Problem-solving
Algebra
Show all working. Partial marks are given for method.
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1**Rectangle dimensions.** A rectangle has length $(x + 3)$ cm and width $(x + 1)$ cm. Its area is $35$ cm². (a) Form an equation in $x$ and expand the brackets. (b) Solve the equation to find $x$. (c) State the dimensions of the rectangle.
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2**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?
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3**Surds in geometry.** A right-angled triangle has legs of length $\sqrt{3}$ cm and $\sqrt{12}$ 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.
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4**Algebraic identities.** (a) Show that $(x+y)^2 - (x-y)^2 = 4xy$. (b) Hence, without a calculator, find the value of $103^2 - 97^2$. (c) Generalise: $a^2 - b^2 = (a+b)(a-b)$. Use this to find $215^2 - 185^2$.
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5**Consecutive integers.** Three consecutive integers have a sum of 102. (a) Set up an equation using $n$ as the smallest integer. (b) Find the three integers. (c) Show that for any three consecutive integers, their sum is divisible by 3.
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6**Mixed factorising.** Factorise each fully. (a) $4x^2 - 25$ (b) $x^3 - 9x$ (c) $2x^2 + 8x + 6$ (d) $x^2 + 4x + 4 - y^2$ *(tricky)*
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7**Two unknown coefficients.** A quadratic $x^2 + bx + c$ has roots 2 and 7. (a) Use the factorised form to find $b$ and $c$. (b) Verify by substituting $x = 2$ into $x^2 + bx + c$.
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8**Rationalising and simplifying.** (a) Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$. (b) Rationalise $\dfrac{4}{\sqrt{2} + 1}$. (c) Hence calculate $(\sqrt{2} + 1)\left(\dfrac{4}{\sqrt{2}+1}\right)$ — verify your answer makes sense.
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9**Triangle perimeter system.** Two sides of an isosceles triangle have length $x + 3$ and the third (the base) has length $2x - 1$. The perimeter is $20$ cm. (a) Form an equation in $x$. (b) Find $x$ and state the side lengths. (c) Could this triangle exist if the perimeter were instead 5 cm? Explain.
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10**Solving a quadratic.** Solve $x^2 - 5x = 6$.
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11**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 $f$ (fixed fee) and $r$ (hourly rate). (b) Solve to find $f$ and $r$. (c) Predict the cost of an 8-hour booking.
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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.
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