Sixth Term Examination Paper (STEP) Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers Syllabus

Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   Coordinate Geometry in the Plane

   5 topics

   • Straight lines: gradient, intersection, distance and perpendicularity
   • The circle: equation, tangents, chords and loci
   • Parametric equations and conversion to Cartesian form
     • Parametric curve sketching
     • Eliminating the parameter
   • Conic sections: parabola, ellipse and hyperbola
     • Foci, directrices and eccentricity
     • Tangents and normals to conics
   • Loci defined by geometric conditions
2. 2

   Trigonometry

   7 topics

   • Radian measure, arc length and sector area
   • Exact values and graphs of trigonometric functions
   • Reciprocal and inverse trigonometric functions
     • Domains and ranges of arcsin, arccos, arctan
   • Pythagorean, addition and double-angle identities
     • Deriving the t = tan(x/2) substitution results
   • Factor formulae and product-to-sum identities
   • Harmonic form R sin(x + a) and its applications
   • Solving trigonometric equations over given intervals
3. 3

   Complex Numbers

   7 topics

   • Cartesian arithmetic and the Argand diagram
   • Modulus, argument and the modulus-argument form
   • Multiplication and division as rotation and scaling
   • De Moivre's theorem and powers/roots of complex numbers
     • nth roots of unity and their geometry
     • Trigonometric identities via De Moivre
   • Euler's relation and the exponential form
   • Loci and regions in the complex plane
     • Circles, perpendicular bisectors and half-lines
   • Roots of polynomials with complex/conjugate pairs

Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers flashcards for Sixth Term Examination Paper (STEP)

20 of 49 cards from the Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers deck — real questions with worked answers.

1. What is the gradient of the line through points $(x_1, y_1)$ and $(x_2, y_2)$?

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ The gradient measures the change in $y$ per unit change in $x$.

2. What is the condition for two lines with gradients $m_1$ and $m_2$ to be perpendicular?

   $m_1 m_2 = -1$ (equivalently $m_2 = -\frac{1}{m_1}$). Parallel lines instead satisfy $m_1 = m_2$.

3. State the perpendicular distance from the point $(x_0, y_0)$ to the line $ax + by + c = 0$.

   $$d = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}$$

4. How do you find the coordinates of the intersection of two non-parallel straight lines?

   Solve their equations simultaneously (e.g. by substitution or elimination); the unique solution $(x, y)$ is the intersection point. Parallel lines give no solution; identical lines give infinitely many.

5. What is the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$?

   $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

6. Give the equation of a circle with centre $(a, b)$ and radius $r$.

   $$(x - a)^2 + (y - b)^2 = r^2$$

7. The general circle equation is $x^2 + y^2 + 2gx + 2fy + c = 0$. State its centre and radius.

   Centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$ (real only when $g^2 + f^2 - c > 0$).

8. What geometric property relates a tangent to a circle and the radius at the point of contact?

   The tangent is perpendicular to the radius drawn to the point of contact. This lets you find the tangent's gradient as the negative reciprocal of the radius's gradient.

9. State the perpendicular-from-centre property for a chord of a circle.

   The perpendicular from the centre to a chord bisects the chord. Hence the perpendicular bisector of any chord passes through the centre.

10. How do you write parametric equations and what does eliminating the parameter achieve?

    Parametric equations give $x = f(t)$ and $y = g(t)$ in terms of a parameter $t$. Eliminating $t$ between them yields the Cartesian relation $F(x,y)=0$.

11. Convert the parametric curve $x = a\cos t$, $y = b\sin t$ to Cartesian form.

    Using $\cos^2 t + \sin^2 t = 1$: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ an ellipse.

12. Give the standard Cartesian and parametric equations of a parabola with vertex at the origin opening rightwards.

    Cartesian: $y^2 = 4ax$. Parametric: $x = at^2$, $y = 2at$. Focus at $(a, 0)$, directrix $x = -a$.

13. State the standard equation of an ellipse and the relationship between its semi-axes and foci.

    $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b$$ Foci at $(\pm c, 0)$ with $c^2 = a^2 - b^2$; eccentricity $e = c/a < 1$.

14. State the standard equation of a hyperbola and its asymptotes.

    $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ Asymptotes $y = \pm \frac{b}{a} x$; foci at $(\pm c, 0)$ with $c^2 = a^2 + b^2$, eccentricity $e = c/a > 1$.

15. How is a conic section classified by its eccentricity $e$?

    $e = 0$: circle; $0 < e < 1$: ellipse; $e = 1$: parabola; $e > 1$: hyperbola. Eccentricity is the ratio of distance to focus over distance to directrix.

16. What locus is defined by the set of points equidistant from a fixed point and a fixed line?

    A parabola. The fixed point is the focus and the fixed line is the directrix; eccentricity equals $1$.

17. What locus is the set of points the sum of whose distances to two fixed points is constant?

    An ellipse, with the two fixed points being its foci. (For the constant difference of distances, the locus is a hyperbola.)

18. Convert $\theta$ radians to degrees and state the radian measure of a full turn.

    Degrees $= \theta \times \frac{180}{\pi}$. A full turn is $2\pi$ radians $= 360^\circ$, so $\pi$ radians $= 180^\circ$.

19. Give the formulas for arc length and sector area of a circle of radius $r$ subtending angle $\theta$ (in radians).

    Arc length $s = r\theta$; sector area $A = \tfrac{1}{2} r^2 \theta$.

20. State the area of a circular segment cut off by a chord subtending angle $\theta$ (radians) at the centre.

    $$A = \tfrac{1}{2} r^2 (\theta - \sin\theta)$$ (sector area minus triangle area).

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