Mock CT3 Paper 3 — Simultaneous Equations & Quadratics

Instructions: Answer all the questions. All answers should be given in their simplest form. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For π, use your calculator value. A Graphic Display Calculator and/or calculator can be used for this paper. Total: 45 marks.

Section A

Algebraic Manipulation & Formulae

[24 marks]

Question 1

Solve the simultaneous equations:

x + y = 7
x² + xy = 28

Question 2

A building is 40 m high. A cable is pulled taut and attached from the top of the building, P, to a point Q on the ground, 75 m from the foot of the building.

Find the length of the cable.

Question 3

Express (

5x+2

−

3x−1

) ÷

(2x−7)(x²+x−2)

as a fraction with a single denominator in its simplest form.

Question 4

Simplify each of the following. Give your answers in their simplest form.

(a) 6p²q × 8pq²
(b)
(3y² − 12)(y + 2)

(a)

(b)

Question 5

Make the stated letter the subject of the formula.

(a) Given that A = 2πr(r + h), make r the subject.
(b) Given that n =
(x+3)(2x−1)
, make x the subject.

(a)

(b)

Question 6

Given that y =

(2x+3)(x−1)

, where x ≠ 1, find:

(a) the value of y when x = 3.
(b) the value of x when y = 4.

(a)

(b)

Question 7

For the equation y = −x² + 6x − 5, find:

(i) the zeros of the equation.
(ii) the y-intercept.
(iii) the maximum y-value.
(iv) the equation of the axis of symmetry.
(v) values of x when y = 3.

(i)

(ii)

(iii)

(iv)

(v)

Question 8

A train travels 120 km from Station X to Station Y at an average speed of v km/h. On the return journey from Y to X, the train's average speed was 10 km/h less than the outward journey.

(i) Write down an expression, in terms of v, for the time taken for the return journey.
(ii) Given that the train took 1 hour longer on the return journey than the outward journey. Show that it reduces to v² − 10v − 1200 = 0.
(iii) Solve the equation v² − 10v − 1200 = 0.

(i)

(ii)

(iii)

Question 9

Solve the following equations:

(a) (3h − 2)(h + 4) = 10.
(b) Hence, find the solutions for the equation (3k² − 2)(k² + 4) = 10.

(a)

(b)

Question 10

Given that

and x + y = 64, find the value of x.

Answer Key

Q1: x = 4, y = 3 or x = −7, y = 14

Q2: 85 m (Pythagoras: √(40²+75²))

Q3:

2(x−1)

Q4: (a) 48p³q³ (b) 3(y−2)

Q5: (a) r = (−h+√(h²+A/π)) (b) x =

(n+3)(2n−1)

Q6: (a) y =

(b) x =

Q7: (i) x = 5 or −1 (ii) y = −5 (iii) y = 4 (iv) x = 3 (v) x = 2 or 4

Q8: (i)

120(v−10)

Q9: (a) h = 0 or −

103

(b) k² = 0 or −

143

, k = 0 only → k = 0

Q10: x = 40

Mock CT3 Paper 3 — Simultaneous Equations & Quadratics

Syllabus Topics Covered

Algebraic Manipulation & Formulae

Answer Key