Mathematics · AQA 7357 · Topic B

Algebra and functions

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

  • 11 sub-topics
  • 6 in Year 12, 3 in Year 13
  • 193 practice questions
  • 11 animated lessons
Practice questions (193) Lessons Worksheets All AQA Mathematics topics

What the specification says

Every sub-topic AQA 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. B1Laws of indices for all rational exponentsY12
  2. B2Surds, including rationalising the denominatorY12
  3. B3Quadratic functions, the discriminant and completing the squareY12
  4. B4Simultaneous equations, one linear and one quadraticY12
  5. B5Linear and quadratic inequalitiesY12
  6. B6Polynomials, the factor theorem and algebraic divisionY12
  7. B7Graphs of functions, including modulus graphs and asymptotesY13
  8. B8Composite and inverse functionsY13
  9. B9Transformations of y = f(x), including combinationsY13
  10. B10Partial fractionsY13
  11. B11Functions in modellingY13

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

  1. The equation x2+kx+9=0x^2 + kx + 9 = 0 has equal roots. Find the possible values of kk.
    (3)
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  1. Equal (repeated) roots happen when the discriminant is zero.b2−4ac=0b^2 - 4ac = 0
  2. Here a=1a = 1, b=kb = k and c=9c = 9.k2−4×1×9k^2 - 4 \times 1 \times 9=0{}= 0⇒k2{}\Rightarrow k^2=36{}= 36
  3. Square-rooting gives two answers: positive and negative.k=6k = 6 or k=−6k = -6

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Algebra and functions

Not needed for Year 12Medium

Partial fractions · Year 13 · 3 marks

  1. Express 5x+1(x−1)(x+2)\dfrac{5x + 1}{(x - 1)(x + 2)} in partial fractions.
    (3)
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  1. Write it as Ax−1\dfrac{A}{x - 1}+Bx+2{}+ \dfrac{B}{x + 2} and multiply through.5x+1=A(x+2)+B(x−1)5x + 1 = A(x + 2) + B(x - 1)
  2. Choose values of xx that make one bracket zero.x=1x = 1: 6=3A⇒A=26 = 3A \Rightarrow A = 2; x=−2x = -2: −9=−3B⇒B=3-9 = -3B \Rightarrow B = 3
  3. 2x−1\dfrac{2}{x - 1}+3x+2{}+ \dfrac{3}{x + 2}

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Algebra and functions

Hard

Roots and the discriminant · Year 12 · 5 marks

  1. Find the values of kk for which kx2+4x+(k−3)=0kx^2 + 4x + (k - 3) = 0 has two distinct real roots.
    (5)
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  1. Two distinct real roots means the discriminant is positive.b2−4ac>0b^2 - 4ac > 0: 16−4k(k−3)>016 - 4k(k - 3) > 0
  2. Divide by −4-4, which reverses the inequality.k2−3k−4<0k^2 - 3k - 4 < 0
  3. Critical values −1-1 and 44; a ∪ shape is negative between them.(k−4)(k+1)<0(k - 4)(k + 1) < 0, so −1<k<4-1 < k < 4
  4. If k=0k = 0 the equation is not a quadratic and has only one root.−1<k<4-1 < k < 4, k≠0k \ne 0

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