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Admissions tests / ESAT / Maths 2 / Algebra, functions and sequences

Test standard. 15 questions, 15 marks, about 22 minutes.

ESAT Mathematics 2: Algebra, functions and sequences, set 2

Laws of indices, surds, quadratics and the discriminant, completing the square, simultaneous equations, inequalities, polynomials, the factor theorem, sequences, series and the binomial expansion.

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  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  1. 11 mark

    Simplify x^(2/3) x x^(1/6) fully, giving your answer as a single power of x.

    1. A x^(1/3)
    2. B x^(5/6)
    3. C x^(1/2)
    4. D x^(1/9)
  2. 21 mark

    Simplify (sqrt(3) + 1)/(sqrt(3) - 1) fully, giving your answer in the form p + qsqrt(3).

    1. A 2
    2. B 2 - sqrt(3)
    3. C 1 + sqrt(3)/2
    4. D 2 + sqrt(3)
  3. 31 mark

    Simplify sqrt(75) - sqrt(12) + sqrt(48) fully, giving your answer in the form ksqrt(3).

    1. A 7sqrt(3)
    2. B 3sqrt(3)
    3. C -sqrt(3)
    4. D 11sqrt(3)
  4. 41 mark

    The equation 2x^2 + 3x + k = 0 has no real roots. Find the range of possible values of k.

    1. A k > 9/2
    2. B k > 3/4
    3. C k > 9/8
    4. D k < 9/8
  5. 51 mark

    By completing the square, find the minimum value of f(x) = x^2 - 8x + 21.

    1. A -16
    2. B 5
    3. C 37
    4. D 4
  6. 61 mark

    The line y = x + 4 and the curve y = x^2 + 2x - 2 intersect at two points. Find the y-coordinates of the points of intersection.

    1. A y = 1 or y = 6
    2. B x = -3 or x = 2
    3. C y = 6 only
    4. D y = -7 or y = -2
  7. 71 mark

    Solve the inequality 7 - 3x >= 1.

    1. A x <= -8/3
    2. B x >= 2
    3. C x <= -2
    4. D x <= 2
  8. 81 mark

    Divide x^4 - 5x^2 + 4 by (x^2 - 4), giving the quotient.

    1. A x^4 - 1
    2. B x^2 + 1
    3. C x^2 - 1
    4. D x^2 - 4
  9. 91 mark

    Given that (x - 1) is a factor of x^3 + ax^2 - 5x + 3, find the value of a.

    1. A 1
    2. B -7
    3. C -3
    4. D -9
  10. 101 mark

    For all real values of x, sqrt(x^2) is equal to:

    1. A x
    2. B |x|
    3. C x^2
    4. D -x
  11. 111 mark

    A sequence has nth term u_n = n^2 + 3n - 2 for n = 1, 2, 3, .... Find the value of n for which u_n = 38.

    1. A n = 10
    2. B n = 5 or n = -8
    3. C n = 8
    4. D n = 5
  12. 121 mark

    Find the sum of the first 25 terms of the arithmetic series 7 + 11 + 15 + 19 + ...

    1. A 525
    2. B 1425
    3. C 1375
    4. D 110
  13. 131 mark

    Find the sum of the first 5 terms of the geometric series 4 - 8 + 16 - 32 + ...

    1. A -20
    2. B 44
    3. C 124
    4. D -44
  14. 141 mark

    The first term of a convergent geometric series is 8 and the second term is 6. Find the sum to infinity of the series.

    1. A 32
    2. B -24
    3. C -8/5
    4. D 32/7
  15. 151 mark

    Find the coefficient of x^2 in the binomial expansion of (2 - x)^5.

    1. A 40
    2. B 160
    3. C -80
    4. D 80

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: B

    1. The law of indices for multiplying powers of the same base is x^m x x^n = x^(m+n), so the exponents must be added.
    2. Write 2/3 with the same denominator as 1/6: 2/3 = 4/6.
    3. Add the exponents: 4/6 + 1/6 = 5/6.
    4. So x^(2/3) x x^(1/6) = x^(5/6), and the answer is B.
    • Why not A: Adds the numerators and denominators of the two fractions separately, treating 2/3 + 1/6 as (2+1)/(3+6) = 3/9 = 1/3, instead of converting to a common denominator first.
    • Why not C: Subtracts the exponents instead of adding them, as if dividing rather than multiplying: 2/3 - 1/6 = 4/6 - 1/6 = 1/2.
    • Why not D: Multiplies the two exponents together instead of adding them, treating x^m x x^n as x^(mn): (2/3) x (1/6) = 2/18 = 1/9.

  2. Question 2Answer: D

    1. Multiply numerator and denominator by the conjugate of the denominator, sqrt(3) + 1.
    2. The denominator becomes (sqrt(3) - 1)(sqrt(3) + 1) = (sqrt(3))^2 - 1^2 = 3 - 1 = 2.
    3. The numerator becomes (sqrt(3) + 1)^2 = (sqrt(3))^2 + 2sqrt(3) + 1^2 = 3 + 2sqrt(3) + 1 = 4 + 2sqrt(3).
    4. Dividing gives (4 + 2sqrt(3))/2 = 2 + sqrt(3), so the answer is D.
    • Why not A: Expands the numerator (sqrt(3) + 1)^2 as (sqrt(3))^2 + 1^2 = 3 + 1 = 4, forgetting the middle 'cross' term 2 x sqrt(3) x 1, and then divides this incomplete result by the correct denominator of 2 to get 4/2 = 2.
    • Why not B: Expands the numerator using the pattern for (a - b)^2 instead of (a + b)^2, computing (sqrt(3))^2 - 2sqrt(3) + 1^2 = 4 - 2sqrt(3), and divides by the correct denominator of 2 to get 2 - sqrt(3).
    • Why not C: Computes the denominator as (sqrt(3))^2 + 1^2 = 3 + 1 = 4, mistaking the difference of two squares for a sum of two squares, instead of the correct (sqrt(3))^2 - 1^2 = 3 - 1 = 2; dividing the correctly expanded numerator 4 + 2sqrt(3) by 4 instead of 2 gives 1 + sqrt(3)/2.

  3. Question 3Answer: A

    1. Simplify each surd by extracting its largest square factor: sqrt(75) = sqrt(25 x 3) = 5sqrt(3).
    2. sqrt(12) = sqrt(4 x 3) = 2sqrt(3), and sqrt(48) = sqrt(16 x 3) = 4sqrt(3).
    3. The expression becomes 5sqrt(3) - 2sqrt(3) + 4sqrt(3), and since all three terms are multiples of sqrt(3), the coefficients combine directly: 5 - 2 + 4 = 7.
    4. So sqrt(75) - sqrt(12) + sqrt(48) = 7sqrt(3), and the answer is A.
    • Why not B: Combines only the first two terms, computing 5sqrt(3) - 2sqrt(3) = 3sqrt(3), and then forgets to add the third term, +4sqrt(3), back in at all.
    • Why not C: Misreads the second '+' sign in the expression, treating it as sqrt(75) - sqrt(12) - sqrt(48) instead of sqrt(75) - sqrt(12) + sqrt(48), giving 5 - 2 - 4 = -1 as the coefficient of sqrt(3).
    • Why not D: Miscalculates sqrt(16) as 8 instead of 4 when simplifying sqrt(48) = sqrt(16 x 3), so treats sqrt(48) as 8sqrt(3) instead of 4sqrt(3), giving 5 - 2 + 8 = 11 as the coefficient of sqrt(3).

  4. Question 4Answer: C

    1. For ax^2 + bx + c = 0 to have no real roots, the discriminant b^2 - 4ac must be strictly negative.
    2. Here a = 2, b = 3 and c = k, so the discriminant is 3^2 - 4(2)(k) = 9 - 8k.
    3. The condition for no real roots is 9 - 8k < 0, which rearranges to 8k > 9, so k > 9/8.
    4. So the answer is C, k > 9/8.
    • Why not A: Uses b^2 - ac instead of the correct b^2 - 4ac for the discriminant, omitting the factor of a itself: 9 - (2)k < 0 gives k > 9/2.
    • Why not B: Miscalculates b^2 as 3 x 2 = 6 instead of 3^2 = 9, then correctly requires 6 - 8k < 0, giving k > 6/8 = 3/4.
    • Why not D: Assumes 'no real roots' corresponds to the discriminant being positive rather than negative, solving 9 - 8k > 0 instead of 9 - 8k < 0, which reverses the direction of the inequality.

  5. Question 5Answer: B

    1. Halve the coefficient of x (which is -8) to get -4, and write x^2 - 8x + 21 as (x - 4)^2 - 16 + 21.
    2. Combine the constants: -16 + 21 = 5, so f(x) = (x - 4)^2 + 5.
    3. Since (x - 4)^2 >= 0 for all real x, with equality when x = 4, the minimum value of f(x) occurs at x = 4.
    4. The minimum value is therefore 5, so the answer is B.
    • Why not A: Correctly writes x^2 - 8x + 21 as (x - 4)^2 - 16 + 21, but stops after the -16 step and forgets to add back the +21, reporting the intermediate constant -16 as the minimum value.
    • Why not C: Confuses the sign in the identity (x - p)^2 = x^2 - 2px + p^2, treating it as x^2 - 2px - p^2, so believes p^2 must be added back rather than subtracted: computes -(-16) + 21 as +16 + 21 = 37.
    • Why not D: Reports the x-coordinate of the minimum point (x = 4, from (x - 4)^2) instead of the minimum value of the function itself.

  6. Question 6Answer: A

    1. Substitute the linear expression for y into the quadratic one: x + 4 = x^2 + 2x - 2.
    2. Rearrange to standard form: 0 = x^2 + 2x - 2 - x - 4, which simplifies to x^2 + x - 6 = 0.
    3. Factorise: the two numbers that multiply to -6 and add to 1 are 3 and -2, giving (x + 3)(x - 2) = 0, so x = -3 or x = 2.
    4. Substituting back into y = x + 4: when x = -3, y = 1; when x = 2, y = 6. So the answer is A.
    • Why not B: Correctly solves the simultaneous equations for x, finding x = -3 or x = 2, but stops there and reports these x-coordinates instead of substituting them back in to find the y-coordinates the question asks for.
    • Why not C: Finds only one root of the resulting quadratic (x = 2) and reports the corresponding y-value, without noticing that the quadratic x^2 + x - 6 = 0 has a second root, x = -3, giving a second intersection point.
    • Why not D: Finds the correct x-values, -3 and 2, but substitutes them into y = x - 4 instead of the given line y = x + 4, subtracting 4 instead of adding it: -3 - 4 = -7 and 2 - 4 = -2.

  7. Question 7Answer: D

    1. Rearrange 7 - 3x >= 1 by subtracting 7 from both sides: -3x >= 1 - 7, which gives -3x >= -6.
    2. Divide both sides by -3. Since -3 is negative, the inequality sign must reverse.
    3. This gives x <= (-6)/(-3) = 2.
    4. So the answer is D, x <= 2.
    • Why not A: Rearranges by adding 7 to the right-hand side instead of subtracting it, computing -3x >= 1 + 7 = 8 instead of -3x >= 1 - 7 = -6, then divides by -3 and correctly flips the inequality to get x <= -8/3.
    • Why not B: Correctly rearranges to -3x >= -6, but forgets to reverse the inequality sign when dividing both sides by the negative number -3, giving x >= 2 instead of x <= 2.
    • Why not C: Miscalculates 1 - 7 as +6 instead of -6, then divides -3x >= 6 by -3 and correctly flips the inequality, arriving at x <= -2 instead of x <= 2.

  8. Question 8Answer: C

    1. Divide the leading terms: x^4 / x^2 = x^2. Multiply x^2 by the divisor: x^2(x^2 - 4) = x^4 - 4x^2.
    2. Subtract this from the original polynomial: (x^4 - 5x^2 + 4) - (x^4 - 4x^2) = -x^2 + 4.
    3. Divide the new leading term: -x^2 / x^2 = -1. Multiply -1 by the divisor: -1(x^2 - 4) = -x^2 + 4, which matches exactly, leaving remainder 0.
    4. The quotient is x^2 - 1, so x^4 - 5x^2 + 4 = (x^2 - 4)(x^2 - 1), and the answer is C.
    • Why not A: Fails to reduce the power of x when dividing the leading terms, treating x^4 divided by x^2 as leaving x^4 unchanged rather than reducing it to x^2, and carries that error through to a final quotient of x^4 - 1.
    • Why not B: Makes a sign error on the final step of the division: after subtracting to reach the remaining term -x^2 + 4, mishandles the sign of this subtraction and produces a next quotient term of +1 instead of the correct -1.
    • Why not D: Misidentifies the quotient as being the same as the divisor itself, without carrying out the division at all.

  9. Question 9Answer: A

    1. By the Factor Theorem, since (x - 1) is a factor, substituting x = 1 into the polynomial must give 0.
    2. Let f(x) = x^3 + ax^2 - 5x + 3. Then f(1) = 1^3 + a(1)^2 - 5(1) + 3 = 1 + a - 5 + 3.
    3. Combine the known constants: 1 - 5 + 3 = -1, so f(1) = a - 1.
    4. Setting f(1) = 0 gives a - 1 = 0, so a = 1, and the answer is A.
    • Why not B: Misapplies the Factor Theorem, evaluating the polynomial at x = -1 instead of at x = 1, since the factor is (x - 1): (-1)^3 + a(-1)^2 - 5(-1) + 3 = -1 + a + 5 + 3 = a + 7, and setting this to 0 gives a = -7.
    • Why not C: Correctly substitutes x = 1, but miscalculates -5(1) + 3 as +2 instead of -2, giving 1 + a + 2 = a + 3 = 0, so a = -3.
    • Why not D: Correctly substitutes x = 1, but makes a sign error on the -5x term, treating -5(1) as +5 instead of -5: 1 + a + 5 + 3 = a + 9 = 0, so a = -9.

  10. Question 10Answer: B

    1. The symbol sqrt always denotes the non-negative square root of a number.
    2. If x is positive, sqrt(x^2) = x, which is already non-negative, so this matches |x|.
    3. If x is negative, for example x = -3, then sqrt(x^2) = sqrt(9) = 3, which is the positive value -x, and this also matches |x|.
    4. Since sqrt(x^2) gives x when x >= 0 and -x when x < 0, it is equal to |x| for every real x, so the answer is B.
    • Why not A: Assumes squaring and square-rooting always cancel to leave x unchanged, which fails for negative x: for example, sqrt((-3)^2) = sqrt(9) = 3, not -3.
    • Why not C: Confuses sqrt(x^2) with squaring x^2 again rather than taking a square root at all, wrongly concluding that no simplification takes place and the expression stays as x^2.
    • Why not D: Overcorrects for the possibility that x is negative by assuming the sign must always flip, which fails for positive x: for example, sqrt(3^2) = sqrt(9) = 3, not -3.

  11. Question 11Answer: D

    1. Set the formula equal to 38: n^2 + 3n - 2 = 38.
    2. Rearrange to standard form: n^2 + 3n - 40 = 0.
    3. Factorise: the two numbers that multiply to -40 and add to 3 are 8 and -5, giving (n + 8)(n - 5) = 0, so n = -8 or n = 5.
    4. Since n must be a positive whole number (a term-number), n = -8 is rejected, leaving n = 5, so the answer is D.
    • Why not A: Misfactorises n^2 + 3n - 40, choosing the factor pair 4 and -10 because they multiply to -40, without checking that they must also sum to the middle coefficient 3 (they sum to -6, not 3); this gives roots n = -4 or n = 10, and the positive one, n = 10, is reported.
    • Why not B: Correctly factorises to (n + 8)(n - 5) = 0 and finds both roots, n = -8 and n = 5, but reports both without discarding the negative one, even though n is a positive term-number and cannot equal -8.
    • Why not C: Reads the factor (n + 8) = 0 as giving the root n = 8 and the factor (n - 5) = 0 as giving the root n = -5, flipping the sign in both cases, and then reports the value that looks positive, n = 8, as the valid term-number.

  12. Question 12Answer: C

    1. This is an arithmetic series with first term a = 7 and common difference d = 4.
    2. The sum of the first n terms is S_n = (n/2)(2a + (n - 1)d). Here n = 25, so (n - 1)d = 24 x 4 = 96.
    3. Then 2a + (n - 1)d = 14 + 96 = 110.
    4. So S_25 = (25/2) x 110 = 25 x 55 = 1375, and the answer is C.
    • Why not A: Computes (n - 1) x d as (n - 1) + d = 24 + 4 = 28 instead of 24 x 4 = 96, adding instead of multiplying, giving 2a + 28 = 14 + 28 = 42, and then S = 25/2 x 42 = 525.
    • Why not B: Uses the formula with nd instead of (n - 1)d, forgetting to subtract 1 from the number of terms: 2a + nd = 14 + 100 = 114, giving S = 25/2 x 114 = 1425.
    • Why not D: Correctly computes the bracket 2a + (n - 1)d = 14 + 96 = 110, but forgets to multiply by n/2 to complete the formula, giving the bracket value itself as the final answer.

  13. Question 13Answer: B

    1. This is a geometric series with first term a = 4, common ratio r = -2, and n = 5 terms.
    2. The sum of the first n terms is S_n = a(r^n - 1)/(r - 1).
    3. Substitute the values: S_5 = 4((-2)^5 - 1)/(-2 - 1) = 4(-32 - 1)/(-3) = 4(-33)/(-3).
    4. 4 x (-33) = -132, and -132/(-3) = 44, so the sum of the first 5 terms is 44, and the answer is B.
    • Why not A: Adds only the first 4 terms (4 - 8 + 16 - 32 = -20), stopping one term early instead of including the 5th term, 64.
    • Why not C: Drops the negative sign of the common ratio, using r = 2 instead of r = -2 throughout: a(r^5 - 1)/(r - 1) = 4(32 - 1)/(2 - 1) = 4 x 31 = 124.
    • Why not D: Mixes the two equivalent forms of the sum formula, using (1 - r^n) in the numerator but (r - 1) in the denominator: 4(1 - (-32))/(-2 - 1) = 4(33)/(-3) = -44, which is the negative of the correct value.

  14. Question 14Answer: A

    1. The common ratio is found by dividing the second term by the first term: r = 6/8 = 3/4.
    2. Since |r| = 3/4 < 1, the series converges and has a sum to infinity, given by S = a/(1 - r).
    3. Here a = 8 and r = 3/4, so 1 - r = 1/4.
    4. S = 8/(1/4) = 8 x 4 = 32, so the answer is A.
    • Why not B: Finds the common ratio 'upside down', dividing the first term by the second instead of the second by the first: r = 8/6 = 4/3, and applies the sum formula despite this value of r not actually giving a convergent series: 8/(1 - 4/3) = 8/(-1/3) = -24.
    • Why not C: Misreads the second term itself as the common ratio, without dividing by the first term at all: using r = 6 gives 8/(1 - 6) = 8/(-5) = -8/5.
    • Why not D: Correctly finds the common ratio r = 3/4, but uses the formula a/(1 + r) instead of the correct a/(1 - r): 8/(1 + 3/4) = 8/(7/4) = 32/7.

  15. Question 15Answer: D

    1. The general term in the expansion of (2 - x)^5 is 5Ck x 2^(5-k) x (-x)^k, for k = 0, 1, ..., 5.
    2. The term in x^2 occurs when k = 2, giving 5C2 x 2^(5-2) x (-x)^2 = 5C2 x 2^3 x x^2.
    3. 5C2 = 5!/(2! x 3!) = (5 x 4)/(2 x 1) = 10, 2^3 = 8, and (-x)^2 = x^2 (squaring a negative gives a positive).
    4. The coefficient of x^2 is therefore 10 x 8 = 80, so the answer is D.
    • Why not A: Uses 2^k instead of the correct 2^(5-k) for the power of 2 attached to each term, computing 2^2 = 4 instead of 2^3 = 8, giving 5C2 x 4 = 10 x 4 = 40.
    • Why not B: Miscalculates 5C2, computing 5 x 4 = 20 instead of dividing by 2! to get the correct value of 10, then multiplies by the correct 2^3 = 8 to give 20 x 8 = 160.
    • Why not C: Wrongly assumes that any term coming from (-x) must carry a negative sign, without checking that (-x)^2 = x^2 is positive because the power 2 is even.

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