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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 10 · Algebra

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Rectangle dimensions. A rectangle has length (x+3)(x + 3) cm and width (x+1)(x + 1) cm. Its area is 3535 cm².

(a) Form an equation in xx and expand the brackets.
(b) Solve the equation to find xx.
(c) State the dimensions of the rectangle.

Working space

2Problem 2 of 12
Coins. Aoife has a mix of 50p and 20p coins. She has 15 coins in total, with a total value of £4.80.

How many of each coin does she have?

Working space

3Problem 3 of 12
Surds in geometry. A right-angled triangle has legs of length 3\sqrt{3} cm and 12\sqrt{12} cm.

(a) Find the hypotenuse, giving the exact answer in simplest surd form.
(b) Find the area of the triangle.
(c) Find the perimeter, giving an exact answer in simplest surd form.

Working space

4Problem 4 of 12
Algebraic identities.

(a) Show that (x+y)2−(x−y)2=4xy(x+y)^2 - (x-y)^2 = 4xy.
(b) Hence, without a calculator, find the value of 1032−972103^2 - 97^2.
(c) Generalise: a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). Use this to find 2152−1852215^2 - 185^2.

Working space

5Problem 5 of 12
Consecutive integers. Three consecutive integers have a sum of 102.

(a) Set up an equation using nn as the smallest integer.
(b) Find the three integers.
(c) Show that for any three consecutive integers, their sum is divisible by 3.

Working space

6Problem 6 of 12
Mixed factorising. Factorise each fully.

(a) 4x2−254x^2 - 25
(b) x3−9xx^3 - 9x
(c) 2x2+8x+62x^2 + 8x + 6
(d) x2+4x+4−y2x^2 + 4x + 4 - y^2 *(tricky)*

Working space

7Problem 7 of 12
Two unknown coefficients. A quadratic x2+bx+cx^2 + bx + c has roots 2 and 7.

(a) Use the factorised form to find bb and cc.
(b) Verify by substituting x=2x = 2 into x2+bx+cx^2 + bx + c.

Working space

8Problem 8 of 12
Rationalising and simplifying.

(a) Simplify 50+18−8\sqrt{50} + \sqrt{18} - \sqrt{8}.
(b) Rationalise 42+1\dfrac{4}{\sqrt{2} + 1}.
(c) Hence calculate (2+1)(42+1)(\sqrt{2} + 1)\left(\dfrac{4}{\sqrt{2}+1}\right) — verify your answer makes sense.

Working space

9Problem 9 of 12
Triangle perimeter system. Two sides of an isosceles triangle have length x+3x + 3 and the third (the base) has length 2x−12x - 1. The perimeter is 2020 cm.

(a) Form an equation in xx.
(b) Find xx and state the side lengths.
(c) Could this triangle exist if the perimeter were instead 5 cm? Explain.

Working space

10Problem 10 of 12
Solving a quadratic. Solve x2−5x=6x^2 - 5x = 6.

Working space

11Problem 11 of 12
Algebraic system in context. A youth-club hires a hall. The cost is a fixed fee plus an hourly rate.

- 3 hours costs £45.
- 5 hours costs £67.

(a) Set up two equations using ff (fixed fee) and rr (hourly rate).
(b) Solve to find ff and rr.
(c) Predict the cost of an 8-hour booking.

Working space

12Problem 12 of 12
Surds and Pythagoras. A square has diagonal length 8 cm.

(a) Find the exact side length of the square, simplifying any surds.
(b) Find the exact area of the square.
(c) Find the perimeter, giving an exact answer.

Working space