Revision Libraryv1.6.8
Choose a subject →
Search
WorksheetsBooksFreePracticeHow it worksPricingFreeAboutFAQ
Choose a subject →

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

Stretch. 15 questions, 15 marks, about 30 minutes.

ESAT Mathematics 2: Algebra, functions and sequences, set 4

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

Download the questions (PDF) Download with worked solutions (PDF)

  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  • This is a stretch set: several questions combine two specification points, and the fastest route is rarely the first one you see.
  1. 11 mark

    Simplify 20^(3/2) fully, giving your answer in the form k sqrt(5).

    1. A 40 sqrt(5)
    2. B 16000 sqrt(5)
    3. C 2 sqrt(5)
    4. D 8 sqrt(5)
  2. 21 mark

    Find the value of c for which the line y = 3x + c is a tangent to the curve y = x^2 + 5x + 7.

    1. A c = 3
    2. B c = -9
    3. C c = 6
    4. D c = -6
  3. 31 mark

    Solve the inequality |x^2 - 9| < 7.

    1. A sqrt(2) < x < 4
    2. B -4 < x < -sqrt(2) or sqrt(2) < x < 4
    3. C -4 < x < 4
    4. D 2 < x < 16
  4. 41 mark

    A sequence is defined by u_1 = 2 and u_(n+1) = u_n + 3n for n >= 1. Find u_8.

    1. A 110
    2. B 84
    3. C 23
    4. D 86
  5. 51 mark

    A geometric series has second term 4 and sum to infinity 18. Find the possible value(s) of the common ratio r.

    1. A r = 2/3 only
    2. B r = 1/3 or r = 2/3
    3. C r = -1/3 or r = -2/3
    4. D r = 2/11
  6. 61 mark

    Find the coefficient of x^2 in the expansion of (1 + x)(1 + 3x)^4.

    1. A 54
    2. B 120
    3. C 30
    4. D 66
  7. 71 mark

    f(x) = x^3 + a x^2 + b x - 4. Given that (x - 1) is a factor of f(x), and that the remainder when f(x) is divided by (x + 1) is 8, find the value of a.

    1. A a = 8
    2. B a = 7
    3. C a = 0
    4. D a = 6
  8. 81 mark

    f(x) = sqrt(x^2 - 6x + 9) for all real x. Write f(x) without a square root sign, for the domain x < 3.

    1. A x - 3
    2. B x^2 - 6x + 9
    3. C 3 - x
    4. D |x - 3|
  9. 91 mark

    Find the range of values of k for which the line y = kx + 1 meets the curve y = x^2 - 3x + 5 at two distinct points.

    1. A k > 1 only
    2. B k < -1 or k > 7
    3. C -7 < k < 1
    4. D k < -7 or k > 1
  10. 101 mark

    Given that 4^x = 8^y and that 2^(x - y) = 16, find the value of x.

    1. A x = 12
    2. B x = -12
    3. C x = 8
    4. D x = 12/5
  11. 111 mark

    An arithmetic series has third term 11, and the sum of its first 10 terms is 185. Find the common difference.

    1. A d = 5
    2. B d = 3
    3. C d = 5/2
    4. D d = -5
  12. 121 mark

    The second term of a geometric series is 6 and the fifth term is 162. Find the common ratio r.

    1. A r = 1/3
    2. B r = 3 sqrt(3)
    3. C r = 27
    4. D r = 3
  13. 131 mark

    Simplify (sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) + (sqrt(5) - sqrt(3))/(sqrt(5) + sqrt(3)), giving your answer as an integer.

    1. A 16
    2. B 2
    3. C 8
    4. D 2 sqrt(15)
  14. 141 mark

    f(x) = x^2 - 4x + 7. Find the smallest value of k such that f is one-to-one on the domain x >= k.

    1. A k = 2
    2. B k = 3
    3. C k = 4
    4. D k = -2
  15. 151 mark

    In the binomial expansion of (1 + x)^n, the sum of all the binomial coefficients is 128. Find the coefficient of x^3.

    1. A 20
    2. B 35
    3. C 56
    4. D 840

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. 20^(3/2) means (20^(1/2))^3, or equivalently 20 x 20^(1/2), since the exponent 3/2 = 1 + 1/2.
    2. First simplify the surd: 20^(1/2) = sqrt(20) = sqrt(4 x 5) = 2sqrt(5).
    3. So 20^(3/2) = 20 x sqrt(20) = 20 x 2sqrt(5) = 40sqrt(5). (Equivalently, (2sqrt(5))^3 = 2^3 x (sqrt(5))^3 = 8 x 5sqrt(5) = 40sqrt(5).)
    4. So the answer is A.
    • Why not B: Misreads the fractional exponent 3/2 as the mixed number 3 and a half, so computes 20^(3 + 1/2) = 20^3 x 20^(1/2) = 8000 x 2sqrt(5) = 16000sqrt(5), instead of treating 3/2 as a single power.
    • Why not C: Applies only the denominator of the exponent, computing 20^(1/2) = sqrt(20) = 2sqrt(5), and forgets that the numerator 3 also needs to be applied.
    • Why not D: Correctly reduces 20^(3/2) to (2sqrt(5))^3, but then cubes only the rational factor 2 to get 8, leaving sqrt(5) unraised instead of also cubing it to 5sqrt(5).

  2. Question 2Answer: C

    1. A line is tangent to a curve exactly when the simultaneous equations have equal roots, so set x^2 + 5x + 7 = 3x + c.
    2. Rearrange to standard form: x^2 + 5x - 3x + 7 - c = 0, so x^2 + 2x + (7 - c) = 0.
    3. For equal roots, the discriminant must be zero: 2^2 - 4(1)(7 - c) = 0, so 4 - 28 + 4c = 0, giving 4c = 24.
    4. So c = 6, and the answer is C.
    • Why not A: Uses the equal-roots condition b^2 = ac instead of the correct b^2 = 4ac, dropping the factor of 4: 4 = 1 x (7 - c) gives c = 3.
    • Why not B: Makes a sign error moving the 3x term across, adding it instead of subtracting it, so combines the x-terms as +8x instead of the correct +2x; setting the discriminant of x^2 + 8x + (7 - c) = 0 to zero, 64 - 4(7 - c) = 0, gives c = -9.
    • Why not D: Makes a sign error moving c across, keeping the equation as x^2 + 2x + (7 + c) = 0 instead of (7 - c); setting its discriminant to zero, 4 - 4(7 + c) = 0, gives c = -6.

  3. Question 3Answer: B

    1. |x^2 - 9| < 7 means -7 < x^2 - 9 < 7. Add 9 throughout: 2 < x^2 < 16.
    2. From x^2 < 16: -4 < x < 4.
    3. From x^2 > 2: x < -sqrt(2) or x > sqrt(2), since both a positive and a negative x with |x| > sqrt(2) give x^2 > 2.
    4. Combining both conditions gives -4 < x < -sqrt(2) or sqrt(2) < x < 4, so the answer is B.
    • Why not A: Correctly reaches x^2 > 2, but takes only the positive square root branch x > sqrt(2), forgetting that a negative x with |x| > sqrt(2) also satisfies x^2 > 2.
    • Why not C: Solves x^2 < 16 correctly but ignores the lower bound of the modulus inequality entirely, treating |x^2 - 9| < 7 as only an upper-bound condition on x^2 - 9.
    • Why not D: Drops the square on x entirely, treating |x^2 - 9| < 7 as though it were |x - 9| < 7, and solves the resulting linear inequality -7 < x - 9 < 7 to get 2 < x < 16.

  4. Question 4Answer: D

    1. Each step adds 3n, so u_8 = u_1 + (increment at n=1) + (increment at n=2) + ... + (increment at n=7) = 2 + 3(1 + 2 + ... + 7).
    2. The sum of the first 7 natural numbers is 7 x 8 / 2 = 28.
    3. So u_8 = 2 + 3 x 28 = 2 + 84 = 86.
    4. So the answer is D.
    • Why not A: Sums the increments from k = 1 to k = n instead of k = 1 to k = n - 1, an off-by-one error: 2 + 3 x (8 x 9)/2 = 2 + 108 = 110.
    • Why not B: Correctly computes the total of the increments, 3 x (7 x 8)/2 = 84, but forgets to add on the starting value u_1 = 2.
    • Why not C: Misreads the recurrence as a simple arithmetic sequence with common difference 3, ignoring that the increment 3n grows with n: u_8 = u_1 + 7 x 3 = 2 + 21 = 23.

  5. Question 5Answer: B

    1. Let the first term be a. Then ar = 4, so a = 4/r, and a/(1 - r) = 18.
    2. Substitute: (4/r)/(1 - r) = 18, so 4 = 18r(1 - r), giving 18r - 18r^2 = 4, or 9r^2 - 9r + 2 = 0.
    3. By the quadratic formula, r = (9 +- sqrt(81 - 72))/18 = (9 +- 3)/18, so r = 2/3 or r = 1/3.
    4. Both satisfy |r| < 1, so both are valid, and the answer is B.
    • Why not A: Correctly solves the quadratic 9r^2 - 9r + 2 = 0 but reports only the larger root, discarding the smaller valid solution.
    • Why not C: Makes a sign error applying the quadratic formula to 9r^2 - 9r + 2 = 0 (where b = -9), using -b = -9 instead of the correct -b = 9, giving r = (-9 +- 3)/18.
    • Why not D: Inverts the sum-to-infinity formula, using a(1 - r) = 18 instead of the correct a/(1 - r) = 18, which combined with ar = 4 gives r = 2/11.

  6. Question 6Answer: D

    1. In (1 + 3x)^4, the coefficient of x^1 is 4C1 x 3 = 12, and the coefficient of x^2 is 4C2 x 3^2 = 6 x 9 = 54.
    2. So (1 + 3x)^4 = 1 + 12x + 54x^2 + ... (higher powers of x do not affect the x^2 term below).
    3. Multiplying by (1 + x), the x^2 term comes from two products: 1 x 54x^2 (from the constant in the first bracket) and x x 12x (from the x in the first bracket).
    4. Adding these gives 54 + 12 = 66, so the coefficient of x^2 is 66 and the answer is D.
    • Why not A: Takes only the x^2 term of (1 + 3x)^4 (which is 54x^2, from multiplying by the constant 1 in the first bracket), and forgets the second contribution that comes from multiplying x by the x^1 term of (1 + 3x)^4.
    • Why not B: Miscalculates 4C2 as 12 instead of 6 (for example by computing 4 x 3 without dividing by 2!), giving an x^2 coefficient of 12 x 9 = 108 from (1 + 3x)^4, then adds the correct cross-term contribution of 12 to get 120.
    • Why not C: Forgets to square the coefficient 3 in the (3x)^2 term, using 4C2 x 3 = 6 x 3 = 18 instead of 4C2 x 3^2 = 54, then adds the correct cross-term contribution of 12 to get 30.

  7. Question 7Answer: A

    1. By the Factor Theorem, (x - 1) a factor means f(1) = 0: 1 + a + b - 4 = 0, so a + b = 3.
    2. By the Remainder Theorem, dividing by (x + 1) gives remainder f(-1): -1 + a - b - 4 = 8, so a - b = 13.
    3. Adding the two equations: 2a = 16, so a = 8.
    4. So the answer is A. (This also gives b = -5, and f(x) = x^3 + 8x^2 - 5x - 4 checks: f(1) = 1+8-5-4 = 0 and f(-1) = -1+8+5-4 = 8.)
    • Why not B: Makes a sign error substituting x = -1 into the cubic term, using (-1)^3 = +1 instead of the correct -1, which turns the remainder equation into a - b - 3 = 8 instead of a - b - 5 = 8.
    • Why not C: Misapplies the Remainder Theorem, setting f(-1) equal to -8 instead of the given remainder +8.
    • Why not D: Correctly substitutes x = -1 into every term of x^3 + ax^2 + bx, but forgets to also include the polynomial's own constant term -4 in the remainder equation.

  8. Question 8Answer: C

    1. x^2 - 6x + 9 is a perfect square: x^2 - 6x + 9 = (x - 3)^2.
    2. So f(x) = sqrt((x - 3)^2) = |x - 3|, since a square root always gives the non-negative value (f is a many-to-one mapping in x, but sqrt itself is single-valued and non-negative).
    3. For x < 3, x - 3 < 0, so |x - 3| = -(x - 3) = 3 - x.
    4. So the answer is C.
    • Why not A: Assumes sqrt(a^2) = a directly, ignoring that a square root always denotes the non-negative root; for x < 3, x - 3 is negative, so it cannot equal sqrt of a square.
    • Why not B: Forgets to take the square root at all, leaving the expression as the original quantity under the root sign.
    • Why not D: Correctly reduces the expression to the modulus form |x - 3|, but does not go on to use the given domain x < 3 to remove the modulus sign.

  9. Question 9Answer: D

    1. Set the equations equal: x^2 - 3x + 5 = kx + 1, so x^2 - (3 + k)x + 4 = 0.
    2. Two distinct intersection points require a positive discriminant: (3 + k)^2 - 4(1)(4) > 0, so (3 + k)^2 > 16.
    3. This means |3 + k| > 4, so 3 + k > 4 or 3 + k < -4, giving k > 1 or k < -7.
    4. So the answer is D.
    • Why not A: Correctly finds the critical values from |3 + k| > 4, but reports only the branch k > 1, missing the second branch k < -7.
    • Why not B: Makes a sign error combining the two x-terms when rearranging, treating -3x - kx as (k - 3)x instead of the correct -(3 + k)x, which shifts the critical values from 1 and -7 to 7 and -1.
    • Why not C: Inverts the logic of the discriminant condition, requiring the discriminant to be negative for two distinct real roots instead of positive, giving the interval between the critical values instead of outside them.

  10. Question 10Answer: A

    1. Since 4 = 2^2 and 8 = 2^3, the first equation 4^x = 8^y becomes 2^(2x) = 2^(3y), so 2x = 3y.
    2. Since 16 = 2^4, the second equation gives x - y = 4.
    3. From 2x = 3y, y = 2x/3. Substituting: x - 2x/3 = 4, so x/3 = 4.
    4. So x = 12, and the answer is A. (This gives y = 8, and both original equations check.)
    • Why not B: Misapplies the index law converting 4^x to base 2, using 4^x = 2^(4x) instead of the correct 2^(2x), which turns the first equation into 4x = 3y.
    • Why not C: Misremembers the power of 2 equal to 8, using 8 = 2^4 instead of the correct 8 = 2^3, which turns the first equation into 2x = 4y.
    • Why not D: Misreads the second equation, treating 2^(x - y) = 16 as x + y = 4 instead of the correct x - y = 4.

  11. Question 11Answer: B

    1. The third term gives a + 2d = 11.
    2. The sum of the first 10 terms gives (10/2)(2a + 9d) = 185, so 2a + 9d = 37.
    3. From the first equation, a = 11 - 2d. Substitute: 2(11 - 2d) + 9d = 37, so 22 + 5d = 37, giving 5d = 15.
    4. So d = 3, and the answer is B.
    • Why not A: Uses a + 3d for the third term instead of the correct a + 2d, an off-by-one indexing error in which term counts as the first.
    • Why not C: Uses S_n = (n/2)(2a + nd) instead of the correct S_n = (n/2)(2a + (n-1)d), an off-by-one error in the sum formula.
    • Why not D: Uses S_n = (n/2)(a + l) with the last term l taken as a + d instead of the correct a + 9d, as though only one common difference had been added by the 10th term.

  12. Question 12Answer: D

    1. The ratio of the fifth term to the second term is (a r^4)/(a r) = r^3, so r^3 = 162/6 = 27.
    2. 27 = 3^3, so r = 3.
    3. (This also gives a = 6/3 = 2, and the terms 2, 6, 18, 54, 162 check against both given values.)
    4. So the answer is D.
    • Why not A: Divides the two given terms in the wrong order, computing 6/162 = 1/27 instead of 162/6 = 27, so solves r^3 = 1/27 and gets r = 1/3.
    • Why not B: Correctly reaches r^3 = 27, but then takes a square root instead of a cube root, giving r = sqrt(27) = 3sqrt(3).
    • Why not C: Correctly reaches r^3 = 27, but forgets to take a cube root at all, treating the cubed value itself as the answer.

  13. Question 13Answer: C

    1. Rationalise the first fraction by multiplying by (sqrt(5)+sqrt(3))/(sqrt(5)+sqrt(3)): the numerator becomes (sqrt(5)+sqrt(3))^2 = 5 + 2sqrt(15) + 3 = 8 + 2sqrt(15), and the denominator becomes 5 - 3 = 2, giving 4 + sqrt(15).
    2. Rationalise the second fraction the same way: the numerator becomes (sqrt(5)-sqrt(3))^2 = 8 - 2sqrt(15), and the denominator is again 2, giving 4 - sqrt(15).
    3. Adding: (4 + sqrt(15)) + (4 - sqrt(15)) = 8, since the sqrt(15) terms cancel.
    4. So the answer is C.
    • Why not A: Correctly expands each numerator as 8 +- 2sqrt(15), but forgets to divide each by its denominator of 2 before adding them, giving (8+2sqrt(15)) + (8-2sqrt(15)) = 16.
    • Why not B: Mixes up the difference of two squares in the denominator, computing (sqrt(5))^2 + (sqrt(3))^2 = 8 instead of the correct (sqrt(5))^2 - (sqrt(3))^2 = 2, in both fractions.
    • Why not D: Correctly rationalises both fractions to get 4 + sqrt(15) and 4 - sqrt(15), but subtracts them instead of adding, as though the two fractions should cancel rather than combine.

  14. Question 14Answer: A

    1. Complete the square: x^2 - 4x + 7 = (x - 2)^2 + 3. This is an upward-opening parabola with vertex (minimum point) at x = 2.
    2. A quadratic is many-to-one either side of its vertex, since points equally spaced either side of x = 2 give equal values of f, so f can only be one-to-one on a domain lying entirely on one side of x = 2.
    3. Restricting to x >= k gives the increasing branch (and so one-to-one) exactly when k >= 2; any k < 2 would include some of the decreasing branch as well.
    4. So the smallest such k is 2, and the answer is A.
    • Why not B: Confuses the minimum value of f (the y-coordinate of the vertex, 3) with the x-coordinate at which that minimum occurs.
    • Why not C: Uses the formula -b/a instead of the correct -b/(2a) for the vertex, forgetting to divide by 2: here -(-4)/1 = 4 instead of -(-4)/2 = 2.
    • Why not D: Makes a sign error completing the square, writing (x + 2)^2 instead of the correct (x - 2)^2, which places the vertex at x = -2 instead of x = 2.

  15. Question 15Answer: B

    1. Setting x = 1 in (1 + x)^n makes every term equal to its binomial coefficient, so the sum of the coefficients is (1 + 1)^n = 2^n.
    2. 2^n = 128 = 2^7, so n = 7.
    3. The coefficient of x^3 in (1 + x)^7 is 7C3 = 7!/(3! x 4!) = (7 x 6 x 5)/(3 x 2 x 1) = 35.
    4. So the answer is B.
    • Why not A: Miscounts by one when solving 2^n = 128, taking n = 6 instead of the correct n = 7, giving a coefficient of 6C3 = 20.
    • Why not C: Miscounts by one the other way when solving 2^n = 128, taking n = 8 instead of the correct n = 7, giving a coefficient of 8C3 = 56.
    • Why not D: Correctly finds n = 7, but computes 7C3 as 7!/3! = 5040/6 = 840, forgetting to also divide by (7-3)! = 4! as the combination formula requires.

More on this strand

More free ESAT practice

Every strand of the published ESAT specification, with worked solutions throughout.

Choose a subject →