CAIE A-Level Mathematics 1.3.5 Graph Intersections

CAIE A-Level Mathematics 1.3.5 Graph Intersections
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise treating graph intersections as simultaneous solutions, eliminating a variable and using roots or discriminants to decide whether curves meet, touch or remain separate.

How this is tested

  • equate graph formulas or eliminate one variable to form an equation for intersection coordinates
  • use the discriminant to prove that two graphs meet, touch or do not intersect
  • interpret real roots or counted crossings within the stated domain as equation solutions

Question 6

[Maximum number: 9]

The equation of a curve is 2x2−kxy+2=02 x^{2}-k x y+2=0 and the equation of a line is y=p x+3, where k and p are constants.

Question 6(a)

(a)

Given that k=2 and p=11, find the coordinates of the points of intersection of the curve and the line.

[ 4 ]

Question 6(b)

(b)

（b）Given instead that p=4 ，find the set of values of k for which the curve and the line do not intersect．

[ 5 ]