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Botswana BGCSE General Mathematics 1236 study materials
Botswana BGCSE General Mathematics 1236 study resources

Botswana BGCSE General Mathematics 1236 practice questions

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Question 1Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out 3/4 + 5/6, giving your answer as a mixed number in its simplest form.

  1. 8/10
  2. 1 1/12
  3. 19/24
  4. 1 7/12 Correct answer.

Explanation

Rewrite both fractions with the lowest common denominator, 12: 3/4 = 9/12 and 5/6 = 10/12. Adding the numerators gives 19/12, which is 1 7/12 as a mixed number, and 7 and 12 share no common factor so it is already simplest.

Question 2Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Write 0.000 042 in standard form.

  1. 4.2 × 10^-6
  2. 4.2 × 10^-5 Correct answer.
  3. 4.2 × 10^5
  4. 4.2 × 10^-4

Explanation

Standard form is A × 10^n where A lies between 1 and 10. Moving the decimal point in 0.000 042 five places to the right gives 4.2, so the power of ten must be negative five.

Question 3Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Express 45 minutes as a fraction of 2 hours, giving the fraction in its simplest form.

  1. 3/8 Correct answer.
  2. 2/3
  3. 1/2
  4. 1/4

Explanation

Both quantities must be in the same unit, and 2 hours is 120 minutes. The fraction is therefore 45/120, and dividing the top and bottom by 15 gives 3/8.

Question 4Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Simplify (2x^3)^2 ÷ x^4.

  1. 4x^2 Correct answer.
  2. 4x^6
  3. 4x^10
  4. 2x^2

Explanation

Square every factor inside the bracket: (2x^3)^2 = 4x^6, since 2^2 = 4 and (x^3)^2 = x^6. Dividing by x^4 subtracts the indices, 6 − 4 = 2, giving 4x^2.

Question 5Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

P450 is shared among three people in the ratio 2 : 3 : 4. How much does the person with the largest share receive?

  1. P100
  2. P200 Correct answer.
  3. P150
  4. P225

Explanation

The ratio contains 2 + 3 + 4 = 9 equal parts, so one part is P450 ÷ 9 = P50. The largest share is the 4-part share, which is 4 × P50 = P200.

Question 6Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Express the recurring decimal 0.454545... as a fraction in its simplest form.

  1. 45/100
  2. 9/20
  3. 5/11 Correct answer.
  4. 9/22

Explanation

Let x = 0.454545..., so 100x = 45.454545.... Subtracting x from 100x gives 99x = 45, hence x = 45/99, which simplifies to 5/11 after dividing both parts by 9.

Question 7Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out (3 × 10^5) × (4 × 10^-3), giving your answer in standard form.

  1. 1.2 × 10^8
  2. 1.2 × 10^2
  3. 1.2 × 10^3 Correct answer.
  4. 12 × 10^3

Explanation

Multiply the number parts, 3 × 4 = 12, and add the indices, 5 + (−3) = 2, giving 12 × 10^2. To reach standard form, rewrite 12 as 1.2 × 10^1, so the answer is 1.2 × 10^3.

Question 8Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Solve the equation 4^(x+1) = 8^(x−1).

  1. x = −1
  2. x = 2
  3. x = 5 Correct answer.
  4. x = 1

Explanation

Write both sides with base 2: 4^(x+1) = 2^(2x+2) and 8^(x−1) = 2^(3x−3). Since the bases now match, equate the indices: 2x + 2 = 3x − 3, which gives x = 5.

Question 9Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

A car bought for P60 000 depreciates by 15% of its value each year. What is its value after 2 years?

  1. P51 000
  2. P39 000
  3. P43 350 Correct answer.
  4. P42 000

Explanation

Each year the car retains 85% of its value, so the value is multiplied by 0.85 once per year. After two years the value is 60 000 × 0.85² = 60 000 × 0.7225 = P43 350.

Question 10Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Five workers take 12 days to complete a job. Working at the same rate, how long would eight workers take to complete the same job?

  1. 4.8 days
  2. 19.2 days
  3. 8 days
  4. 7.5 days Correct answer.

Explanation

Workers and time are inversely proportional, so the total amount of work stays fixed at 5 × 12 = 60 worker-days. Sharing those 60 worker-days among 8 workers gives 60 ÷ 8 = 7.5 days.

Question 11Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Round 8.476 to 2 decimal places.

  1. 8.50
  2. 8.46
  3. 8.47
  4. 8.48 Correct answer.

Explanation

The third decimal digit is 6, so the second decimal digit 7 is rounded up. Replacing the later digits by zero gives 8.48.

Question 12Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Express 3/8 as a decimal.

  1. 0.3
  2. 0.375 Correct answer.
  3. 0.38
  4. 0.83

Explanation

Divide the numerator 3 by the denominator 8. Since 3 ÷ 8 = 0.375, the decimal terminates after three places.

Question 13Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Find 15% of P240.

  1. P48
  2. P36 Correct answer.
  3. P16
  4. P24

Explanation

Write 15% as 15/100 and multiply by 240. 0.15 × 240 = 36, so 15% of P240 is P36.

Question 14Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

P2 500 is invested at 6% simple interest per year. How much interest is earned in 3 years?

  1. P750
  2. P150
  3. P450 Correct answer.
  4. P2 950

Explanation

Simple interest is principal × rate × time. Thus I = 2 500 × 0.06 × 3 = P450.

Question 15Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

A map has a scale of 1 : 50 000. A road is 6 cm long on the map. What is its actual length in kilometres?

  1. 0.3 km
  2. 30 km
  3. 3 km Correct answer.
  4. 300 km

Explanation

Scale 1 : 50 000 means 1 cm represents 50 000 cm. So 6 cm represents 300 000 cm; dividing by 100 000 gives 3 km.

Question 16Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out (6 × 10^8) ÷ (1.5 × 10^3), giving your answer in standard form.

  1. 4 × 10^11
  2. 4 × 10^5 Correct answer.
  3. 4 × 10^24
  4. 4.5 × 10^5

Explanation

Divide the coefficients and subtract the indices: 6 ÷ 1.5 = 4 and 10^8 ÷ 10^3 = 10^5. So the answer is 4 × 10^5.

Question 17Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Express 360 as a product of its prime factors in index form.

  1. 2^3 × 3 × 15
  2. 2^2 × 3^2 × 10
  3. 2^4 × 3 × 5
  4. 2^3 × 3^2 × 5 Correct answer.

Explanation

Dividing 360 repeatedly by primes gives 360 = 2 × 2 × 2 × 3 × 3 × 5. Grouping equal factors gives 2^3 × 3^2 × 5.

Question 18Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out 2/3 ÷ 4/9 + 1/2, giving your answer in its simplest form.

  1. 2 Correct answer.
  2. 1 1/2
  3. 2 1/2
  4. 7/6

Explanation

First divide by multiplying by the reciprocal: 2/3 × 9/4 = 3/2. Then add 1/2 to get 2.

Question 19Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Evaluate (27/8)^(−2/3).

  1. −4/9
  2. 4/9 Correct answer.
  3. 9/4
  4. 8/27

Explanation

A negative exponent gives the reciprocal, and the 1/3 power takes a cube root: (27/8)^(−2/3) = (8/27)^(2/3). Then (∛(8/27))^2 = (2/3)^2 = 4/9.

Question 20Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

A town's population is 2 500. It increases by 8% each year. What is the population after 2 years?

  1. 2 900
  2. 2 700
  3. 3 000
  4. 2 916 Correct answer.

Explanation

Each year's population is multiplied by 1.08. After two years, 2 500 × 1.08^2 = 2 500 × 1.1664 = 2 916.

Question 21Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Which of the following lists the numbers 0.7, 0.07, 0.707 and 0.77 in ascending order?

  1. 0.7, 0.07, 0.707, 0.77
  2. 0.07, 0.707, 0.7, 0.77
  3. 0.07, 0.7, 0.77, 0.707
  4. 0.07, 0.7, 0.707, 0.77 Correct answer.

Explanation

Compare place values from left to right. Only 0.07 has 0 tenths, so it is smallest; the rest all start with 7 tenths. Writing 0.7 as 0.700 shows it is less than 0.707, and 0.707 is less than 0.77 because 0 hundredths is smaller than 7 hundredths.

Question 22Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Write the fraction 7/20 as a percentage.

  1. 28%
  2. 70%
  3. 14%
  4. 35% Correct answer.

Explanation

To change a fraction into a percentage, multiply it by 100. So 7/20 × 100 = 700/20 = 35, giving 35%. Equivalently, 7/20 = 0.35 = 35%.

Question 23Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out the value of 2^(−3).

  1. −8
  2. −6
  3. 1/8
  4. 1/6 Correct answer.

Explanation

A negative index means take the reciprocal of the positive power: 2^(−3) = 1/2^3. Since 2^3 = 8, the value is 1/8, not a negative number.

Question 24Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Money is shared between two people in the ratio 5 : 3. The larger share is P240 more than the smaller share. What is the total amount shared?

  1. P1 920
  2. P960 Correct answer.
  3. P640
  4. P1 200

Explanation

The difference between the shares is 5 − 3 = 2 ratio parts, and these equal P240, so one part is P120. The total is 5 + 3 = 8 parts, giving 8 × P120 = P960.

Question 25Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

A trader buys a radio for P450 and sells it for P540. What is the percentage profit on the cost price?

  1. 20% Correct answer.
  2. 25%
  3. 90%
  4. 16.7%

Explanation

Profit = P540 − P450 = P90. Percentage profit is always found on the cost price, so it is 90/450 × 100 = 20%.

Question 26Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out (4.5 × 10^5) + (3.5 × 10^4), giving your answer in standard form.

  1. 4.85 × 10^5 Correct answer.
  2. 8.0 × 10^5
  3. 7.75 × 10^5
  4. 4.85 × 10^4

Explanation

Write both numbers with the same power of ten: 3.5 × 10^4 = 0.35 × 10^5. Then 4.5 × 10^5 + 0.35 × 10^5 = 4.85 × 10^5, which is already in standard form because 4.85 lies between 1 and 10.

Question 27Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

A farmer has 250 goats. He sells 3/5 of them and then sells 1/4 of the goats that remain. How many goats are left?

  1. 60
  2. 25
  3. 75 Correct answer.
  4. 100

Explanation

First sale: 3/5 × 250 = 150 goats, leaving 100. Second sale: 1/4 × 100 = 25 goats, leaving 100 − 25 = 75 goats. Each fraction must be applied to the amount remaining at that stage.

Question 28Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

After a 15% increase, a worker's monthly wage became P4 830. What was the monthly wage before the increase?

  1. P4 350
  2. P4 200 Correct answer.
  3. P5 550
  4. P4 100

Explanation

An increase of 15% means the new wage is 115% of the old wage. Divide by 1.15: 4 830 ÷ 1.15 = 4 200, so the original wage was P4 200. Check: 15% of 4 200 is 630, and 4 200 + 630 = 4 830.

Question 29Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Simplify (16x^8)^(3/4) ÷ (2x^2).

  1. 8x^4
  2. 4x^4 Correct answer.
  3. 4x^6
  4. 2x^4

Explanation

Apply the power 3/4 to each factor: 16^(3/4) = (2^4)^(3/4) = 2^3 = 8 and (x^8)^(3/4) = x^6. So the expression becomes 8x^6 ÷ 2x^2 = 4x^4, dividing coefficients and subtracting indices.

Question 30Number and Numeration (including fractions, decimals, percentages, ratio, proportion, indices and standard form)

Work out (2.4 × 10^(−3))^2, giving your answer in standard form.

  1. 5.76 × 10^(−5)
  2. 5.76 × 10^(−6) Correct answer.
  3. 4.8 × 10^(−6)
  4. 4.8 × 10^(−5)

Explanation

Square each part separately: 2.4^2 = 5.76 and (10^(−3))^2 = 10^(−6) because indices are multiplied. The result is 5.76 × 10^(−6), which is valid standard form since 5.76 lies between 1 and 10.

Question 31Sets, Functions and Relations

The set A is written as A = {2, 4, 6, 8, 10}. Which one of the following statements is true?

  1. A has exactly four elements
  2. 4 ∉ A
  3. 10 ∉ A
  4. 6 ∈ A Correct answer.

Explanation

The symbol ∈ means 'is an element of', so 6 ∈ A is true because 6 appears in the list inside the braces. The symbol ∉ means the opposite, so saying 4 ∉ A or 10 ∉ A is false since both numbers are listed. Counting the entries shows that A has five elements, not four.

Question 32Sets, Functions and Relations

Given that P = {1, 3, 5, 7} and Q = {3, 5, 9}, which set represents P ∩ Q?

  1. { }
  2. {1, 3, 5, 7, 9}
  3. {1, 7, 9}
  4. {3, 5} Correct answer.

Explanation

The intersection P ∩ Q contains only the elements that appear in both P and Q. Comparing the two lists, the shared members are 3 and 5, so P ∩ Q = {3, 5}. Elements such as 1, 7 and 9 belong to only one of the sets and are therefore excluded.

Question 33Sets, Functions and Relations

A function is defined by f(x) = 3x − 2. What is the value of f(5)?

  1. 9
  2. 12
  3. 17
  4. 13 Correct answer.

Explanation

To evaluate f(5), substitute x = 5 into the rule: f(5) = 3(5) − 2. Multiplying gives 15, and subtracting 2 gives 13. The order of operations requires the multiplication to be done before the subtraction.

Question 34Sets, Functions and Relations

The universal set is ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}. Which set represents (A ∪ B)′?

  1. {3, 5, 7}
  2. {4, 6, 8, 10} Correct answer.
  3. {4, 6, 8}
  4. {1, 2, 9, 10}

Explanation

First form the union: A ∪ B = {1, 2, 3, 5, 7, 9}. The complement then contains every member of the universal set that is missing from this union. Those members are 4, 6, 8 and 10, so (A ∪ B)′ = {4, 6, 8, 10}.

Question 35Sets, Functions and Relations

The set S is written as S = {a, b, c, d}. How many subsets does S have altogether?

  1. 8
  2. 4
  3. 32
  4. 16 Correct answer.

Explanation

A set with n elements has 2ⁿ subsets, because each element is either included in a subset or left out. Here n = 4, so the number of subsets is 2⁴ = 16. These sixteen subsets include the empty set and the whole set S itself.

Question 36Sets, Functions and Relations

In a class of 30 students, 18 study French, 15 study Spanish and 6 study both languages. How many students study neither French nor Spanish?

  1. 0
  2. 3 Correct answer.
  3. 6
  4. 9

Explanation

Adding the two groups double-counts the 6 students who study both, so the number studying at least one language is 18 + 15 − 6 = 27. The students left over are 30 − 27 = 3. Hence 3 students study neither language.

Question 37Sets, Functions and Relations

Two functions are defined by f(x) = 2x + 1 and g(x) = x². What is f(g(x))?

  1. 2x² + 1 Correct answer.
  2. (2x + 1)²
  3. 2x² + 2x + 1
  4. 4x² + 1

Explanation

The notation f(g(x)) means that g is applied first and its output is fed into f. Since g(x) = x², substituting this into f gives f(x²) = 2(x²) + 1 = 2x² + 1. The squaring applies only to x, not to the whole expression 2x + 1.

Question 38Sets, Functions and Relations

The function f is defined by f(x) = 4x − 8. Which expression represents the inverse function f⁻¹(x)?

  1. (x − 8)/4
  2. (x + 8)/4 Correct answer.
  3. 1/(4x − 8)
  4. 4x + 8

Explanation

Write y = 4x − 8 and rearrange to make x the subject: add 8 to both sides and divide by 4, giving x = (y + 8)/4. Replacing y with x gives f⁻¹(x) = (x + 8)/4. Checking with x = 1 confirms that f(1) = −4 and f⁻¹(−4) = 1.

Question 39Sets, Functions and Relations

In a survey of 40 students, 20 like Mathematics, 18 like Physics, 15 like Chemistry, 8 like Mathematics and Physics, 6 like Mathematics and Chemistry, 5 like Physics and Chemistry, and 3 like all three subjects. How many students like none of the three subjects?

  1. 4
  2. 3 Correct answer.
  3. 8
  4. 6

Explanation

By inclusion–exclusion the number liking at least one subject is 20 + 18 + 15 − 8 − 6 − 5 + 3 = 37. The students liking none are therefore 40 − 37 = 3. The 3 who like all three subjects must be added back because they were removed three times in the subtractions.

Question 40Sets, Functions and Relations

The function f is defined by f(x) = 1/√(x − 5). For which values of x is f defined?

  1. x ≥ 0
  2. x ≥ 5
  3. x > 5 Correct answer.
  4. x ≠ 5

Explanation

The expression under the square root must not be negative, so x − 5 ≥ 0, that is x ≥ 5. Because the root also appears in the denominator, its value cannot be zero, which rules out x = 5. Combining the two conditions gives x > 5.

Question 41Sets, Functions and Relations

The set {3, 6, 9, 12} is described exactly by which set-builder notation?

  1. {n² : n ∈ {1, 2, 3}}
  2. {3n : n ∈ {1, 2, 3, 4}} Correct answer.
  3. {n : n is even and n < 13}
  4. {3n : n ∈ ℕ, n ≤ 5}

Explanation

The notation {3n : n ∈ {1, 2, 3, 4}} lists 3×1, 3×2, 3×3 and 3×4, giving 3, 6, 9 and 12 exactly. The other options either include extra values such as 15 or describe a different collection of numbers.

Question 42Sets, Functions and Relations

Which set of ordered pairs represents a function?

  1. {(1, 2), (2, 3), (1, 4), (3, 5)}
  2. {(1, 2), (1, 3), (2, 4), (3, 5)}
  3. {(1, 2), (2, 2), (2, 3), (3, 4)}
  4. {(1, 2), (2, 3), (3, 4), (4, 5)} Correct answer.

Explanation

A function assigns exactly one second coordinate to each first coordinate. In the correct set, the first coordinates 1, 2, 3 and 4 each appear only once, so every input has a single output.

Question 43Sets, Functions and Relations

The universal set is ξ = {a, b, c, d, e, f} and A = {a, c, f}. Which set represents the complement A′?

  1. {b, d, e} Correct answer.
  2. {a, c, f}
  3. {a, b, c, d, e, f}
  4. {b, c, d}

Explanation

The complement of A contains every element of the universal set that is not in A. Removing a, c and f from {a, b, c, d, e, f} leaves b, d and e.

Question 44Sets, Functions and Relations

Given A = {1, 2, 3, 4}, B = {3, 4, 5} and C = {2, 4, 6}, which set represents (A ∪ B) ∩ C?

  1. {2, 3, 4}
  2. {2, 4} Correct answer.
  3. {4}
  4. {1, 2, 3, 4, 5, 6}

Explanation

First form A ∪ B = {1, 2, 3, 4, 5}. Then intersect this with C = {2, 4, 6}; only 2 and 4 occur in both.

Question 45Sets, Functions and Relations

How many subsets of the set {p, q, r, s, t} contain the element r?

  1. 16 Correct answer.
  2. 32
  3. 24
  4. 8

Explanation

A subset containing r is determined by choosing, for each of the other four elements, whether to include it or not. This gives 2^4 = 16 such subsets.

Question 46Sets, Functions and Relations

The function f is defined by f(x) = (x + 1)/(x − 3). What is the domain of f?

  1. all real numbers except 3 Correct answer.
  2. all real numbers except −1
  3. all real numbers except 0
  4. all real numbers

Explanation

A rational function is undefined where its denominator equals zero. Here x − 3 = 0 when x = 3, so every real number except 3 is allowed.

Question 47Sets, Functions and Relations

Two functions are defined by f(x) = x + 4 and g(x) = 3x. What is the value of g(f(2))?

  1. 18 Correct answer.
  2. 10
  3. 30
  4. 12

Explanation

Work from the inside out: f(2) = 2 + 4 = 6. Then g(f(2)) = g(6) = 3 × 6 = 18.

Question 48Sets, Functions and Relations

In a class of 40 students, 25 study History, 20 study Geography, and every student studies at least one of the two subjects. How many students study both subjects?

  1. 10
  2. 45
  3. 15
  4. 5 Correct answer.

Explanation

Using inclusion–exclusion, n(H ∪ G) = n(H) + n(G) − n(H ∩ G). Since everyone studies at least one subject, 40 = 25 + 20 − x, so x = 5.

Question 49Sets, Functions and Relations

The function f is defined by f(x) = x² − 4 for x ≥ 0. Which expression represents the inverse function f⁻¹(x)?

  1. x² + 4, x ≥ 0
  2. √(x − 4), x ≥ 4
  3. ±√(x + 4), x ≥ −4
  4. √(x + 4), x ≥ −4 Correct answer.

Explanation

Write y = x² − 4 and solve for x: x² = y + 4. Since the domain of f is x ≥ 0, take the positive root x = √(y + 4), with y ≥ −4. Thus f⁻¹(x) = √(x + 4), x ≥ −4.

Question 50Sets, Functions and Relations

For sets X and Y, the symmetric difference is X △ Y = (X ∪ Y) \ (X ∩ Y). If n(X) = 12, n(Y) = 9 and n(X ∩ Y) = 4, what is n(X △ Y)?

  1. 13 Correct answer.
  2. 5
  3. 17
  4. 21

Explanation

First find the union: n(X ∪ Y) = 12 + 9 − 4 = 17. The symmetric difference removes the 4 common elements, so n(X △ Y) = 17 − 4 = 13.