TMUA Paper 2: Reasoning with algebra and number, set 1

• Answer all questions. No calculator is allowed.
• Each question has exactly one correct answer.
• This strand tests reasoning about algebra and number results, not just computing them: check every statement or option against the working, rather than trusting a first impression.

1. 11 mark

   Consider the following four statements, where x and y are positive real numbers and m and n are rational numbers.

   (i) x^m * x^n = x^(m+n)

   (ii) x^m * y^m = (xy)^m

   (iii) x^m + x^n = x^(m+n)

   (iv) (x^m)^n = x^(mn)

   How many of these four statements are true for all positive real x, y and all rational m, n?

   1. A 3
   2. B 2
   3. C 4
   4. D 1
2. 21 mark

   Consider the statement: 'For every positive integer n, the value of n^2 + n + 41 is a prime number.' Which of the following values of n provides a counterexample to this statement?

   1. A n = 40
   2. B n = 1
   3. C n = 10
   4. D n = 39
3. 31 mark

   For real numbers x, let P be the statement 'x > 2' and let Q be the statement 'x^2 > 4'. Which of the following is true?

   1. A P is necessary and sufficient for Q
   2. B P is neither necessary nor sufficient for Q
   3. C P is necessary but not sufficient for Q
   4. D P is sufficient but not necessary for Q
4. 41 mark

   How many of the following four equations have two distinct real roots?

   (1) x^2 - 4x + 4 = 0

   (2) x^2 - 5x + 6 = 0

   (3) x^2 + 2x + 5 = 0

   (4) 2x^2 - 3x - 2 = 0

   1. A 4
   2. B 2
   3. C 3
   4. D 1
5. 51 mark

   A sequence is defined by x_1 = 3 and x_(n+1) = 2x_n - 1 for n >= 1. Which of the following statements about this sequence is true?

   1. A The sequence is arithmetic with common difference 2
   2. B The sequence itself is geometric with common ratio 2
   3. C The nth term is given by x_n = 2n + 1
   4. D The sequence x_n - 1 is geometric with common ratio 2
6. 61 mark

   How many of the following four statements are true?

   (1) The sum of the first n positive integers is n(n+1)/2.

   (2) The geometric series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.

   (3) The geometric series 2 + 3 + 4.5 + 6.75 + ..., with common ratio 1.5, has a finite sum to infinity.

   (4) The arithmetic series 1 + 4 + 7 + 10 + ... has a finite sum to infinity.

   1. A 2
   2. B 1
   3. C 3
   4. D 4
7. 71 mark

   Let n be a positive integer. Which of the following statements about the binomial expansion of (1+x)^n is NOT always true?

   1. A The expansion has exactly n+1 terms
   2. B The coefficients of x^k and x^(n-k) are equal for every 0 <= k <= n
   3. C The sum of all the coefficients in the expansion equals 2^n
   4. D The coefficient of x^n in the expansion is n
8. 81 mark

   Let P be the statement 'x = 3' and let Q be the statement '|x| = 3'. Which of the following is true?

   1. A P is neither necessary nor sufficient for Q
   2. B P is necessary but not sufficient for Q
   3. C P is necessary and sufficient for Q
   4. D P is sufficient but not necessary for Q
9. 91 mark

   Which of the following statements about surds is true for all positive real numbers a and b?

   1. A sqrt(a-b) = sqrt(a) - sqrt(b)
   2. B sqrt(a) + sqrt(b) = sqrt(a+b)
   3. C sqrt(a) * sqrt(b) = sqrt(ab)
   4. D sqrt(a)/sqrt(b) + sqrt(b)/sqrt(a) = 1
10. 101 mark

    Let f(x) = x^3 - 2x^2 - 5x + 6. How many of the following four statements are true?

    (1) f(3) = 0.

    (2) (x - 3) is a factor of f(x).

    (3) When f(x) is divided by (x - 1), the remainder is 0.

    (4) (x - 2) is a factor of f(x).

    1. A 1
    2. B 2
    3. C 4
    4. D 3
11. 111 mark

    A line has equation y = x + c, where c is a constant, and a curve has equation y = x^2 - 4x + 7. For which of the following sets of values of c does the line meet the curve at two distinct points?

    1. A c >= 3/4
    2. B c = 3/4
    3. C c > 3/4
    4. D c < 3/4
12. 121 mark

    How many of the following four statements are true for all integers n?

    (1) If n is divisible by 6, then n is divisible by 3.

    (2) If n^2 is divisible by 4, then n is divisible by 4.

    (3) If n is a prime number greater than 2, then n is odd.

    (4) If n is divisible by 2 and by 3, then n is divisible by 12.

    1. A 1
    2. B 3
    3. C 2
    4. D 4

Worked solutions

Every question below carries the reasoning, not just the answer. The official material for this test publishes a correct option letter and nothing else.

1. Question 1Answer: A

   1. Statement (i): x^m * x^n = x^(m+n) is the standard product rule for indices, true for all rational m, n when x > 0.
   2. Statement (ii): x^m * y^m = (xy)^m is the power-of-a-product rule, also true for all rational m when x, y > 0.
   3. Statement (iii): x^m + x^n = x^(m+n) is not a valid law. Taking x = 2, m = 1, n = 2 as a check, the left side is 2 + 4 = 6 but the right side is 2^3 = 8, so statement (iii) is false.
   4. Statement (iv): (x^m)^n = x^(mn) is the power-of-a-power rule, true for all rational m, n when x > 0.
   5. So three of the four statements, (i), (ii) and (iv), are true, which is option A.
   • Why not B: This treats (x^m)^n as x^(m+n) instead of x^(mn), confusing the power-of-a-power rule with the product rule, so statement (iv) is wrongly judged false, leaving only (i) and (ii) counted as true, giving 2.
   • Why not C: This over-generalises the product rule x^m * x^n = x^(m+n) to addition, wrongly accepting statement (iii) x^m + x^n = x^(m+n) as true, so all four statements are wrongly counted as true, giving 4.
   • Why not D: This assumes the product rule and the power-of-a-power rule only hold when m and n are positive integers, so statements (ii) and (iv) are wrongly rejected for non-integer rational exponents, leaving only statement (i) counted as true, giving 1.

2. Question 2Answer: A

   1. Evaluate the expression at n = 40: 40^2 + 40 + 41 = 1600 + 40 + 41 = 1681.
   2. Recognise 1681 = 41 x 41 = 41^2, so 1681 is composite, not prime.
   3. The statement claims the expression is always prime, but n = 40 produces a composite number, so n = 40 is a counterexample.
   4. Checking the other options confirms they do not work: n = 1 gives 43 (prime), n = 10 gives 151 (prime) and n = 39 gives 1601 (prime), so only n = 40 breaks the pattern, giving answer A.
   • Why not B: Misreads n^2 as 2n, computing 2(1) + 1 + 41 = 44 for n = 1, an even number, and wrongly concludes this is the counterexample; correctly evaluated, 1^2 + 1 + 41 = 43, which is prime.
   • Why not C: Adds 100 + 10 + 41 incorrectly as 150 instead of 151, and 150 is clearly composite (divisible by 2), wrongly making n = 10 look like a counterexample; the correct value 151 is prime.
   • Why not D: Computes 39^2 + 39 + 41 as 1521 + 79 = 1600 (an off-by-one slip on 39 + 41, which is actually 80), and since 1600 = 40^2 is obviously composite, wrongly selects n = 39; the correct total is 1521 + 80 = 1601, which is prime.

3. Question 3Answer: D

   1. Suppose P holds, i.e. x > 2. Since x > 2 > 0, squaring both sides of a positive inequality preserves its direction, so x^2 > 4. Hence P implies Q, so P is sufficient for Q.
   2. To check necessity, ask whether Q implies P: does x^2 > 4 force x > 2? Take x = -3: x^2 = 9 > 4, so Q holds, but x = -3 is not greater than 2, so P fails.
   3. So Q can be true while P is false, meaning P is not necessary for Q.
   4. Combining both parts, P is sufficient but not necessary for Q: answer D.
   • Why not A: This assumes squaring an inequality preserves an 'if and only if' relationship, ignoring that Q is also satisfied by negative values such as x = -3 (since (-3)^2 = 9 > 4) where P fails, so P is not necessary for Q.
   • Why not B: This overlooks that P does guarantee Q: for any x > 2, squaring preserves the inequality because both sides are positive, so x^2 > 4 always follows; P does satisfy sufficiency, so P is not 'neither'.
   • Why not C: This reverses sufficiency and necessity: it treats the existence of values satisfying Q without P (such as x = -3) as showing P does not guarantee Q, rather than correctly showing only that P is not necessary; in fact whenever x > 2 it is always true that x^2 > 4, so P is sufficient for Q, not merely necessary.

4. Question 4Answer: B

   1. Equation (1): discriminant = (-4)^2 - 4(1)(4) = 16 - 16 = 0, so it has one repeated real root, not two distinct ones.
   2. Equation (2): discriminant = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0, so it has two distinct real roots (it factorises as (x-2)(x-3) = 0).
   3. Equation (3): discriminant = 2^2 - 4(1)(5) = 4 - 20 = -16 < 0, so it has no real roots.
   4. Equation (4): discriminant = (-3)^2 - 4(2)(-2) = 9 + 16 = 25 > 0, so it has two distinct real roots (it factorises as (2x+1)(x-2) = 0).
   5. Exactly two of the four equations, (2) and (4), have two distinct real roots, so the answer is B.
   • Why not A: This counts all four equations by miscomputing the discriminant of equation (3) with a sign slip, working out b^2 - 4ac as 4 - 4(1)(5) = 4 + 20 = 24 instead of 4 - 20 = -16, wrongly treating (3) as having real roots, on top of a repeated-root confusion over equation (1) that also inflates the count.
   • Why not C: This correctly identifies equations (2) and (4), but also counts equation (1), x^2 - 4x + 4 = 0, whose discriminant is 16 - 16 = 0; a discriminant of exactly zero gives one repeated root, not two distinct ones, so (1) should not be counted.
   • Why not D: This correctly rejects equation (3) but also wrongly rejects equation (4): computing its discriminant with the wrong sign as b^2 + 4ac = 9 + 4(2)(-2) = 9 - 16 = -7 instead of b^2 - 4ac = 9 - 4(2)(-2) = 9 + 16 = 25, so (4) is wrongly judged to have no real roots, leaving only equation (2) counted.

5. Question 5Answer: D

   1. Compute the first few terms: x1 = 3, x2 = 2(3) - 1 = 5, x3 = 2(5) - 1 = 9, x4 = 2(9) - 1 = 17.
   2. The differences between consecutive terms are 2, 4, 8, which are not constant, so the sequence itself is not arithmetic.
   3. The ratios of consecutive terms are 5/3, 9/5, 17/9, which are not constant either, so the sequence itself is not geometric.
   4. Consider y_n = x_n - 1: y1 = 2, y2 = 4, y3 = 8, y4 = 16. Each term is double the previous one, so y_n is geometric with common ratio 2.
   5. Therefore the true statement is D: the sequence x_n - 1 is geometric with common ratio 2.
   • Why not A: This mistakes the first two differences, x2 - x1 = 5 - 3 = 2, for a constant common difference, without checking further; the next difference is x3 - x2 = 9 - 5 = 4, which is not equal to 2, so the sequence is not arithmetic.
   • Why not B: This assumes the sequence itself has a constant ratio because consecutive terms roughly double, without computing exactly; x2/x1 = 5/3 but x3/x2 = 9/5, which are not equal, so the sequence x_n itself is not geometric (it is x_n - 1 that is geometric).
   • Why not C: This fits a linear formula 2n + 1 using only the first two terms, x1 = 3 and x2 = 5, without checking it against later terms; x3 = 9 but 2(3) + 1 = 7, so this formula is not correct beyond n = 2.

6. Question 6Answer: A

   1. Statement (1): the sum of the first n positive integers is a standard formula, S_n = n(n+1)/2 (for example, with n = 4: 1+2+3+4 = 10 = 4x5/2 = 10), so this is true.
   2. Statement (2): the series 1 + 1/2 + 1/4 + ... is geometric with first term a = 1 and common ratio r = 1/2, and since |r| < 1 it converges to a/(1-r) = 1/(1 - 1/2) = 2, so this is true.
   3. Statement (3): the series 2 + 3 + 4.5 + ... is geometric with common ratio r = 1.5, and since |r| = 1.5 is not less than 1, this series diverges and has no finite sum to infinity, so this statement is false.
   4. Statement (4): the series 1 + 4 + 7 + 10 + ... is arithmetic (not geometric) with common difference 3; an arithmetic series with a nonzero common difference grows without bound as more terms are added, so it also has no finite sum to infinity, making this statement false.
   5. Exactly two of the four statements, (1) and (2), are true, so the answer is A.
   • Why not B: This wrongly rejects statement (2) by computing the sum to infinity as a(1-r) instead of a/(1-r), giving 1(1 - 1/2) = 0.5 instead of 1/(1 - 1/2) = 2, so only statement (1) is left counted as true.
   • Why not C: This wrongly accepts statement (3), mistaking the condition for convergence as 'the common ratio is less than 2' rather than the correct condition |r| < 1; since r = 1.5 is greater than 1, this geometric series actually diverges.
   • Why not D: This wrongly accepts statement (3) as well, mistaking the condition for convergence as 'the common ratio is less than 2' rather than the correct condition |r| < 1, so the series with r = 1.5 is wrongly judged to converge; it also wrongly accepts statement (4), confusing the existence of a formula for the sum of the first n terms of an arithmetic series with convergence to a finite limit, when in fact the partial sums grow without bound as n increases because the common difference is nonzero. Combined with correctly accepting (1) and (2), this wrongly counts all four statements as true.

7. Question 7Answer: D

   1. Statement A: the expansion is the sum from k=0 to n of nCk x^k, which has one term for each integer k from 0 to n, giving n+1 terms in total - always true.
   2. Statement B: binomial coefficients satisfy nCk = nC(n-k) for every k, so the coefficients of x^k and x^(n-k) are always equal - always true.
   3. Statement C: setting x = 1 in (1+x)^n = sum of nCk x^k gives (1+1)^n = 2^n on the left, and the sum of every coefficient nCk on the right, so this identity always holds - always true.
   4. Statement D: the coefficient of x^n is the k=n term, nCn, which equals 1 for every n (there is exactly one way to choose all n items), not n itself; for example when n = 3, the coefficient of x^3 is 3C3 = 1, not 3.
   5. So statement D is the one that is NOT always true, and the answer is D.
   • Why not A: This misjudges the number of terms; the expansion sum from k=0 to n of nCk x^k has one term for each value of k from 0 to n inclusive, giving n+1 terms in total - this count is always correct, so rejecting this statement comes from missing that k starts at 0, not 1.
   • Why not B: This misjudges the symmetry property; the identity nCk = nC(n-k) holds for every binomial coefficient (for example, with n = 4, 4C1 = 4 and 4C3 = 4), so the coefficients of x^k and x^(n-k) are always equal - rejecting this statement may come from confusing it with a claim about the terms themselves, which differ in their power of x, rather than their coefficients.
   • Why not C: This misjudges the sum of coefficients; substituting x = 1 into (1+x)^n = sum of nCk x^k gives (1+1)^n = 2^n on one side and the sum of every coefficient nCk on the other, so this identity always holds - rejecting it may come from substituting x = 1 into only the general term nCk x^k rather than summing over all k.

8. Question 8Answer: D

   1. Suppose P holds, i.e. x = 3. Then |x| = |3| = 3, so Q holds. Hence P implies Q, so P is sufficient for Q.
   2. To check necessity, ask whether Q implies P: does |x| = 3 force x = 3? Take x = -3: |x| = 3, so Q holds, but x = -3 is not equal to 3, so P fails.
   3. So Q can be true while P is false, meaning P is not necessary for Q.
   4. Combining both parts, P is sufficient but not necessary for Q: answer D.
   • Why not A: This overlooks that P does guarantee Q: for x = 3, |x| = |3| = 3 follows immediately, so P is sufficient for Q; P is not 'neither'.
   • Why not B: This reverses sufficiency and necessity: it correctly notes that Q can be true without P (e.g. x = -3 gives |x| = 3), but wrongly concludes from this that P fails to be sufficient, rather than only failing to be necessary; whenever x = 3, |x| = 3 always follows, so P is sufficient.
   • Why not C: This assumes taking the absolute value preserves an 'if and only if' relationship, ignoring that Q is also satisfied by x = -3 (since |-3| = 3) where P fails, so P is not necessary for Q.

9. Question 9Answer: C

   1. Test option A with a = 4, b = 1: sqrt(4) - sqrt(1) = 2 - 1 = 1, but sqrt(4-1) = sqrt(3), which is not 1, so A is false.
   2. Test option B with a = b = 1: sqrt(1) + sqrt(1) = 1 + 1 = 2, but sqrt(1+1) = sqrt(2), which is approximately 1.414, so B is false.
   3. Test option C directly using the product law for surds: sqrt(a) * sqrt(b) = sqrt(a x b) is a standard, always-true identity for a, b >= 0, since both sides square to give ab.
   4. Test option D with a = 4, b = 1: sqrt(4)/sqrt(1) + sqrt(1)/sqrt(4) = 2 + 1/2 = 2.5, not 1, so D is false.
   5. Only option C holds for all positive a and b, so the answer is C.
   • Why not A: This wrongly extends the product law to subtraction in the same way; taking a = 4, b = 1, sqrt(4) - sqrt(1) = 2 - 1 = 1, but sqrt(4-1) = sqrt(3), which is not 1, so this identity also fails (and is undefined whenever b > a).
   • Why not B: This wrongly extends the product law sqrt(a) * sqrt(b) = sqrt(ab) to addition; taking a = b = 1 as a check, sqrt(1) + sqrt(1) = 2 but sqrt(1+1) = sqrt(2), which is approximately 1.414, not 2, so this identity fails.
   • Why not D: This assumes the two fractions sqrt(a)/sqrt(b) and sqrt(b)/sqrt(a) must sum to 1, as if they were reciprocals that always total 1, but sqrt(a)/sqrt(b) + sqrt(b)/sqrt(a) = (a+b)/sqrt(ab), which equals 1 only in the special case a+b = sqrt(ab), not for all positive a, b - for example a = 4, b = 1 gives 2 + 1/2 = 2.5, not 1.

10. Question 10Answer: D

    1. Statement (1): f(3) = 3^3 - 2(3)^2 - 5(3) + 6 = 27 - 18 - 15 + 6 = 0, so this is true.
    2. Statement (2): by the factor theorem, f(k) = 0 if and only if (x-k) is a factor of f(x); since f(3) = 0, (x-3) is indeed a factor, so this is true.
    3. Statement (3): by the remainder theorem, dividing f(x) by (x-1) leaves remainder f(1) = 1 - 2 - 5 + 6 = 0, so this is true (and (x-1) is also a factor).
    4. Statement (4): f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 8 - 8 - 10 + 6 = -4, which is not 0, so (x-2) is not a factor and this statement is false.
    5. Exactly three of the four statements, (1), (2) and (3), are true, so the answer is D.
    • Why not A: This correctly counts (1) as true but wrongly rejects (2), treating the factor theorem as only applying in one direction (that a factor implies a zero, but not that a zero implies a factor), when in fact it works both ways: f(3) = 0 if and only if (x-3) is a factor. It also wrongly rejects (3), perhaps not realising the remainder theorem generalises the factor theorem to any divisor, factor or not.
    • Why not B: This accepts (1) and (2) as true (correctly linking f(3) = 0 to (x-3) being a factor) but wrongly rejects (3), perhaps assuming the remainder theorem only applies to the specific value already tested (k=3) rather than to any value of k, missing that f(1) = 1 - 2 - 5 + 6 = 0 shows the remainder on division by (x-1) is also 0.
    • Why not C: This wrongly accepts (4) by miscalculating f(2): computing 2^3 - 2(2^2) - 5(2) + 6 by dropping the middle term's coefficient of 2, working out 8 - 4 - 10 + 6 = 0 instead of the correct 8 - 2(4) - 10 + 6 = 8 - 8 - 10 + 6 = -4, wrongly concluding (x-2) is a factor when the true remainder is -4.

11. Question 11Answer: C

    1. Substitute y = x + c into y = x^2 - 4x + 7 to find where the line meets the curve: x + c = x^2 - 4x + 7, which rearranges to x^2 - 5x + (7-c) = 0.
    2. For the line to meet the curve at two distinct points, this quadratic in x needs two distinct real roots, which requires its discriminant to be strictly positive.
    3. The discriminant is (-5)^2 - 4(1)(7-c) = 25 - 28 + 4c = 4c - 3.
    4. Setting 4c - 3 > 0 and solving gives c > 3/4.
    5. So the line meets the curve at two distinct points exactly when c > 3/4, which is option C.
    • Why not A: This includes the boundary value c = 3/4, where the discriminant is exactly 0 and the line is tangent to the curve (touching at exactly one point), rather than intersecting at two distinct points; the boundary should be excluded with a strict inequality.
    • Why not B: This gives only the single tangency value where the discriminant equals zero (one repeated intersection point), rather than the whole range of c for which there are two distinct intersection points.
    • Why not D: This reverses the inequality direction obtained from 4c - 3 > 0, incorrectly solving it as c < 3/4, as if dividing by 4 also flipped the inequality, which only happens when dividing by a negative number.

12. Question 12Answer: C

    1. Statement (1): 6 = 2 x 3, so 3 is a factor of 6; any n divisible by 6 is therefore also divisible by 3. This is true.
    2. Statement (2): try n = 2. Then n^2 = 4, which is divisible by 4, but n = 2 is not divisible by 4. This single counterexample makes (2) false.
    3. Statement (3): every even integer greater than 2 has a factor of 2 in addition to 1 and itself, so it cannot be prime; 2 is the only even prime, so every prime greater than 2 must be odd. This is true.
    4. Statement (4): try n = 6. It is divisible by both 2 and 3, but 6 is not divisible by 12. This counterexample makes (4) false.
    5. Exactly two of the four statements, (1) and (3), are true, so the answer is C.
    • Why not A: This correctly accepts (1) but wrongly rejects (3), perhaps by treating 2 itself as a counterexample even though the statement explicitly restricts to primes greater than 2; every prime greater than 2 is in fact odd, since 2 is the only even prime.
    • Why not B: This wrongly accepts (2) as well as (1) and (3), using n = 2 to test it but concluding n^2 = 4 divisible by 4 'confirms' the statement rather than checking whether n itself is divisible by 4 - since 2 is not divisible by 4, n = 2 is actually a counterexample that makes (2) false.
    • Why not D: This wrongly accepts statement (2) as well, testing n = 2 and taking n^2 = 4 divisible by 4 as 'confirming' the statement rather than checking whether n itself is divisible by 4 (it is not, so n = 2 is actually a counterexample); it also wrongly accepts statement (4), confusing 'divisible by 2 and by 3' with 'divisible by 2 x 3 x 2 = 12', when in fact the lowest common multiple of 2 and 3 is 6, not 12, and n = 6 satisfies the hypothesis of (4) but is not divisible by 12, so (4) is actually false. Combined with correctly accepting (1) and (3), this wrongly counts all four statements as true.