 3 has discriminant −11. Worth connecting to 2.7. |
| 1 on question 3 | Found x = 1 and stopped trying integers. The habit is to try 1, −1, 2 and −2, all four. |
| 1 on question 4 | Counted x = 0 as a solution. At x = 0 the two sides are 1 and 0, so it is not one, and substituting settles it. |
| −3.183 | Gave the largest actual solution. Correct number, wrong question: the restriction was x ≥ 0. |
"Can I just use the solver for everything?" On Paper 2, for the ones with no structure, yes. Two reasons not to make it the habit. Paper 1 has no calculator and the equations there all have structure, so the technique is examined directly. And a solver returns one root with no indication of how many there are, which is exactly the mark students lose on ex = sin x.
"How do I know when there is no analytic method?" Honestly, by having tried the list: substitution, factorising, same base, logarithms. The useful thing is that the list is short, so working through it takes under a minute and the conclusion is a decision rather than a guess. Mixed function types with the unknown in two different roles, as in ex = sin x, is the usual signature.
"Why do we reject u = −3?" Because ex = −3 has no solution: an exponential is never negative. The rejection is a mark and it needs the reason, not just a crossing out. Worth setting one question where a substitution produces a negative u so they meet it before an exam does.
"Are the crossings exactly 2π apart?" Two questions hide in that one, and the first is how many there are. Two per cycle, not one. The roots sit near the multiples of π, because that is where the sine is zero and the exponential has all but vanished: −3.183, −6.281, −9.425, −12.566, so consecutive crossings are about π apart. Crossings a full cycle apart are 6.242, against 2π = 6.283, a little under because the exponential is still shrinking across the cycle. A sharp class will ask, and the honest answer names both numbers.
| Demonstrate | Graph ex and sin x as two separate functions rather than their difference, with x from −12 to 2. Then count the crossings out loud with the class, walking left with the intersection tool. Four is enough for them to see it never stops. Graphing the difference hides the mechanism, which is the exponential shrinking past a fixed oscillation. |
|---|---|
| Where they stick | Radians. In degrees the sine curve has a period of 360 and looks almost flat on this window, so the crossings vanish and students conclude there are none. Set Angle to Rad before anything is drawn. Also the solver's starting guess: without one, both machines return whichever root they find first, and students read that as the only one. |
| The check | Substitute each answer back into the ORIGINAL equation, not into the substituted one. x = 0 in e2x − 5ex + 4 gives 1 − 5 + 4 = 0. That catches a u reported as an x, which is the commonest error on the page. |
Analysis Paper 1 has no calculator and the structured equations live there. Split the practice deliberately: hidden quadratics and integer-root polynomials with machines away, the graphical ones with machines on.
| Step | What |
|---|---|
| 1 | All three equations on the board. Ask which is hardest. Record the vote. |
| 2 | The hidden quadratic, with u = ex written out and undone twice. |
| 3 | Figure view 1, and the x = 0 root named as the one that gets lost. |
| 4 | The quartic: try 1, −1, 2, −2 as a class. Two hits. |
| 5 | Divide out and look at the remaining quadratic's discriminant. |
| 6 | ex = sin x: ask for an analytic route. Let them fail for a minute. |
| 7 | The size argument for x ≥ 0, then the graph for the negatives. |
| 8 | The five-line decision list, written up, with the vote from step 1 revisited. |
Do not say "if it looks hard, graph it". It is the instinct this page exists to correct, and the quartic is the counterexample: it looks hard and has integer roots that a graph would give you as 1.00 and −2.00 with no exactness. Say to run the list, which takes a minute and ends in a decision either way.
Do not say "there is always a method". There is not, and pretending otherwise leaves students hunting for a rearrangement that does not exist while the clock runs. ex = sin x has no closed-form solution, the syllabus says so explicitly, and knowing that is part of the content.