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Algebra: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 4 computed question types on this topic.

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Questions

1. Solve for x: 3x − 7 = 11.

  1. 4/3
  2. 6
  3. −6
  4. 18
Show the answer and solution

Answer: B. 6

Add 7 to both sides: 3x = 18, so x = 18/3 = 6.

2. The roots of 2x² − 8x + 3 = 0 are r and s. What is r² + s²?

  1. 16
  2. 13
  3. 10
  4. 19
Show the answer and solution

Answer: B. 13

By Vieta, r + s = 8/2 = 4 and rs = 3/2. Then r² + s² = (r + s)² − 2rs = 16 − 3 = 13.

3. Solve log₂ x + log₂(x − 2) = 3.

  1. x = 4
  2. x = −2
  3. x = 4 or x = −2
  4. x = 5
Show the answer and solution

Answer: A. x = 4

Combine: log₂[x(x − 2)] = 3, so x² − 2x − 8 = 0 and x = 4 or −2. x = −2 is rejected because the logarithm of a negative number is undefined.

4. What is the sum of the first 20 positive odd integers?

  1. 420
  2. 380
  3. 400
  4. 210
Show the answer and solution

Answer: C. 400

The sum of the first n odd integers is n², so 20² = 400.

5. Find the sum of the infinite geometric series 8 + 4 + 2 + 1 + …

  1. 16
  2. 15
  3. 12
  4. 32
Show the answer and solution

Answer: A. 16

a = 8 and r = 1/2, so S = a/(1 − r) = 8/0.5 = 16.

6. What is the coefficient of x³ in the expansion of (2x − 1)⁵?

  1. −80
  2. 40
  3. 160
  4. 80
Show the answer and solution

Answer: D. 80

The x³ term is C(5,3)(2x)³(−1)² = 10 × 8x³ × 1 = 80x³.

7. Pipe A fills a tank in 6 h and pipe B fills it in 9 h, while a drain empties the full tank in 12 h. With both pipes and the drain open, how long does it take to fill the empty tank?

  1. 2.77 h
  2. 3.60 h
  3. 9.00 h
  4. 5.14 h
Show the answer and solution

Answer: D. 5.14 h

Net rate = 1/6 + 1/9 − 1/12 = (6 + 4 − 3)/36 = 7/36 tank per hour, so t = 36/7 = 5.14 h. Adding the drain rate instead of subtracting it gives 2.77 h.

8. How many litres of pure water must be added to 20 L of a 30% acid solution to dilute it to 12% acid?

  1. 50 L
  2. 12 L
  3. 30 L
  4. 20 L
Show the answer and solution

Answer: C. 30 L

The acid stays at 0.30 × 20 = 6 L. Final volume = 6/0.12 = 50 L, so add 50 − 20 = 30 L of water.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. What is the larger root of x² − 4x − 21 = 0?

  1. 7
  2. -3
  3. -7
  4. -4
Show the answer and solution

Answer: A. 7

x² − (sum)x + (product): the roots multiply to -21 and add to 4, so they are -3 and 7; the larger is 7.

10. An arithmetic progression starts at 11 and increases by 4. What is the sum of its first 33 terms?

  1. 4587
  2. 139
  3. 2541
  4. 2475
Show the answer and solution

Answer: D. 2475

S = n/2·[2a + (n − 1)d] = 33/2 × [22 + 32×4] = 2475.

11. Find the sum of the infinite geometric series whose first term is 8 and common ratio is 0.4.

  1. 11.2
  2. 5.714
  3. 20
  4. 13.33
Show the answer and solution

Answer: D. 13.33

For |r| < 1 the sum is a/(1 − r) = 8/(1 − 0.4) = 13.33.

12. Solve 3^x = 279 for x.

  1. 5.631
  2. 93
  3. 7.337
  4. 5.126
Show the answer and solution

Answer: D. 5.126

x = log(279)/log(3) = 5.126.

Other topics in the same subjects

Trigonometry · Analytic geometry · Differential calculus · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

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