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UI Post-UTME Mathematics 2013 past questions and answers

17 questions. Try each one, then open the answer to check yourself and read the explanation.

Question 1

Surds

Solve the equation √(x + 8) + √(x + 1) = 7

  1. A.3
  2. B.4
  3. C.8
  4. D.9
Show the answer and explanation

Answer: C (8)

Explanation

Check x = 8: √16 + √9 = 4 + 3 = 7. (Squaring twice also leads to √(x + 1) = 3, so x = 8.)

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Question 2

Linear and Simultaneous Equations

The sum of the digits of a 2-digit number is 8. When the digits are interchanged, the new number exceeds the original number by 18. Find the original number?

  1. A.35
  2. B.26
  3. C.17
  4. D.71
Show the answer and explanation

Answer: A (35)

Explanation

A + B = 8 and 9(B − A) = 18, so B − A = 2: A = 3, B = 5, giving 35 (53 − 35 = 18).

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Question 3

Linear and Simultaneous Equations

Solve the simultaneous equations; x − y = −5 and x2 − y2 = 35

  1. A.x = 1, y = −5
  2. B.x = −3, y = 2
  3. C.x = −7, y = −2
  4. D.x = −6, y = −1
Show the answer and explanation

Answer: D (x = −6, y = −1)

Explanation

x² − y² = (x − y)(x + y) = −5(x + y) = 35, so x + y = −7. With x − y = −5: x = −6, y = −1.

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Question 5

Fractions, Decimals and Approximation

Simplify 2¼/(4 ÷ ⅛ of 2⅔ × 3)

  1. A.1/16
  2. B.9/16
  3. C.3/16
  4. D.9/32
Show the answer and explanation

Answer: A (1/16)

Explanation

Work the bottom first: ⅛ of 2⅔ = ⅛ × 8/3 = ⅓. Then 4 ÷ ⅓ = 12 and 12 × 3 = 36. So 2¼ ÷ 36 = 9/4 × 1/36 = 1/16.

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Question 6

Fractions, Decimals and Approximation

If m and n are distinct natural numbers which of the following is not necessarily a natural number?

  1. A.m + n
  2. B.m2 + n2
  3. C.m − n
  4. D.mn
Show the answer and explanation

Answer: C (m − n)

Explanation

Sums and products of natural numbers are natural, but m − n is negative when m < n (e.g. 2 − 7 = −5).

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Question 7

Algebraic Expressions

If x, y, z are distinct prime numbers, find the HCF of x2y3z4, xy5z3, x3y4z2

  1. A.x2y3z4
  2. B.xyz
  3. C.xy3z2
  4. D.x2y2z2
Show the answer and explanation

Answer: C (xy3z2)

Explanation

Take the lowest power of each prime: x¹, y³, z², giving xy³z².

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Question 8

Number Bases

Simplify (1111112 + 1001012) ÷ 101002

  1. A.1112
  2. B.1012
  3. C.1002
  4. D.112
Show the answer and explanation

Answer: B (1012)

Explanation

63 + 37 = 100, and 10100₂ = 20; 100 ÷ 20 = 5 = 101₂.

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Question 9

Binary Operations

If a binary operation * is defined as x * y = x2 + y2 − √(xy). Evaluate 15 * (3 * 3)

  1. A.435
  2. B.420
  3. C.425
  4. D.520
Show the answer and explanation

Answer: A (435)

Explanation

3 * 3 = 9 + 9 − 3 = 15; 15 * 15 = 225 + 225 − 15 = 435.

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Question 10

Sets

Out of 12 Physics teacher in a school, 7 also teach Mathematics while 6 also teach Integrated Science. If 3 teachers teach all three subjects, use the information to answer the next three questions.
How many teach Physics only?

  1. A.None
  2. B.2
  3. C.4
  4. D.6
Show the answer and explanation

Answer: B (2)

Explanation

Physics & Maths only = 7 − 3 = 4, Physics & Integrated Science only = 6 − 3 = 3, all three = 3; Physics only = 12 − 10 = 2.

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Question 11

Sets

Out of 12 Physics teacher in a school, 7 also teach Mathematics while 6 also teach Integrated Science. If 3 teachers teach all three subjects, use the information to answer the next three questions.
How many teach Physics and Mathematics only?

  1. A.None
  2. B.2
  3. C.4
  4. D.6
Show the answer and explanation

Answer: C (4)

Explanation

7 physics teachers also teach Mathematics, and 3 of them teach all three, so 7 − 3 = 4 teach Physics and Mathematics only.

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Question 13

Commercial Arithmetic

An article was sold for ₦280 at a profit of 40%. For how much should it have been sold to make a profit of 50%

  1. A.₦290
  2. B.₦300
  3. C.₦320
  4. D.₦340
Show the answer and explanation

Answer: B (₦300)

Explanation

Cost = 280 ÷ 1.4 = ₦200, and 150% of 200 = ₦300.

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Question 14

Ratio, Percentages and Rates

A student was asked to correct 0.02539 to two significant figures gave its value to two decimal places. Calculate the percentage error

  1. A.20%
  2. B.16⅔%
  3. C.15%
  4. D.10%
Show the answer and explanation

Answer: A (20%)

Explanation

2 s.f. gives 0.025 but 2 d.p. gives 0.03; error = 0.005, and 0.005/0.025 × 100% = 20%.

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Question 15

Indices

Solve for x in 4x − ½ = 2x²

  1. A.½
  2. B.1
  3. C.2
  4. D.3
Show the answer and explanation

Answer: B (1)

Explanation

2^(2x − 1) = 2^(x²), so x² − 2x + 1 = 0 and x = 1.

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Question 16

Logarithms

Simplify log0.258

  1. A.−3/2
  2. B.3/2
  3. C.2/3
  4. D.−2/3
Show the answer and explanation

Answer: A (−3/2)

Explanation

0.25 = 2⁻² and 8 = 2³, so log₀.₂₅8 = 3/(−2) = −3/2.

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Question 17

Logarithms

Simplify log2√8 + log3√3

  1. A.4
  2. B.3
  3. C.2
  4. D.1
Show the answer and explanation

Answer: C (2)

Explanation

log₂√8 = log₂2^(3/2) = 3/2 and log₃√3 = 1/2; the sum is 2.

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Question 18

Number Bases

Find x if 212three = 1x3four

  1. A.3
  2. B.2
  3. C.1
  4. D.0
Show the answer and explanation

Answer: C (1)

Explanation

212₃ = 18 + 3 + 2 = 23 and 1x3₄ = 16 + 4x + 3, so 4x = 4 and x = 1.

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Question 19

Quadratic Equations

Find the value of k for which the equation 2x2 − kx + 8 = 0 has equal roots.

  1. A.16
  2. B.±3
  3. C.±8
  4. D.±4
Show the answer and explanation

Answer: C (±8)

Explanation

k² − 4(2)(8) = 0 gives k² = 64, so k = ±8.

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