Why isn't the answer to TMUA-P1X-MM3-05 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P1X-MM3-05 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P1X-MM3-05?

No — calculators are banned in the TMUA. TMUA-P1X-MM3-05 is set for a no-calculator target of about 240 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P1X-MM3-05?

TMUA-P1X-MM3-05 is Paper P1 · Coordinate geometry in the (x,y)-plane (specification code MM3), level 3. Drill more of the same topic from this page.

TMUA-P1X-MM3-05 — Paper P1 · Coordinate geometry in the (x,y)-plane · Level 3

P1MM3Level 3~240s target

Spec: MM3 · No calculator · ~240s

The line

y = 2x + k

with

k > 0

is tangent to the circle

x^2 + y^2 = 20

. What is

k

Options

$5$
$10$
$20$
$2\sqrt{5}$
$4\sqrt{5}$

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: tangency-discriminant, discriminant-zero. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Substituting gives

5x^2 + 4kx + k^2 - 20 = 0

; tangency needs discriminant zero:

(4k)^2 - 20(k^2 - 20) = 400 - 4k^2 = 0

, so

k = 10

(taking

k > 0

). Trap:

2\sqrt{5}

is the radius — plausible if you equate

k

itself to the circle's radius.

Keep practising

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Browse more

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Keep practising

Same topic · same level

P1MM3Level 3

TMUA-P1-MM3-010

The line y = x + 1 crosses the circle x^2 + y^2 = 25 at two points. What is the length of the chord joining them?

Coordinate geometry in the (x,y)-plane

Open TMUA-P1-MM3-010 →

Same trap: discriminant-zero

P1MM1Level 2

TMUA-P1-MM1-001

The equation x^2 + kx + 9 = 0 has two equal real roots. What is the sum of all possible values of k?

Algebra and functions

Open TMUA-P1-MM1-001 →

TMUA-P1X-MM3-05 — Paper P1 · Coordinate geometry in the (x,y)-plane · Level 3

Options

Keep practising

Master Coordinate geometry in the (x,y)-plane

TMUA-P1X-MM3-05 — Paper P1 · Coordinate geometry in the (x,y)-plane · Level 3

Options

Keep practising

Master Coordinate geometry in the (x,y)-plane