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ESAT Maths 1 Algebra questions

12 original ESAT Maths 1 algebra questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: Solve simultaneous equations (substitute or eliminate)Change the subject: collect the terms, then factoriseRationalise the denominator with the conjugateSolve an inequality with a sign table, then count integersWrite everything as powers of one baseComplete the square for turning points and exact roots

Question 1ESAT Maths 1Warm-up

x−y=−10x - y = -10 and 5x+3y=−185x + 3y = -18. What is the value of xyxy?
  1. A−24-24
  2. B−23-23
  3. C−22-22
  4. D−2-2
  5. E2424
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Answer: A · Method: Solve simultaneous equations (substitute or eliminate)

  1. Eliminate one variable: scale the equations so one coefficient matches, then subtract.
  2. This gives x=−6x = -6 and y=4y = 4, so xy=−24xy = -24.

Why the tempting wrong answers are wrong

  • E: Check the sign when you substitute back.

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Question 2ESAT Maths 1Warm-up

Given that −1+63−6=p+q6\dfrac{-1 + \sqrt{6}}{3 - \sqrt{6}} = p + q\sqrt{6}, where pp and qq are rational, what is qq?
  1. A−23-\dfrac{2}{3}
  2. B−13-\dfrac{1}{3}
  3. C215\dfrac{2}{15}
  4. D23\dfrac{2}{3}
  5. E11
  6. F53\dfrac{5}{3}
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Answer: D · Method: Rationalise the denominator with the conjugate

  1. Multiply top and bottom by the conjugate 3+63 + \sqrt{6}. The bottom becomes 9−6=39 - 6 = 3.
  2. The top becomes (−1+6)(3+6)=3+26(-1 + \sqrt{6})(3 + \sqrt{6}) = 3 + 2\sqrt{6}.
  3. So p=1p = 1 and q=23q = \dfrac{2}{3}.

Why the tempting wrong answers are wrong

  • C: (3−6)(3+6)=9−6(3 - \sqrt{6})(3 + \sqrt{6}) = 9 - 6, not 9+69 + 6.
  • E: That's pp, the rational part.

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Question 3ESAT Maths 1Warm-up

How many integers xx satisfy x2−8x+12<0x^2 - 8x + 12 < 0?
  1. A11
  2. B22
  3. C33
  4. D44
  5. E55
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Answer: C · Method: Solve an inequality with a sign table, then count integers

  1. Factorise: (x−2)(x−6)<0(x - 2)(x - 6) < 0, so 2<x<62 < x < 6.
  2. That's 33 integers.

Why the tempting wrong answers are wrong

  • E: The inequality is strict, so the roots themselves don't count.

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Question 4ESAT Maths 1Warm-up

Solve 3x−4×92x=272x+13^{x - 4} \times 9^{2x} = 27^{2x + 1}.
  1. A−9-9
  2. B−8-8
  3. C−7-7
  4. D−6-6
  5. E−5-5
  6. F−4-4
  7. G55
  8. H77
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Answer: C · Method: Write everything as powers of one base

  1. Write everything as a power of 33: 3x−4×34x=36x+33^{x - 4} \times 3^{4x} = 3^{6x + 3}.
  2. Add the powers on the left and compare: 5x−4=6x+35x - 4 = 6x + 3.
  3. So −x=7-x = 7 and x=−7x = -7.

Why the tempting wrong answers are wrong

  • G: Write 9=329 = 3^2 and 27=3327 = 3^3 before comparing powers.

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Question 5ESAT Maths 1Warm-up

The curve y=x2+2x+8y = x^2 + 2x + 8 has a turning point at (a,b)(a, b). What is bb?
  1. A55
  2. B66
  3. C77
  4. D88
  5. E99
  6. F1010
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Answer: C · Method: Complete the square for turning points and exact roots

  1. Complete the square: y=(x+1)2+7y = (x + 1)^2 + 7.
  2. The turning point is (−1,7)(-1, 7), so b=7b = 7.

Why the tempting wrong answers are wrong

  • D: That's the yy-intercept, where x=0x = 0.

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Question 6ESAT Maths 1Warm-up

Solve 2x−3×42x=83x−22^{x - 3} \times 4^{2x} = 8^{3x - 2}.
  1. A−34-\dfrac{3}{4}
  2. B−14-\dfrac{1}{4}
  3. C34\dfrac{3}{4}
  4. D74\dfrac{7}{4}
  5. E114\dfrac{11}{4}
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Answer: C · Method: Write everything as powers of one base

  1. Write everything as a power of 22: 2x−3×24x=29x−62^{x - 3} \times 2^{4x} = 2^{9x - 6}.
  2. Add the powers on the left and compare: 5x−3=9x−65x - 3 = 9x - 6.
  3. So −4x=−3-4x = -3 and x=34x = \dfrac{3}{4}.

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Question 7ESAT Maths 1Standard

The line y=2x+3y = 2x + 3 meets the curve y=x2−5x+13y = x^2 - 5x + 13 at two points. What is the sum of the yy-coordinates of these two points?
  1. A−20-20
  2. B77
  3. C1717
  4. D1919
  5. E2020
  6. F2121
  7. G2222
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Answer: E · Method: Solve simultaneous equations (substitute or eliminate)

  1. Set them equal: x2−5x+13=2x+3x^2 - 5x + 13 = 2x + 3, so x2−7x+10=0x^2 - 7x + 10 = 0, i.e. (x−2)(x−5)=0(x - 2)(x - 5) = 0.
  2. So x=2x = 2 or x=5x = 5, giving y=7y = 7 and y=13y = 13. The sum is 2020.

Why the tempting wrong answers are wrong

  • B: That's the sum of the xx-coordinates. Put them back into the line.
  • C: Each point gets its own +3+3: add it twice.

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Question 8ESAT Maths 1Standard

Make xx the subject of y=3x−44x+2y = \dfrac{3x - 4}{4x + 2}.
  1. Ax=−4−2y3−4yx = \dfrac{-4 - 2y}{3 - 4y}
  2. Bx=2y−3−4−4yx = \dfrac{2y - 3}{-4 - 4y}
  3. Cx=−4+2y4y−3x = \dfrac{-4 + 2y}{4y - 3}
  4. Dx=4+2y4−3yx = \dfrac{4 + 2y}{4 - 3y}
  5. Ex=−4−2y4y−3x = \dfrac{-4 - 2y}{4y - 3}
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Answer: E · Method: Change the subject: collect the terms, then factorise

  1. Multiply up: y(4x+2)=3x−4y(4x + 2) = 3x - 4, so 4xy+2y=3x−44xy + 2y = 3x - 4.
  2. Collect the xx terms: x(4y−3)=−4−2yx(4y - 3) = -4 - 2y.
  3. Divide: x=−4−2y4y−3x = \dfrac{-4 - 2y}{4y - 3}.

Why the tempting wrong answers are wrong

  • A: Check the signs when you move the terms across.

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Question 9ESAT Maths 1Standard

The line y=−x−5y = -x - 5 meets the curve y=x2−8x+7y = x^2 - 8x + 7 at two points. What is the sum of the yy-coordinates of these two points?
  1. A−17-17
  2. B−16-16
  3. C−12-12
  4. D77
  5. E1717
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Answer: A · Method: Solve simultaneous equations (substitute or eliminate)

  1. Set them equal: x2−8x+7=−x−5x^2 - 8x + 7 = -x - 5, so x2−7x+12=0x^2 - 7x + 12 = 0, i.e. (x−3)(x−4)=0(x - 3)(x - 4) = 0.
  2. So x=3x = 3 or x=4x = 4, giving y=−8y = -8 and y=−9y = -9. The sum is −17-17.

Why the tempting wrong answers are wrong

  • C: Each point gets its own +−5+-5: add it twice.
  • D: That's the sum of the xx-coordinates. Put them back into the line.

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Question 10ESAT Maths 1Standard

Make xx the subject of y=x−63x−2y = \dfrac{x - 6}{3x - 2}.
  1. Ax=−2y−1−6−3yx = \dfrac{-2y - 1}{-6 - 3y}
  2. Bx=−6+2y3y−1x = \dfrac{-6 + 2y}{3y - 1}
  3. Cx=−6−2y3y−1x = \dfrac{-6 - 2y}{3y - 1}
  4. Dx=6−2y3−yx = \dfrac{6 - 2y}{3 - y}
  5. Ex=−6+2y1−3yx = \dfrac{-6 + 2y}{1 - 3y}
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Answer: B · Method: Change the subject: collect the terms, then factorise

  1. Multiply up: y(3x−2)=x−6y(3x - 2) = x - 6, so 3xy−2y=x−63xy - 2y = x - 6.
  2. Collect the xx terms: x(3y−1)=−6+2yx(3y - 1) = -6 + 2y.
  3. Divide: x=−6+2y3y−1x = \dfrac{-6 + 2y}{3y - 1}.

Why the tempting wrong answers are wrong

  • E: Check the signs when you move the terms across.

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Question 11ESAT Maths 1Standard

How many integers xx with −10≤x≤10-10 \le x \le 10 satisfy (x+6)(x+2)(x−3)<0(x + 6)(x + 2)(x - 3) < 0?
  1. A66
  2. B77
  3. C88
  4. D99
  5. E1010
  6. F1111
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Answer: C · Method: Solve an inequality with a sign table, then count integers

  1. The product changes sign at x=−6,−2,3x = -6, -2, 3. It's positive for large xx, so it's negative for x<−6x < -6 and for −2<x<3-2 < x < 3.
  2. In the range, that's 88 integers.

Why the tempting wrong answers are wrong

  • E: Check the sign table: for large xx the product is positive, so it's negative below the smallest root and between the middle two.
  • F: The inequality is strict, so the roots don't count.

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Question 12ESAT Maths 1Standard

The solutions of x2+6x+4=0x^2 + 6x + 4 = 0 can be written as x=m±nx = m \pm \sqrt{n}, where mm and nn are integers. What is m+nm + n?
  1. A00
  2. B11
  3. C22
  4. D33
  5. E44
  6. F55
  7. G88
  8. H1717
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Answer: C · Method: Complete the square for turning points and exact roots

  1. Complete the square: (x+3)2−9+4=0(x + 3)^2 - 9 + 4 = 0, so (x+3)2=5(x + 3)^2 = 5.
  2. So x=−3±5x = -3 \pm \sqrt{5}: m+n=2m + n = 2.

Why the tempting wrong answers are wrong

  • G: Check the sign of mm: (x−m)2(x - m)^2 has −2m-2m as the xx coefficient.
  • H: With the quadratic formula, b2−4ac\sqrt{b^2 - 4ac} is then divided by 2: simplify fully.

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