CAIE A-Level Mathematics 1.1 Quadratics Question Bank

CAIE A-Level Mathematics 1.1 Quadratics Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise completing the square, solving quadratic equations and inequalities, using the discriminant, eliminating simultaneous variables and recognising quadratic form.

Exam points

  • complete the square to expose the vertex, range or exact roots of a quadratic
  • use b² − 4ac with strict or non-strict inequalities to classify real roots
  • substitute or eliminate variables and solve the resulting quadratic for every valid solution

Covered syllabus points

1.1.1—Completing the square1.1.2—Quadratic discriminant1.1.3—Quadratic equations1.1.4—Simultaneous equations1.1.5—Quadratic-form equations

Question 1

[Maximum number: 4]

Find the set of values of the constant k for which the quadratic equation

3kx2+(k+8)x+3=03 k x^{2}+(k+8) x+3=0

has two distinct real roots.

Question 1

Question 1(a)

(a)

Express 9x2−36x+89 x^{2}-36 x+8 in the form p(x+q)2+rp(x+q)^{2}+r, where p, q and r are constants.

[ 2 ]

Question 1(b)

(b)

Hence find the set of values of the constant k for which the equation 9x2−36x+8=k9 x^{2}-36 x+8=k has no real roots.

[ 1 ]

Question 1(c)

(c)

Find the exact roots of the equation 9x2−36x+8=−159 x^{2}-36 x+8=-15.

[ 2 ]

Question 1

Question 1(a)

(a)

Express 3y2−12y−153 y^{2}-12 y-15 in the form 3(y+a)2+b3(y+a)^{2}+b, where a and b are constants.

[ 2 ]

Question 1(b)

(b)

Hence find the exact solutions of the equation 3x4−12x2−15=03 x^{4}-12 x^{2}-15=0.

[ 3 ]

Question 2

[Maximum number: 4]

Find the coordinates of the points of intersection of the curve and the line with equations

2xy+5y2=24 and 2x+y+4=0.2 x y+5 y^{2}=24 \text { and } 2 x+y+4=0 .

Question 3

Question 3(a)

(a)

Express 4x2+10x+64 x^{2}+10 x+6 in the form a(x+b)2+ca(x+b)^{2}+c, where a, b and c are rational constants to be determined.

[ 2 ]

Question 3(b)

(b)

The curve with equation y=4x2+10x+6y=4 x^{2}+10 x+6 and the line y=k have exactly one point of intersection.

Using your answer to part (a) or otherwise, state the value of the constant k.

[ 1 ]