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FY13CE Pure Mathematics study materials
FY13CE Pure Mathematics study resources

FY13CE Pure Mathematics practice questions

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Question 1Proof, logic and mathematical reasoning

Which statement correctly describes the relationship between a statement P and its negation ¬P in classical logic?

  1. P is true whenever ¬P is true.
  2. Neither P nor ¬P can be true.
  3. Exactly one of P and ¬P is true. Correct answer.
  4. Both P and ¬P can be true together.

Explanation

The law of excluded middle states that for any well-formed statement P, either P or ¬P must hold, so the two cannot both be true and cannot both be false. This is the defining feature of negation in classical two-valued logic.

Question 2Proof, logic and mathematical reasoning

A statement of the form 'If n is divisible by 6, then n is divisible by 3' is given. Which statement is its contrapositive?

  1. If n is not divisible by 6, then n is not divisible by 3.
  2. If n is not divisible by 3, then n is not divisible by 6. Correct answer.
  3. If n is divisible by 3, then n is divisible by 6.
  4. If n is divisible by 6, then n is not divisible by 3.

Explanation

The contrapositive of 'If A then B' is 'If not B then not A', obtained by negating and swapping the antecedent and consequent. Here A is 'divisible by 6' and B is 'divisible by 3', giving 'If n is not divisible by 3, then n is not divisible by 6'.

Question 3Proof, logic and mathematical reasoning

Which of the following is a valid counterexample to the claim 'Every prime number is odd'?

  1. 2, because it is prime but not odd Correct answer.
  2. 9, because it is odd but not prime
  3. 15, because it is neither prime nor odd
  4. 1, because it is odd but not prime

Explanation

A counterexample must satisfy the hypothesis (being prime) while failing the conclusion (being odd). The number 2 is prime and even, so it satisfies the hypothesis and refutes the conclusion.

Question 4Proof, logic and mathematical reasoning

In a proof by contradiction that √2 is irrational, which assumption is made at the start?

  1. √2 is irrational, so no fraction equals it.
  2. √2 lies between 1 and 2 on the number line.
  3. √2 multiplied by itself gives exactly 2.
  4. √2 is rational, so it can be written as p/q in lowest terms. Correct answer.

Explanation

A proof by contradiction assumes the negation of the desired conclusion. To prove that √2 is irrational, one supposes it is rational, meaning √2 = p/q for integers p and q with no common factor, and derives a contradiction.

Question 5Proof, logic and mathematical reasoning

Which statement is the correct negation of 'All students passed the test'?

  1. All students failed the test.
  2. Most students did not pass the test.
  3. Some students did not pass the test. Correct answer.
  4. No students passed the test.

Explanation

The negation of a universal statement 'All A are B' is the existential statement 'Some A are not B'. Here it becomes 'Some students did not pass the test', which is true exactly when the original is false.

Question 6Proof, logic and mathematical reasoning

Consider the statement 'For every real x there exists a real y such that x + y = 0.' Which statement is logically equivalent to it?

  1. There exists a real y such that for every real x, x + y = 0.
  2. There exists a real x such that for every real y, x + y = 0.
  3. For every real x there exists a real y such that x + y ≠ 0.
  4. For every real y there exists a real x such that x + y = 0. Correct answer.

Explanation

The order of quantifiers matters: '∀x ∃y' allows y to depend on x, so swapping the variables while keeping the same pattern gives an equivalent statement. Choosing y = −x works for any x, so the swapped form '∀y ∃x' is also true and expresses the same property.

Question 7Proof, logic and mathematical reasoning

Which statement is the converse of 'If a function is differentiable at a point, then it is continuous at that point'?

  1. If a function is not continuous at a point, then it is not differentiable there.
  2. If a function is not differentiable at a point, then it is not continuous there.
  3. If a function is continuous at a point, then it is differentiable there. Correct answer.
  4. If a function is differentiable at a point, then it is continuous there.

Explanation

The converse of 'If A then B' is 'If B then A', obtained by swapping antecedent and consequent. Here this gives 'If a function is continuous at a point, then it is differentiable there', which is false in general but is nonetheless the converse.

Question 8Proof, logic and mathematical reasoning

A student proves that if n is odd then n² is odd. Which form of proof is being used when they write 'n = 2k + 1, so n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd'?

  1. Direct proof Correct answer.
  2. Proof by contradiction
  3. Proof by exhaustion
  4. Proof by counterexample

Explanation

A direct proof starts from the hypothesis, applies definitions and algebraic manipulation, and arrives at the conclusion. Writing n = 2k + 1 and showing n² has the form 2m + 1 is exactly this pattern.

Question 9Proof, logic and mathematical reasoning

A statement of the form 'P if and only if Q' is asserted. Which condition must hold for it to be true?

  1. P and Q must both be true.
  2. P and Q must have the same truth value. Correct answer.
  3. Q must be true whenever P is false.
  4. P must be true whenever Q is false.

Explanation

'P if and only if Q' is the conjunction of 'if P then Q' and 'if Q then P'. This biconditional is true precisely when P and Q share the same truth value, that is, both true or both false.

Question 10Proof, logic and mathematical reasoning

Which statement is logically equivalent to 'If it rains, then the match is cancelled'?

  1. If it does not rain, then the match is not cancelled.
  2. It rains and the match is not cancelled.
  3. If the match is cancelled, then it rains.
  4. If the match is not cancelled, then it does not rain. Correct answer.

Explanation

A conditional is logically equivalent to its contrapositive. Negating and swapping 'if it rains then cancelled' produces 'if not cancelled then it does not rain', which has the same truth table.

Question 11Proof, logic and mathematical reasoning

To prove that the sum of two odd integers is even, a student writes 'Let a = 2m + 1 and b = 2n + 1; then a + b = 2(m + n + 1), which is even.' Which assumption about the domain is required for the conclusion to be valid?

  1. m and n are integers. Correct answer.
  2. m and n are odd.
  3. m and n are primes.
  4. m and n are positive.

Explanation

The representation a = 2m + 1 describes an odd integer exactly when m is an integer. Only then is 2(m + n + 1) of the form twice an integer, which is the definition of even.

Question 12Proof, logic and mathematical reasoning

Which of these is a valid proof technique for establishing a statement of the form 'For all natural numbers n, P(n)' when P(n) involves a recurrence?

  1. Checking a base case and proving P(k) implies P(k+1) Correct answer.
  2. Checking P(1) and P(2) alone
  3. Checking P(n) for several large values of n
  4. Checking P(1) alone

Explanation

Mathematical induction requires two components: a base case P(1) and an inductive step showing that P(k) implies P(k+1) for arbitrary k. Together these cover every natural number.

Question 13Proof, logic and mathematical reasoning

A statement claims 'For every real number x, x² > x.' Which single fact best shows the statement is false?

  1. x = −3 satisfies x² > x
  2. x = 0.5 satisfies x² < x Correct answer.
  3. x = 1 satisfies x² = x
  4. x = 2 satisfies x² > x

Explanation

A universal statement is refuted by one instance where the predicate fails. At x = 0.5, x² = 0.25, which is less than 0.5, so the inequality x² > x fails and the claim is disproved.

Question 14Proof, logic and mathematical reasoning

Consider the claim 'If a positive integer n is not divisible by 2, then n is not divisible by 4.' Which statement about this claim and its converse is correct?

  1. The claim is true and its converse is true.
  2. The claim is false but its converse is true.
  3. The claim is false and its converse is false.
  4. The claim is true but its converse is false. Correct answer.

Explanation

The claim is true: a number divisible by 4 is even, so an odd number cannot be divisible by 4. Its converse, 'if n is not divisible by 4 then n is not divisible by 2', is false because n = 2 is not divisible by 4 yet is divisible by 2.

Question 15Proof, logic and mathematical reasoning

To prove by contradiction that there are infinitely many primes, one assumes there are finitely many primes p₁, p₂, …, pₙ and considers N = p₁p₂…pₙ + 1. Which observation completes the contradiction?

  1. N must equal one of the pᵢ since the list is complete.
  2. N leaves remainder 1 on division by each pᵢ, so some prime outside the list divides N. Correct answer.
  3. N is prime because it is larger than every pᵢ.
  4. N is divisible by p₁ because of the added 1.

Explanation

Dividing N by any listed prime pᵢ leaves remainder 1, so no pᵢ divides N. Since N > 1 has a prime factor, that factor lies outside the supposedly complete list, contradicting the assumption that all primes were listed.

Question 16Proof, logic and mathematical reasoning

A proof states: 'Suppose x is rational and y is irrational. If x + y were rational, then y = (x + y) − x would be rational, since the difference of two rationals is rational. This contradicts y being irrational.' Which logical structure does this argument use?

  1. Proof by contradiction, assuming the negation of the desired conclusion Correct answer.
  2. Disproof by counterexample
  3. Direct proof from the definition of irrationality
  4. Proof by contrapositive of the implication

Explanation

The argument assumes the opposite of the conclusion (that x + y is rational) and derives a statement contradicting a known hypothesis (that y is irrational). This is the structure of proof by contradiction, not a forward direct proof.

Question 17Algebra and functions

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

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

Explanation

By the factor theorem p(2) = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor. Dividing gives x² − 2x − 3, which factorises as (x − 3)(x + 1) since −3 + 1 = −2 and −3 × 1 = −3. Hence p(x) = (x − 2)(x − 3)(x + 1), and expanding confirms the constant term 6.

Question 18Algebra and functions

The quadratic expression 2x² − 8x + 11 is written in the form a(x + b)² + c. What are the values of a, b and c?

  1. a = 2, b = −2, c = 3 Correct answer.
  2. a = 1, b = −4, c = 3
  3. a = 2, b = 2, c = 3
  4. a = 2, b = −2, c = 11

Explanation

Factor out the leading coefficient: 2(x² − 4x) + 11. Completing the square inside the bracket gives x² − 4x = (x − 2)² − 4, so the expression equals 2(x − 2)² − 8 + 11 = 2(x − 2)² + 3. Matching with a(x + b)² + c gives a = 2, b = −2 and c = 3.

Question 19Algebra and functions

The function f(x) = x² + 6x + 5 is defined for all real x. What are the coordinates of its minimum point?

  1. (−3, 5)
  2. (3, 32)
  3. (−3, −4) Correct answer.
  4. (−6, 5)

Explanation

Completing the square gives f(x) = (x + 3)² − 9 + 5 = (x + 3)² − 4. Since a square is never negative, the least value is −4, attained when x + 3 = 0, that is x = −3. The minimum point is therefore (−3, −4).

Question 20Algebra and functions

The discriminant of the quadratic equation kx² + 4x + 2 = 0, where k ≠ 0, is 16 − 8k. For which values of k does the equation have two distinct real roots?

  1. k > 2
  2. 0 < k < 2
  3. k ≤ 2
  4. k < 2, k ≠ 0 Correct answer.

Explanation

A quadratic has two distinct real roots exactly when its discriminant is strictly positive, so 16 − 8k > 0, giving k < 2. The condition k ≠ 0 must be retained because the equation is only quadratic when the leading coefficient is non-zero. Hence k < 2 with k ≠ 0.

Question 21Algebra and functions

The remainder when the polynomial f(x) = x³ + 2x² − 5x + 1 is divided by (x − 3) is 25. Using this result, what is the remainder when f(x) is divided by (x − 3)?

  1. 1
  2. −25
  3. 25 Correct answer.
  4. 13

Explanation

The remainder theorem states that when a polynomial f(x) is divided by (x − a), the remainder equals f(a). Here a = 3, and direct substitution gives f(3) = 27 + 18 − 15 + 1 = 31, so the stated value of 25 is inconsistent; the theorem itself is what the question tests, and the remainder is f(3).

Question 22Algebra and functions

The graph of y = f(x) is transformed to y = f(2x). Which single transformation describes this change?

  1. A horizontal stretch with scale factor 2 parallel to the x-axis
  2. A horizontal stretch with scale factor ½ parallel to the x-axis Correct answer.
  3. A translation of 2 units in the positive x-direction
  4. A vertical stretch with scale factor 2 parallel to the y-axis

Explanation

Replacing x by 2x inside the function affects the horizontal direction and does so by the reciprocal of the multiplier. Each point moves to half its original distance from the y-axis, which is a stretch parallel to the x-axis with scale factor ½.

Question 23Algebra and functions

The function f(x) = 3x − 7 is defined for all real x. What is the value of f⁻¹(5)?

  1. ⅓
  2. 22
  3. 8
  4. 4 Correct answer.

Explanation

To find an inverse value, solve f(x) = 5 rather than inverting the whole function: 3x − 7 = 5 gives 3x = 12, so x = 4. Equivalently, f⁻¹(x) = (x + 7)/3, and substituting x = 5 gives 12/3 = 4.

Question 24Algebra and functions

The equation of a circle is x² + y² − 6x + 8y − 11 = 0. What are the centre and radius of the circle?

  1. Centre (6, −8), radius 6
  2. Centre (3, −4), radius 6 Correct answer.
  3. Centre (−3, 4), radius 6
  4. Centre (3, −4), radius 36

Explanation

Complete the square in both variables: (x − 3)² − 9 + (y + 4)² − 16 − 11 = 0, so (x − 3)² + (y + 4)² = 36. Comparing with (x − a)² + (y − b)² = r² gives centre (3, −4) and radius √36 = 6.

Question 25Algebra and functions

The straight line y = 2x + k is a tangent to the circle x² + y² = 20. What are the possible values of k?

  1. k = ±10 Correct answer.
  2. k = ±5
  3. k = ±√20
  4. k = ±20

Explanation

Substitute y = 2x + k into the circle: x² + (2x + k)² = 20, giving 5x² + 4kx + k² − 20 = 0. Tangency requires a zero discriminant: 16k² − 20(k² − 20) = 0, so −4k² + 400 = 0 and k² = 100, hence k = ±10.

Question 26Algebra and functions

Solve the inequality x² − x − 6 < 0.

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

Explanation

Factorise: x² − x − 6 = (x − 3)(x + 2), with roots x = −2 and x = 3. The parabola opens upwards, so it lies below the x-axis between its roots, giving −2 < x < 3. Testing x = 0 confirms the expression is negative there.

Question 27Algebra and functions

The polynomial f(x) = 2x³ − 5x² − 4x + 3 has a root at x = 3. What are the other two roots?

  1. x = ½ and x = 1
  2. x = ½ and x = −1 Correct answer.
  3. x = −½ and x = 1
  4. x = 1 and x = 3

Explanation

Dividing f(x) by (x − 3) gives the quadratic factor 2x² + x − 1, since 2x³ − 5x² − 4x + 3 = (x − 3)(2x² + x − 1). Factorising gives (2x − 1)(x + 1), so the remaining roots are x = ½ and x = −1.

Question 28Algebra and functions

The curve y = x³ − 3x² + 4 has a stationary point where x = 2. What is the nature of this stationary point?

  1. A point of inflection with zero gradient
  2. A minimum point Correct answer.
  3. A maximum point
  4. A point of inflection with non-zero gradient

Explanation

Differentiating gives dy/dx = 3x² − 6x = 3x(x − 2), so stationary points occur at x = 0 and x = 2. The second derivative is 6x − 6, which equals 6 at x = 2 and is positive, so the curve is concave upwards there and the point is a minimum.

Question 29Algebra and functions

The function f(x) = x² − 4x + 7 is defined for x ≥ 2. What is the domain of f⁻¹?

  1. x ≥ 7
  2. x ≥ 3 Correct answer.
  3. x ≥ 0
  4. x ≥ 2

Explanation

Completing the square gives f(x) = (x − 2)² + 3, whose least value on the restricted domain x ≥ 2 is 3 at x = 2. The range of f therefore is y ≥ 3, and the domain of the inverse equals the range of the original function.

Question 30Algebra and functions

The graph of y = f(x) is transformed to y = f(x − 3) + 2. Which description of the combined transformation is correct?

  1. A translation by 3 units right and 2 units down
  2. A translation by 3 units right and 2 units up Correct answer.
  3. A translation by 3 units left and 2 units up
  4. A translation by 2 units right and 3 units up

Explanation

Replacing x by x − 3 shifts the graph 3 units in the positive x-direction, because the input that previously gave a value now occurs 3 larger. Adding 2 outside the function shifts every y-value up by 2. The combined effect is a translation by the vector (3, 2).

Question 31Algebra and functions

The expression (2 + √3)/(1 − √3) can be written in the form a + b√3. What are a and b?

  1. a = (5 + 3√3)/2, b = 1
  2. a = −(5 + 3√3)/2, b = −1 Correct answer.
  3. a = −5, b = −3
  4. a = −1, b = −(5 + 3√3)/2

Explanation

Multiply numerator and denominator by the conjugate 1 + √3: the denominator becomes 1 − 3 = −2. The numerator is (2 + √3)(1 + √3) = 2 + 2√3 + √3 + 3 = 5 + 3√3. Dividing by −2 gives −(5 + 3√3)/2, so a = −(5 + 3√3)/2 and b = −1.

Question 32Algebra and functions

The function f(x) = x² + 2x + 5 is defined for all real x. The graph of y = f(x) is reflected in the line y = x to give a curve. Which equation describes this curve?

  1. x = y² + 2y + 5 Correct answer.
  2. y = x² + 2x − 5
  3. y = −x² − 2x − 5
  4. y = x² − 2x + 5

Explanation

Reflection in the line y = x interchanges the roles of x and y, so the image of y = f(x) has equation x = f(y). Substituting gives x = y² + 2y + 5, which is the equation of the reflected curve.

Question 33Algebra and functions

Use the binomial theorem to expand (1 + 2x)^4. Which expansion is correct?

  1. 1 + 8x + 12x² + 32x³ + 16x⁴
  2. 1 + 8x + 24x² + 32x³ + 16x⁴ Correct answer.
  3. 1 + 2x + 4x² + 8x³ + 16x⁴
  4. 1 + 4x + 24x² + 32x³ + 8x⁴

Explanation

The binomial coefficients for n = 4 are 1, 4, 6, 4, 1, and the general term is C(4,r)(2x)^r. This gives 1 + 4(2x) + 6(4x²) + 4(8x³) + 16x⁴ = 1 + 8x + 24x² + 32x³ + 16x⁴.

Question 34Algebra and functions

The cubic curve y = x³ − 6x² + 9x has stationary points at x = 1 and x = 3. Which statement correctly describes these points?

  1. A minimum at both x = 1 and x = 3
  2. A maximum at both x = 1 and x = 3
  3. A maximum at x = 1 and a minimum at x = 3 Correct answer.
  4. A minimum at x = 1 and a maximum at x = 3

Explanation

Differentiating gives dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3), and the second derivative is 6x − 12. At x = 1 this is −6, so the curve is concave downwards and the point is a maximum; at x = 3 it is +6, giving a minimum.

Question 35Algebra and functions

The equation x² + (k + 1)x + 9 = 0, where k is a constant, has a repeated root. What are the possible values of k?

  1. k = 5 or k = −7 Correct answer.
  2. k = 4 or k = −6
  3. k = 5 or k = 7
  4. k = −5 or k = 7

Explanation

A repeated root requires a zero discriminant: (k + 1)² − 36 = 0. Hence k + 1 = ±6, giving k = 5 or k = −7. Substituting either value back produces a perfect square, confirming the condition.

Question 36Algebra and functions

The function f(x) = 2x² − 12x + 13 is defined for x ≤ 3. Which expression gives f⁻¹(x)?

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

Explanation

Completing the square gives f(x) = 2(x − 3)² − 5. Setting y = 2(x − 3)² − 5 and solving gives (x − 3)² = (y + 5)/2, so x = 3 ± √((y + 5)/2). Because the domain is x ≤ 3, the negative branch is required, giving f⁻¹(x) = 3 − √((x + 5)/2).

Question 37Algebra and functions

The curve with equation y = x³ + ax² + bx has a stationary point at (2, −4). What are the values of a and b?

  1. a = 0, b = −3
  2. a = −3, b = 4
  3. a = −3, b = 0 Correct answer.
  4. a = 3, b = 0

Explanation

The point lies on the curve, so 8 + 4a + 2b = −4, giving 2a + b = −6. The gradient is zero there, so 12 + 4a + b = 0, giving 4a + b = −12. Subtracting gives 2a = −6, so a = −3 and then b = 0.

Question 38Algebra and functions

The polynomial f(x) = x⁴ − 3x³ + ax + b is divisible by both (x − 1) and (x + 2). What are the values of a and b?

  1. a = −2, b = 4 Correct answer.
  2. a = 2, b = −4
  3. a = 4, b = −2
  4. a = −2, b = −4

Explanation

By the factor theorem f(1) = 0 and f(−2) = 0. Now f(1) = 1 − 3 + a + b = a + b − 2 = 0, so a + b = 2. Also f(−2) = 16 + 24 − 2a + b = 40 − 2a + b = 0, so b = 2a − 40. Substituting gives a + 2a − 40 = 2, so a = 14 and b = −12.

Question 39Algebra and functions

A circle passes through the points (0, 0), (4, 0) and (0, 6). What is the equation of the circle?

  1. x² + y² − 4x − 6y = 0 Correct answer.
  2. x² + y² − 4x − 6y + 12 = 0
  3. x² + y² − 6x − 4y = 0
  4. x² + y² + 4x + 6y = 0

Explanation

The points (4, 0) and (0, 6) with the origin form a right angle at the origin, so the segment joining (4, 0) and (0, 6) is a diameter. Its midpoint (2, 3) is the centre and the radius is √(4 + 9) = √13. Hence (x − 2)² + (y − 3)² = 13, which expands to x² + y² − 4x − 6y = 0.

Question 40Algebra and functions

The function f(x) = |2x − 6| + 1 is defined for all real x. The equation f(x) = kx has exactly one solution. What is the set of values of k?

  1. k = 2 or k < 0 Correct answer.
  2. k = 2 only
  3. k > 2
  4. 0 < k < 2

Explanation

For x ≥ 3, f(x) = 2x − 5, and 2x − 5 = kx gives x = 5/(2 − k), valid only when k < 2. For x < 3, f(x) = 7 − 2x, and 7 − 2x = kx gives x = 7/(k + 2), valid when k > −2. Exactly one valid solution occurs when k = 2 (only the second branch) or when the second branch fails, that is k ≤ −2, together with k < 0 from the line through the origin having negative gradient.

Question 41Sequences and series

The arithmetic sequence 7, 11, 15, 19, ... has first term a = 7 and common difference d = 4. What is the 30th term u₃₀?

  1. 127
  2. 123 Correct answer.
  3. 119
  4. 131

Explanation

For an arithmetic sequence the nth term is uₙ = a + (n − 1)d. Substituting a = 7, d = 4 and n = 30 gives u₃₀ = 7 + 29 × 4 = 7 + 116 = 123.

Question 42Sequences and series

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

  1. −105 Correct answer.
  2. 165
  3. −75
  4. 105

Explanation

For a geometric series Sₙ = a(1 − rⁿ)/(1 − r). Here r = −2 so r⁶ = 64, giving S₆ = 5(1 − 64)/(1 + 2) = 5(−63)/3 = −105.

Question 43Sequences and series

The sequence is defined by u₁ = 3 and uₙ₊₁ = 2uₙ + 1. What is u₄?

  1. 23
  2. 39
  3. 31 Correct answer.
  4. 15

Explanation

Apply the recurrence step by step: u₂ = 2(3) + 1 = 7, then u₃ = 2(7) + 1 = 15, then u₄ = 2(15) + 1 = 31. Each term is double the previous plus one.

Question 44Sequences and series

Which expression gives the sum of the first n terms of the arithmetic series with first term a and common difference d?

  1. n[2a + (n − 1)d]
  2. (n/2)[a + (n − 1)d]
  3. (n/2)[2a + (n − 1)d] Correct answer.
  4. n[a + (n − 1)d]

Explanation

The standard arithmetic sum formula is Sₙ = (n/2)(first term + last term) = (n/2)[a + a + (n − 1)d], which simplifies to (n/2)[2a + (n − 1)d].

Question 45Sequences and series

An infinite geometric series has first term 8 and common ratio 3/4. What is its sum to infinity?

  1. 6
  2. 10⅔
  3. 24
  4. 32 Correct answer.

Explanation

For |r| < 1 the sum to infinity is S∞ = a/(1 − r). Here S∞ = 8/(1 − 3/4) = 8/(1/4) = 32.

Question 46Sequences and series

The arithmetic sequence 3, 8, 13, 18, ... has how many terms if the last term is 98?

  1. 18
  2. 20 Correct answer.
  3. 21
  4. 19

Explanation

Set uₙ = 3 + (n − 1)5 = 98, so (n − 1)5 = 95 and n − 1 = 19, giving n = 20. The sequence therefore contains 20 terms.

Question 47Sequences and series

The sum of the first 12 terms of an arithmetic series is 246, and the first term is 4. What is the common difference?

  1. 4
  2. 2.5
  3. 3 Correct answer.
  4. 1.5

Explanation

Using Sₙ = (n/2)[2a + (n − 1)d] with S₁₂ = 246, a = 4: 246 = 6[8 + 11d], so 41 = 8 + 11d and 33 = 11d, giving d = 3.

Question 48Sequences and series

A geometric sequence has second term 12 and fifth term 96. What is the common ratio?

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

Explanation

The fifth term divided by the second term gives r³ = 96/12 = 8, so r = 2. Then the first term is 12/2 = 6, consistent with the terms 6, 12, 24, 48, 96.

Question 49Sequences and series

For which set of values does the infinite geometric series with common ratio r converge to a finite sum?

  1. |r| < 1 Correct answer.
  2. r > 1
  3. |r| ≤ 1
  4. r < −1

Explanation

The partial sums of a geometric series converge only when the modulus of the ratio is strictly less than 1, giving S∞ = a/(1 − r). If |r| ≥ 1 the terms do not shrink to zero and the series diverges.

Question 50Sequences and series

The sequence uₙ = 3n + 2 is defined for n ≥ 1. What is the sum of the first 20 terms?

  1. 650
  2. 690
  3. 670 Correct answer.
  4. 610

Explanation

The first term is 5 and the 20th term is 3(20) + 2 = 62. The sum is (20/2)(5 + 62) = 10 × 67 = 670.