Sixth Term Examination Paper: a Full Guide

The Sixth Term Examination Papers (STEP) are advanced maths exams used by several top UK universities, primarily the University of Cambridge, as part of their admissions process for mathematics programs. 

There are two STEP papers—STEP 2 and STEP 3. STEP 1 was previously a third paper, but it was discontinued in 2020 according to the Cambridge site.

For Cambridge specifically, achieving certain grades in STEP 2 and 3 is typically part of the conditional offer requirements to read mathematics there. However, other universities like Warwick and Imperial may only require one of the two papers or make them optional.

Some students choose to take the STEP papers simply for the intellectual challenge, even if not required by the universities they apply to. For those applying to the institutions mentioned, they will typically need to take either STEP 2 or STEP 3, or potentially both papers, depending on the specific entry requirements.

The STEP papers consist of 12 questions each—8 pure maths, 2 mechanics, and 2 statistics/probability questions. However, your final score is based on just your 6 highest scoring responses out of the 12, with each question worth 20 points max for a total of 120 possible points.

In terms of content level:

STEP 2 is based on A-level Math and AS Further Maths material.

STEP 3 is based on a typical A-level Further Math syllabus and intended to be of similar difficulty to STEP 2.

The Why Behind the STEP

Let’s elucidate the key reasons why achieving a strong STEP performance is so highly valued and integrated into the Cambridge mathematics offer process.

STEP serves as an elegant litmus test for identifying those students who possess the refined mathematical mindset and problem-solving prowess to truly thrive in the rarefied environment of a Cambridge mathematics education. It goes beyond just computational skill.

You see, the STEP questions are meticulously crafted to probe a candidate’s ability to think deeply, rigorously, and creatively about rich conceptual mathematics. Executing stellar solutions demands fluently navigating complex logical reasoning, making insightful connections across domains, and grappling with novel, paradigm-stretching problem formulations.

In essence, STEP separates the mathematical “translators” from the pioneering explorers and abstractors. The latter category—those who can blaze conceptual trails rather than just follow coded procedures—is precisely what Cambridge aims to cultivate among its tight-knit cohort of mathematical wunderkinds.

By making STEP mastery an integral part of their admissions criteria, Cambridge upholds a crucial filter for identifying the rare students who can match the university’s trademark pedagogy of nurturing audacious mathematical thinking and preparing the field’s future vanguards.

It’s a hallmark of Cambridge’s steadfast commitment to producing not just technically skilled mathematicians, but iconoclastic theoretical arthritis who will inexorably expand the frontiers of mathematical understanding itself. Those who conquer STEP’s uniquely formidable challenges demonstrate the intellectual mettle to join such rarefied ranks.

STEP Preparation Advice

With only around 5% of students achieving a passing grade, it certainly represents a formidable challenge even for the mathematically gifted. There is also a critical importance of rigorous preparation, especially for those without access to dedicated school resources.

The STEP papers serve as a refining crucible to identify those students who possess the true grit and mathematical maturity to thrive in the famously rigorous theoretical climate of the Cambridge mathematics program.

Succeeding on these exams signals one’s ability to wrestle with fundamentally abstract concepts and untangle multi-layered logical knots. It’s a glimpse into how well a student can enter the mental universe of pure mathematics.

While working through STEP past papers (Sixth Term Examination Paper Past Papers in Mathematics) is the foundational recommendation, allow me to share some additional insightful strategies that can complement this approach when working independently.

First advise here, seek to deeply internalise and apply core mathematical principles and methods rather than simply memorising techniques. STEP rigorously tests conceptual understanding over rote procedures. Mastering profound theoretical frameworks will reveal unconventional solution pathways.

Next, consciously cultivate an appetite for intellectual struggle. Embrace being stuck initially—the cognitive chaos preceding an inspired breakthrough. Develop persistence and creativity in reframing problems from multiple representations.

Moreover, seek out enriching dialogue with fellow travellers on this arduous journey. Forming a virtuous cycle of posing conundrums, proposing partial solutions, and receiving critical feedback forges crucial instincts.

Finally, approach each practice paper as an opportunity for joyful cerebral exploration, not just mechanical preparation. Savour the delicious perplexities! An attitude of curiosity vitalizes insights.

In essence, comprehensive mastery of STEP demands combining resolute solo problem-wrestling with the illuminating role of collaboration and an unwavering zest for intellectual adventure. Approach it as both a sublime personal vision-quest and an energising communion of united mathematical minds.

Moreover, by requiring top scores across the broad spectrum of pure maths, mechanics, stats, and probability represented in STEP, Cambridge filters for those with comprehensive fundamentals and the scintillating flashes of insight needed to weave between discrete domains. The best mathematicians are polymaths, able to discern the underlying connections.

There’s also an opportunity to showcase one’s creative problem-solving flair on STEP. The constructed response format prizes rigorous logical reasoning but also a poet’s inspirational leaps. Long after facts have faded, those instincts to discern novel solutions from first principles are what elevate the mathematical mind.

In essence, excelling at STEP is a hallmark of possessing that indescribable je ne sais quoi—the weird flex of elite mathematical intelligence prized in the rarefied realms of a Cambridge mathematics education. Those who conquer have demonstrated their enrollment amongst the vanguard, poised to reshape what’s possible at the outermost frontiers.

Sample STEP Questions

Let’s start with a pretty gnarly one from STEP 2 involving some sneaky algebra.

If x^3 + y^3 = 6 and xy = 2, express x^6 + y^6 in terms of x and y

On the surface, it seems straightforward enough. But quickly you realise there’s some nasty exponential and variable juggling required. My first instinct would be to try using the given equations to isolate x or y, then substitute that into the other equation. But that path looks rough.

Instead, let’s get a little clever. Notice that (x^3)^2 + (y^3)^2 = (x^3 + y^3)^2 – 2x^3y^3. Substituting the given equations, this becomes 36 – 12xy. But we’re also given xy = 2, so the whole expression reduces to 36 – 24 = 12. 

Recognizing that the left side is (x^6 + y^6), we’ve already solved for the desired quantity! A bit of an “aha” moment where the solution was hiding in plain sight. That’s the STEP vibe—expecting the unexpected.

 Let’s keep this curiosity-sparking vibe rolling with a wild one from STEP 3.

If a circle has equation

x^2 + y^2 – 2ax – 2by + c = 0, show that its radius is √(a^2 + b^2 – c).

Alright, this one requires some creative circle geometry chops. We’re given the circle’s equation in standard form, but with those pesky extra 2a and 2b terms usually reserved for translating the centre.

My first hunch is that we’ll want to complete the square on both the x and y variables to optimise that circle equation. But having those coefficients on the x and y terms makes it trickier.

What if we tried substituting u = x – a and v = y – b? That would eliminate those coefficients, right? The equation then becomes u^2 + v^2 = a^2 + b^2 – c. 

Yes, that’s looking more recognizable as the equation for a circle centred at the origin with radius √(a^2 + b^2 – c)! By that nifty substitution, we’ve transformed it to an easier-to-recognize form.

From there, we just need to reason through the reverse substitution to confirm that 

x^2 + y^2 – 2ax – 2by + c = (x – a)^2 + (y – b)^2

which gives the radius in terms of the original coefficients.

When you approach maths from a place of genuine passion and delight in the intellectual journey, then even the thorniest obstacles become invigorating riddles to untangle. The STEP questions are crafted to put your creative problem-solving to the test, but for the mathematically inquisitive mind, that’s an irresistible siren call.

Leave a Comment