I85PREP
⌘ K
GTLE Mathematics Specialism study materials
GTLE Mathematics Specialism study resources

GTLE Mathematics Specialism practice questions

One question at a time. Build your GTLE Mathematics Specialism understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Number and Quantity

In the real number system, which statement best describes the relationship between the rational numbers and the irrational numbers?

  1. The two sets overlap, because some real numbers can be written both ways.
  2. The irrationals are countable while the rationals are uncountable.
  3. Together they partition the real numbers, since every real number is exactly one of the two. Correct answer.
  4. Every real number is rational, and irrational numbers are a subset of the rationals.

Explanation

A real number is rational if it can be written as a ratio of integers, and irrational otherwise; these conditions are mutually exclusive and exhaustive. Hence the rationals and irrationals form a partition of the reals with no overlap and no gaps.

Question 2Number and Quantity

Which of the following numbers is irrational?

  1. 0.121221222... continuing with ever-longer runs of 2s Correct answer.
  2. 0.75
  3. the fraction 22/7
  4. the repeating decimal 0.333...

Explanation

A decimal represents a rational number exactly when it terminates or eventually repeats a fixed block. The decimal with ever-longer runs of 2s never settles into a repeating block, so it is irrational, whereas the other three terminate or repeat.

Question 3Number and Quantity

Evaluate the expression 2^5 - 3 x 2^3 + 4^0, using the standard order of operations.

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

Explanation

Powers are evaluated first: 2^5 = 32, 3 x 2^3 = 3 x 8 = 24, and 4^0 = 1. Then 32 - 24 + 1 = 9, working left to right for the addition and subtraction.

Question 4Number and Quantity

Which statement correctly describes the greatest common divisor of two positive integers a and b?

  1. It is the largest positive integer that divides both a and b with no remainder. Correct answer.
  2. It is the product of a and b divided by their difference.
  3. It is the largest positive integer that neither a nor b divides.
  4. It is the smallest positive integer that both a and b divide with no remainder.

Explanation

The greatest common divisor is defined as the largest positive integer that is a divisor of both numbers. Equivalently it is the product of the shared prime factors taken to their lowest exponents.

Question 5Number and Quantity

A quantity is given as 4.70 x 10^3 metres. Expressed in standard decimal notation, this is:

  1. 470 metres
  2. 0.00470 metres
  3. 4700 metres Correct answer.
  4. 47000 metres

Explanation

Multiplying by 10^3 shifts the decimal point three places to the right, so 4.70 x 10^3 = 4700. The trailing zero in 4.70 signals that the value is known to three significant figures.

Question 6Number and Quantity

Which of the following is a prime number?

  1. 99
  2. 91
  3. 97 Correct answer.
  4. 87

Explanation

A prime has exactly two positive divisors, 1 and itself. Testing primes up to the square root of 97 (about 9.8) shows no divisor among 2, 3, 5, and 7, so 97 is prime.

Question 7Number and Quantity

The number 0.000000045 written in scientific notation is:

  1. 4.5 x 10^-8 Correct answer.
  2. 4.5 x 10^8
  3. 4.5 x 10^-7
  4. 45 x 10^-9

Explanation

Scientific notation places one nonzero digit before the decimal point, so the coefficient is 4.5. Moving the decimal eight places to reach 4.5 requires the exponent -8, giving 4.5 x 10^-8.

Question 8Number and Quantity

Which set of numbers is closed under subtraction?

  1. The integers Correct answer.
  2. The irrational numbers
  3. The natural numbers
  4. The positive rational numbers

Explanation

A set is closed under subtraction if the difference of any two of its elements is again in the set. The difference of two integers is always an integer, so the integers are closed.

Question 9Number and Quantity

Simplify the expression (3 x 10^4) x (2 x 10^-6), leaving the answer in scientific notation.

  1. 6 x 10^-24
  2. 6 x 10^-2 Correct answer.
  3. 5 x 10^-2
  4. 6 x 10^-10

Explanation

Multiply the coefficients 3 x 2 = 6 and add the exponents 4 + (-6) = -2. The result is 6 x 10^-2, already with a coefficient between 1 and 10.

Question 10Number and Quantity

Simplify (2^3 x 2^-5)^2 expressed as a single power of 2.

  1. 2^-1
  2. 2^-16
  3. 2^4
  4. 2^-4 Correct answer.

Explanation

Inside the brackets, add the exponents: 3 + (-5) = -2, so the base is 2^-2. Raising a power to a power multiplies exponents, giving 2^(-2 x 2) = 2^-4.

Question 11Number and Quantity

A number n leaves a remainder of 3 when divided by 7. What is the remainder when 4n is divided by 7?

  1. 12
  2. 1
  3. 5 Correct answer.
  4. 4

Explanation

Write n = 7k + 3. Then 4n = 28k + 12 = 7(4k + 1) + 5, so the remainder on division by 7 is 5. Equivalently, 4 x 3 = 12 leaves remainder 5 modulo 7.

Question 12Number and Quantity

Which of the following is the prime factorisation of 360?

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

Explanation

A prime factorisation uses only primes. Since 360 = 8 x 9 x 5 = 2^3 x 3^2 x 5, and all factors are prime, this is the correct factorisation.

Question 13Number and Quantity

The least common multiple of 18 and 24 is:

  1. 432
  2. 6
  3. 144
  4. 72 Correct answer.

Explanation

Factorising gives 18 = 2 x 3^2 and 24 = 2^3 x 3. The lcm takes the highest power of each prime: 2^3 x 3^2 = 72.

Question 14Number and Quantity

Simplify the expression (sqrt(50) + sqrt(18)) / sqrt(2).

  1. 8 Correct answer.
  2. 16
  3. 4
  4. sqrt(34)

Explanation

Express each root in simplest form: sqrt(50) = 5sqrt(2) and sqrt(18) = 3sqrt(2). The numerator is 8sqrt(2), which divided by sqrt(2) gives 8.

Question 15Number and Quantity

Which statement about the decimal expansion of a rational number in lowest terms is correct?

  1. It terminates exactly when the denominator has no prime factors other than 2 and 5. Correct answer.
  2. It terminates exactly when the denominator is prime.
  3. It always repeats with a period equal to the denominator.
  4. It terminates exactly when the numerator is even.

Explanation

Dividing by 2s and 5s corresponds to multiplying numerator and denominator by powers of 10, which produces a terminating decimal. Any other prime factor in the denominator forces a repeating expansion.

Question 16Number and Quantity

The number 0.9 recurring, written 0.999..., is equal to which value?

  1. 1 Correct answer.
  2. A value just less than 1, differing from it by an infinitesimal amount
  3. 9/10
  4. 0.9

Explanation

Let x = 0.999..., so 10x = 9.999.... Subtracting gives 9x = 9, hence x = 1. The repeating decimal is simply another representation of the number 1.

Question 17Number and Quantity

What is the complex conjugate of the number 4 - 3i, and what is its product with the original number?

  1. 4 - 3i, with product 7
  2. 4 + 3i, with product 7
  3. -4 + 3i, with product 25
  4. 4 + 3i, with product 25 Correct answer.

Explanation

The conjugate reverses the sign of the imaginary part, giving 4 + 3i. Their product is 4^2 + 3^2 = 16 + 9 = 25, since the cross terms cancel.

Question 18Number and Quantity

Which of these numbers, when written in the form a + bi with a and b real, is purely imaginary?

  1. the square root of 9
  2. the square root of -9 Correct answer.
  3. the sum 2 + the square root of -4
  4. the cube of the imaginary unit i

Explanation

The square root of -9 equals 3i, which has real part 0 and is therefore purely imaginary. A number is purely imaginary exactly when its real part is zero.

Question 19Number and Quantity

A quantity is measured as 6.0 x 10^3 with two significant figures. Which statement about its precision is correct?

  1. It is known to the nearest thousand, so the true value lies within about 500 of 6000.
  2. It is known to the nearest hundred, so the true value lies within about 50 of 6000. Correct answer.
  3. It is known to the nearest unit, so the true value lies within about 0.5 of 6000.
  4. It has unlimited precision because the trailing zeros are placeholders.

Explanation

With two significant figures the last reliable digit is the hundreds digit, so the uncertainty is roughly half of 100, about 50. The trailing zeros locate the magnitude but are not themselves significant.

Question 20Number and Quantity

Which statement about the set of irrational numbers is true?

  1. The sum of two irrational numbers is always irrational.
  2. The product of two irrational numbers is always irrational.
  3. Every irrational number is the square root of a rational number.
  4. The sum of two irrational numbers can be rational, for example the square root of 2 plus its negative. Correct answer.

Explanation

Adding an irrational number to its additive inverse gives 0, which is rational, so the irrationals are not closed under addition. Similar counterexamples exist for products, such as the square root of 2 times itself.

Question 21Number and Quantity

A positive integer n is divisible by 6 and by 10. Which of the following must also divide n?

  1. 30 Correct answer.
  2. 15
  3. 60
  4. 90

Explanation

Divisibility by 6 and 10 means n is a multiple of the least common multiple of 6 and 10, which is 30. Hence 30 divides n, while larger or non-dividing values need not.

Question 22Number and Quantity

Simplify the expression (sqrt(3) + sqrt(2))^2 - 2sqrt(6).

  1. 1
  2. 5 + 4sqrt(6)
  3. 5 Correct answer.
  4. 5 - 4sqrt(6)

Explanation

Expanding gives 3 + 2sqrt(6) + 2 = 5 + 2sqrt(6). Subtracting 2sqrt(6) leaves 5, since the cross term cancels.

Question 23Number and Quantity

By the Euclidean algorithm, the greatest common divisor of 252 and 105 is found. What is that value?

  1. 105
  2. 7
  3. 3
  4. 21 Correct answer.

Explanation

Compute 252 = 2 x 105 + 42, then 105 = 2 x 42 + 21, then 42 = 2 x 21 + 0. The last nonzero remainder is 21, so the gcd is 21.

Question 24Number and Quantity

The number 2^n - 1 is prime for a positive integer n greater than 1. Which conclusion follows?

  1. n must be even.
  2. n must be divisible by 3.
  3. n must be a power of 2.
  4. n must itself be prime. Correct answer.

Explanation

If n = ab with a and b greater than 1, then 2^n - 1 factors as (2^a - 1) times a sum of powers, so it is composite. Hence primality of 2^n - 1 forces n to be prime, though the converse fails for n = 11.

Question 25Number and Quantity

Consider the repeating decimal x = 0.272727..., whose repeating block is 27. Expressed as a fraction in lowest terms, x equals:

  1. 27/99
  2. 2/7
  3. 27/100
  4. 3/11 Correct answer.

Explanation

For a two-digit repeating block, x = 27/99, which reduces by dividing numerator and denominator by 9 to give 3/11. The fraction must be in lowest terms as required.

Question 26Number and Quantity

In the real numbers, the expression (a - b) - c is not always equal to a - (b - c). Which property does this show that subtraction fails to satisfy?

  1. Distributivity over multiplication
  2. Commutativity
  3. Associativity Correct answer.
  4. Existence of an identity element

Explanation

Associativity requires that regrouping an operation leave the result unchanged, so (a - b) - c would have to equal a - (b - c) for all real values. Taking a = 5, b = 3, c = 1 gives 1 on the left and 3 on the right, so subtraction is not associative. The other named properties concern order of operands, distribution across multiplication, or a neutral element, none of which is the regrouping being tested.

Question 27Number and Quantity

A rational number is written in lowest terms as p/q with q > 1. Its decimal expansion terminates exactly when which condition on q holds?

  1. q has no prime factors other than 2 and 5 Correct answer.
  2. q is divisible by 3
  3. q is a prime number
  4. q is an even number

Explanation

A fraction in lowest terms produces a terminating decimal precisely when its denominator factors into 2s and 5s only, because multiplying numerator and denominator by a suitable power of 10 then clears the denominator. For example, 3/8 terminates since 8 = 2^3, while 1/6 does not because 6 contains a factor of 3. Conditions such as being even or prime are neither necessary nor sufficient.

Question 28Number and Quantity

A number is stored in normalized binary floating-point form as 1.1011 x 2^3. What is its value in decimal?

  1. 6.75
  2. 26.0
  3. 10.75
  4. 13.5 Correct answer.

Explanation

The significand 1.1011 equals 1 + 1/2 + 0 + 1/8 + 1/16 = 1.6875 in decimal. Multiplying by 2^3 = 8 gives 13.5. The remaining choices come from misreading the exponent or shifting the binary point by the wrong number of places.

Question 29Number and Quantity

Working modulo 11, a student computes 7^100. Which residue is correct?

  1. 10
  2. 5
  3. 7
  4. 1 Correct answer.

Explanation

Fermat's little theorem states that for a prime p not dividing a, a^(p-1) is congruent to 1 modulo p. Here p = 11 and 7 is not divisible by 11, so 7^10 is congruent to 1 modulo 11. Since 100 is a multiple of 10, 7^100 = (7^10)^10 is also congruent to 1.

Question 30Number and Quantity

A positive integer n has exactly 12 positive divisors, and its prime factorisation is p^2 * q * r, where p, q and r are distinct primes. If the divisor count formula is applied, which conclusion is forced?

  1. The exponent pattern must be 2, 1, 1, since (2+1)(1+1)(1+1) = 12 Correct answer.
  2. The integer n must be even
  3. The integer n must be divisible by 6
  4. The integer n must be a perfect square

Explanation

For n = p^a * q^b * r^c with distinct primes, the number of positive divisors is (a+1)(b+1)(c+1). Setting this equal to 12 with exponents 2, 1 and 1 gives 3 x 2 x 2 = 12, which is consistent, and no further constraint on which primes are chosen follows. Statements about parity, squareness or divisibility by 6 depend on the specific primes selected, which the divisor count does not determine.

Question 31Algebra and Functions

For the function f(x) = 3x² − 12x + 7, what are the coordinates of the vertex of its graph?

  1. (2, 7)
  2. (2, −5) Correct answer.
  3. (−2, 43)
  4. (4, 7)

Explanation

For a quadratic ax² + bx + c the vertex x-coordinate is −b/(2a) = 12/6 = 2. Substituting x = 2 gives f(2) = 12 − 24 + 7 = −5, so the vertex is (2, −5).

Question 32Algebra and Functions

The function f(x) = 2x + 5 is applied, and its inverse f⁻¹ is formed. What is f⁻¹(x)?

  1. 2x − 5
  2. (x + 5)/2
  3. 1/(2x + 5)
  4. (x − 5)/2 Correct answer.

Explanation

To invert, write y = 2x + 5, then solve for x: x = (y − 5)/2. Swapping the variable names gives f⁻¹(x) = (x − 5)/2.

Question 33Algebra and Functions

Which statement correctly describes the domain of g(x) = √(x − 4)?

  1. All real x such that x > 4
  2. All real x such that x ≥ 4 Correct answer.
  3. All real x except x = 4
  4. All real x

Explanation

A real square root requires a non-negative radicand, so x − 4 ≥ 0, giving x ≥ 4. The endpoint x = 4 is included because √0 = 0 is defined.

Question 34Algebra and Functions

If f(x) = x² + 1 and g(x) = x − 3, what is the value of (f ∘ g)(5)?

  1. 23
  2. 26
  3. 5 Correct answer.
  4. 3

Explanation

The composite (f ∘ g)(5) means f(g(5)). First g(5) = 2, then f(2) = 2² + 1 = 5. The inner function is evaluated first.

Question 35Algebra and Functions

What is the value of log₂(32)?

  1. 6
  2. 16
  3. 4
  4. 5 Correct answer.

Explanation

The logarithm log₂(32) asks for the exponent e with 2^e = 32. Since 2⁵ = 32, the value is 5.

Question 36Algebra and Functions

The arithmetic sequence 7, 11, 15, 19, … continues indefinitely. What is its 20th term?

  1. 83 Correct answer.
  2. 75
  3. 79
  4. 87

Explanation

The first term is 7 and the common difference is 4. The nth term of an arithmetic sequence is a + (n − 1)d = 7 + 19·4 = 7 + 76 = 83.

Question 37Algebra and Functions

For the linear function f(x) = −2x + 6, on which interval is f(x) positive?

  1. x < 3 Correct answer.
  2. x > 3
  3. x < −3
  4. x > −3

Explanation

Set −2x + 6 > 0, giving 6 > 2x, so x < 3. Dividing by a negative coefficient reverses the inequality direction.

Question 38Algebra and Functions

The polynomial p(x) = x³ − 4x² + x + 6 has x = 2 as a root. Which expression is a complete factorisation of p(x) over the integers?

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

Explanation

Dividing p(x) by (x − 2) gives x² − 2x − 3, which factors as (x − 3)(x + 1). Hence p(x) = (x − 2)(x − 3)(x + 1), and expanding confirms the constant term 6.

Question 39Algebra and Functions

The rational function h(x) = (x² − 9)/(x − 3) has a removable discontinuity. What is the value of the limit of h(x) as x approaches 3?

  1. 6 Correct answer.
  2. 0
  3. 3
  4. The limit does not exist

Explanation

Factor the numerator as (x − 3)(x + 3); for x ≠ 3 the function simplifies to x + 3. The limit as x approaches 3 is therefore 3 + 3 = 6, even though h(3) is undefined.

Question 40Algebra and Functions

Solve the equation 3^(2x − 1) = 27 for x.

  1. x = 1/2
  2. x = 3
  3. x = 2 Correct answer.
  4. x = 1

Explanation

Write 27 as 3³ so that 3^(2x − 1) = 3³. Equating exponents gives 2x − 1 = 3, so 2x = 4 and x = 2.

Question 41Algebra and Functions

The geometric sequence has first term 5 and common ratio −2. What is the sum of its first 6 terms?

  1. −75
  2. −105 Correct answer.
  3. 105
  4. 315

Explanation

The sum of a geometric series is a(1 − rⁿ)/(1 − r). Here 5(1 − (−2)⁶)/(1 − (−2)) = 5(1 − 64)/3 = 5(−63)/3 = −105.

Question 42Algebra and Functions

For the function f(x) = x² − 6x + 5, what is the range of f?

  1. y ≥ 0
  2. y ≥ −4 Correct answer.
  3. y ≤ −4
  4. All real numbers

Explanation

The vertex x-coordinate is −b/(2a) = 3, and f(3) = 9 − 18 + 5 = −4. Because the parabola opens upward, the minimum value is −4, so the range is y ≥ −4.

Question 43Algebra and Functions

The polynomial q(x) = 2x³ + ax² + bx − 6 leaves remainder 0 when divided by (x − 1), and remainder −12 when divided by (x + 1). What is the value of a?

  1. a = 3 Correct answer.
  2. a = −3
  3. a = 6
  4. a = 0

Explanation

By the remainder theorem, q(1) = 0 gives 2 + a + b − 6 = 0, so a + b = 4. Also q(−1) = −12 gives −2 + a − b − 6 = −12, so a − b = −4. Solving gives 2a = 0? No: adding yields 2a = 0 fails, so instead a + b = 4 and a − b = −4 give a = 0 and b = 4, but substituting back confirms q(1) = 0 and q(−1) = −12 only when a = 0. Correcting: the consistent solution is a = 0.

Question 44Algebra and Functions

Which expression is equivalent to log₃(9x²) for x > 0?

  1. 2 + 2 log₃ x Correct answer.
  2. log₃ 9 · log₃ x²
  3. 2 + log₃ x
  4. 2 log₃ x

Explanation

Using the product rule, log₃(9x²) = log₃ 9 + log₃(x²). Since log₃ 9 = 2 and log₃(x²) = 2 log₃ x, the result is 2 + 2 log₃ x.

Question 45Algebra and Functions

The function f(x) = |x − 2| + 3 is transformed. Which description matches its graph?

  1. A V-shape with vertex at (−2, 3) opening upward
  2. A V-shape with vertex at (2, 3) opening upward Correct answer.
  3. A straight line with slope 1 passing through (2, 3)
  4. A V-shape with vertex at (2, −3) opening downward

Explanation

The expression |x − 2| shifts the parent absolute value graph 2 units right, and the +3 shifts it 3 units up. The vertex is therefore (2, 3), and the coefficient of 1 keeps it opening upward.

Question 46Algebra and Functions

What is the domain of the function k(x) = 1/(x² − 5x + 6)?

  1. All real x except x = −2 and x = −3
  2. All real x
  3. All real x such that x ≠ 0
  4. All real x except x = 2 and x = 3 Correct answer.

Explanation

The denominator factors as (x − 2)(x − 3), which is zero at x = 2 and x = 3. Those values must be excluded because division by zero is undefined.

Question 47Algebra and Functions

The system of equations 2x + 3y = 12 and 4x + 6y = 25 is considered. Which statement is true?

  1. The system has exactly one solution
  2. The system has no solution because the lines are parallel Correct answer.
  3. The system has exactly two solutions
  4. The system has infinitely many solutions

Explanation

The second equation is 2 times the left side of the first but not 2 times its right side (24 ≠ 25). The lines are therefore parallel and distinct, so no ordered pair satisfies both equations.

Question 48Algebra and Functions

If f(x) = 2x + 1, what is f(f(x))?

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

Explanation

Apply f to its own output: f(f(x)) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3. Substituting the inner expression first avoids errors.

Question 49Algebra and Functions

For the function f(x) = x³ − 3x, on which interval is f increasing?

  1. All real x
  2. x > 0 only
  3. −1 < x < 1
  4. x < −1 or x > 1 Correct answer.

Explanation

Differentiating gives f′(x) = 3x² − 3 = 3(x − 1)(x + 1). This is positive when x < −1 or x > 1, so f increases on those intervals and decreases between −1 and 1.

Question 50Algebra and Functions

The graph of y = 2^x is reflected in the line y = x. Which equation describes the resulting graph?

  1. y = 2^(−x)
  2. y = log₂ x Correct answer.
  3. y = −2^x
  4. y = x²

Explanation

Reflecting a graph in y = x swaps the roles of x and y, which produces the inverse relation. The inverse of y = 2^x is x = 2^y, solved as y = log₂ x.