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IEB practice · Paper I

IEB Algebra practice

6 newly generated questions across all 8 algebra topics the IEB examines in Paper I, 83 marks in all. Work each one, then open its answer underneath to check yourself. Nothing here is taken from a past paper.

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Exponential equations and exponent laws

2 questions · 28 marks

116 marks
  1. (a)
    Solve for x in terms of y: 5^(x − 4) · 25^y = 5^y
    (3)
  2. (b)
    Solve for x: √(x + 22) = x + 2
    (5)
  3. (c)
    Solve for x: log_3 (x^2 − 2x + 12) = 3
    (4)
  4. (d)
    Solve for x: 5^(2x) − 30(5^x) + 125 = 0
    (4)
Show the answers

(a) Answer: x = 4 − y

  1. 5^(x − 4) · 5^(2y) = 5^(y)
  2. x − 4 + 2y = y
  3. x = 4 − y

(b) Answer: x = 3

  1. x + 22 = x^2 + 4x + 4
  2. x^2 + 3x − 18 = 0
  3. (x − 3)(x + 6) = 0
  4. check x = −6: √(−6 + 22) = 4 but −6 + 2 = −4, so x = −6 is rejected
  5. x = 3

(c) Answer: x = −3 or x = 5

  1. x^2 − 2x + 12 = 3^3
  2. x^2 − 2x − 15 = 0
  3. (x + 3)(x − 5) = 0
  4. x = −3 or x = 5

(d) Answer: x = 1 or x = 2

  1. let t = 5^x: t^2 − 30t + 125 = 0
  2. (t − 5)(t − 25) = 0
  3. 5^x = 5, so x = 1
  4. 5^x = 25, so x = 2
212 marks
  1. (a)
    Solve for x, correct to two decimal places: 3^(3x) = 2
    (3)
  2. (b)
    Solve for x: log_3 (x^2 − 5x + 3) = 1
    (4)
  3. (c)
    Solve for x: x − √(x − 5) = 7
    (5)
Show the answers

(a) Answer: x = 0.21

  1. 3x = log_3 2
  2. x = (log 2)/(3 log 3)
  3. x = 0.21

(b) Answer: x = 0 or x = 5

  1. x^2 − 5x + 3 = 3^1
  2. x^2 − 5x = 0
  3. (x)(x − 5) = 0
  4. x = 0 or x = 5

(c) Answer: x = 9

  1. √(x − 5) = x − 7
  2. x − 5 = x^2 − 14x + 49
  3. x^2 − 15x + 54 = 0
  4. (x − 9)(x − 6) = 0
  5. check x = 6: √(6 − 5) = 1 but 6 − 7 = −1, so x = 6 is rejected
  6. x = 9

Quadratic equations

2 questions · 26 marks

313 marks
  1. (a)
    Solve for x: 2x(x − 4)(3x + 7) = 0
    (3)
  2. (b)
    Solve for x, correct to two decimal places, by completing the square: x^2 + 10x + 5 = 0
    (4)
  3. (c)
    Solve for x: x(x + 1) ≥ 12
    (3)
  4. (d)
    Solve for x in terms of p: x^2 + 2p^2 = −3px
    (3)
Show the answers

(a) Answer: x = −7/3 or x = 0 or x = 4

  1. x = 0 or x − 4 = 0 or 3x + 7 = 0
  2. x = −7/3 or x = 0 or x = 4

(b) Answer: x = −9.47 or x = −0.53

  1. x^2 + 10x = −5
  2. x^2 + 10x + 25 = −5 + 25
  3. (x + 5)^2 = 20
  4. x + 5 = ±√(20)
  5. x = −5 ± √(20)
  6. x = −9.47 or x = −0.53

(c) Answer: x ≤ −4 or x ≥ 3

  1. x^2 + x − 12 ≥ 0
  2. (x + 4)(x − 3) ≥ 0
  3. critical values: x = −4 and x = 3
  4. x ≤ −4 or x ≥ 3

(d) Answer: x = −2p or x = −p

  1. x^2 + 3px + 2p^2 = 0
  2. (x + 2p)(x + p) = 0
  3. x = −2p or x = −p
413 marks
  1. (a)
    Solve for x in terms of p: x^2 − 8p^2 = −2px
    (3)
  2. (b)
    If x^2 − 2x + 20 = (x − m)^2 + p, determine the values of m and p.
    (3)
  3. (c)
    Hence write down the minimum value of x^2 − 2x + 20.
    (1)
  4. (d)
    Solve for x: x^2 ≥ 3x + 18
    (3)
  5. (e)
    For which values of k will the roots of x^2 − 7x + k = 0 be non-real?
    (3)
Show the answers

(a) Answer: x = −4p or x = 2p

  1. x^2 + 2px − 8p^2 = 0
  2. (x + 4p)(x − 2p) = 0
  3. x = −4p or x = 2p

(b) Answer: m = 1 and p = 19

  1. (x − m)^2 + p = x^2 − 2mx + m^2 + p
  2. −2m = −2, so m = 1
  3. m^2 + p = 20, so p = 19

(c) Answer: 19

  1. (x − m)^2 ≥ 0, so the least value is p = 19

(d) Answer: x ≤ −3 or x ≥ 6

  1. x^2 − 3x − 18 ≥ 0
  2. (x + 3)(x − 6) ≥ 0
  3. critical values: x = −3 and x = 6
  4. x ≤ −3 or x ≥ 6

(e) Answer: k > 49/4

  1. non-real roots: Δ < 0
  2. Δ = (−7)^2 − 4(1)(k) = 49 − 4k
  3. 49 − 4k < 0
  4. k > 49/4

Simultaneous equations

2 questions · 29 marks

514 marks
  1. (a)
    Solve for x and y if 5^(x + y) = 25 and 2^(x − y) = 1/16.
    (5)
  2. (b)
    For which values of k will the line y = −3x + k never meet the parabola y = x^2 − x + 5?
    (4)
  3. (c)
    A tank holds 391 litres and is filled at x litres per minute. If the tap delivered 6 litres per minute more, the tank would fill 6 minutes sooner. Determine x.
    (5)
Show the answers

(a) Answer: x = −1, y = 3

  1. 5^(x + y) = 5^(2), so x + y = 2
  2. 2^(x − y) = 2^(−4), so x − y = −4
  3. adding: 2x = −2, so x = −1
  4. y = 2 − (−1) = 3

(b) Answer: k < 4

  1. x^2 − x + 5 = −3x + k
  2. x^2 + 2x + 5 − k = 0
  3. no point of intersection: Δ < 0
  4. (2)^2 − 4(1)(5 − k) < 0
  5. k < 4

(c) Answer: x = 17

  1. 391/x − 391/(x + 6) = 6
  2. 391(x + 6) − 391x = 6x(x + 6)
  3. x^2 + 6x − 391 = 0
  4. (x − 17)(x + 23) = 0
  5. x = 17 (x = −23 is rejected: a rate is positive)
615 marks
  1. (a)
    Solve for x and y if y + 3x = −2 and y = x^2 − 2x − 14.
    (6)
  2. (b)
    For which values of k will the line y = −2x + k never meet the parabola y = x^2 + 2x + 1?
    (4)
  3. (c)
    An amount of R19 800 is shared equally among a group of people. If there were 2 more people in the group, each person would receive R75 less. How many people are in the group?
    (5)
Show the answers

(a) Answer: x = −4, y = 10 or x = 3, y = −11

  1. y = −3x − 2
  2. −3x − 2 = x^2 − 2x − 14
  3. x^2 + x − 12 = 0
  4. (x − 3)(x + 4) = 0
  5. x = −4 or x = 3
  6. y = 10 or y = −11

(b) Answer: k < −3

  1. x^2 + 2x + 1 = −2x + k
  2. x^2 + 4x + 1 − k = 0
  3. no point of intersection: Δ < 0
  4. (4)^2 − 4(1)(1 − k) < 0
  5. k < −3

(c) Answer: 22 people

  1. let there be n people: 19800/n − 19800/(n + 2) = 75
  2. 19800(n + 2) − 19800n = 75n(n + 2)
  3. n^2 + 2n − 528 = 0
  4. (n − 22)(n + 24) = 0
  5. n = 22 (n = −24 is rejected: the number of people is positive)

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Every question on this page was generated for this site by a computer engine, and its answer was confirmed by an independent method at the moment the question was made.

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