Mathematics · OCR H240 · Topic 1.02

Algebra and functions

The algebra toolkit: indices, surds, quadratics, simultaneous equations, inequalities, polynomials, graph sketching, functions, transformations and partial fractions.

11 sub-topics
5 in Year 12, 3 in Year 13
193 practice questions
11 animated lessons

Practice questions (193) Lessons Worksheets coming soon All OCR Mathematics topics

What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

1.02aIndicesY12
1.02bSurdsY12
1.02cSimultaneous equationsY12
1.02d-fQuadratic functionsY12
1.02g-iInequalitiesY12
1.02j-kPolynomialsY13
1.02l-tThe modulus function and curve sketchingY13
1.02u-vFunctionsY13
1.02w-xGraph transformationsY13
1.02yPartial fractionsY13
1.02zModels in contextY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 193 questions in this topic, free, no account needed.

Algebra and functions

Easy

Equal roots · Year 12 · 3 marks

The equation $x^2 + kx + 9 = 0$ has equal roots. Find the possible values of $k$ .
(3)

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Equal (repeated) roots happen when the discriminant is zero. $b^2 - 4ac = 0$
Here $a = 1$ , $b = k$ and $c = 9$ . $k^2 - 4 \times 1 \times 9$ ${}= 0$ ${}\Rightarrow k^2$ ${}= 36$
Square-rooting gives two answers: positive and negative. $k = 6$ or $k = -6$

Total3

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Algebra and functions

Not needed for Year 12Medium

Partial fractions · Year 13 · 3 marks

Express $\dfrac{5x + 1}{(x - 1)(x + 2)}$ in partial fractions.
(3)

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Write it as $\dfrac{A}{x - 1}$ ${}+ \dfrac{B}{x + 2}$ and multiply through. $5x + 1 = A(x + 2) + B(x - 1)$
Choose values of $x$ that make one bracket zero. $x = 1$ : $6 = 3A \Rightarrow A = 2$ ; $x = -2$ : $-9 = -3B \Rightarrow B = 3$
$\dfrac{2}{x - 1}$ ${}+ \dfrac{3}{x + 2}$

Total3

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Algebra and functions

Hard

Roots and the discriminant · Year 12 · 5 marks

Find the values of $k$ for which $kx^2 + 4x + (k - 3) = 0$ has two distinct real roots.
(5)

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Two distinct real roots means the discriminant is positive. $b^2 - 4ac > 0$ : $16 - 4k(k - 3) > 0$
Divide by $-4$ , which reverses the inequality. $k^2 - 3k - 4 < 0$
Critical values $-1$ and $4$ ; a ∪ shape is negative between them. $(k - 4)(k + 1) < 0$ , so $-1 < k < 4$
If $k = 0$ the equation is not a quadratic and has only one root. $-1 < k < 4$ , $k \ne 0$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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