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OAU Post-UTME Mathematics 2011 past questions and answers

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

Question 3

Coordinate Geometry

Find the equation of the circle with centre (−1,3) and radius 4

  1. A.x2 + y2 − 6x + 4y = 6
  2. B.x2 + y2 + 2x − 6y = 16
  3. C.x2 + y2 − 6x + 2y = 16
  4. D.x2 + y2 + 2x − 6y = 6
Show the answer and explanation

Answer: D (x2 + y2 + 2x − 6y = 6)

Explanation

(x + 1)² + (y − 3)² = 16 expands to x² + y² + 2x − 6y + 10 = 16, i.e. x² + y² + 2x − 6y = 6.

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Question 4

Differentiation

Find dy/dx, if y = 3/√x

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

Answer: A (−(3/2)x−3/2)

Explanation

y = 3x−1/2, so dy/dx = 3 × (−½)x−3/2 = −(3/2)x−3/2.

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Question 5

Integration

Integrate 1/(2x)

  1. A.Not defined
  2. B.0
  3. C.½ ln x + C
  4. D.¼x2 + C
Show the answer and explanation

Answer: C (½ ln x + C)

Explanation

Take out the ½: ½∫(1/x)dx = ½ ln x + C.

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Question 6

Probability

A die is tossed twice. What is the probability of obtaining a total of 6 if both numbers are odd?

  1. A.1/12
  2. B.1/18
  3. C.5/36
  4. D.1/6
Show the answer and explanation

Answer: A (1/12)

Explanation

Of the 36 outcomes, the ones with both numbers odd and a total of 6 are (1,5), (3,3), (5,1): 3/36 = 1/12.

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Question 7

Measures of Central Tendency

If the mean of the numbers a, b, c, d, e is x, find the mean of numbers a + k, b + 2k, c − k, d − 2k, e

  1. A.x
  2. B.x + k
  3. C.x − k
  4. D.2x
Show the answer and explanation

Answer: A (x)

Explanation

The k terms add to k + 2k − k − 2k = 0, so the total is still 5x and the mean is still x.

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Question 8

Algebraic Expressions

Factorize: a2 − b2 + (a + b)2

  1. A.2a2
  2. B.2a(a − b)
  3. C.2a(a + b)
  4. D.2b(b − a)
Show the answer and explanation

Answer: C (2a(a + b))

Explanation

a² − b² = (a − b)(a + b). Take out (a + b): (a + b)[(a − b) + (a + b)] = 2a(a + b).

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Question 9

Quadratic Equations

Let α and β be roots of quadratic equation x2 + 2x − 3 = 0 then αβ is

  1. A.−3
  2. B.−2
  3. C.2
  4. D.6
Show the answer and explanation

Answer: A (−3)

Explanation

Product of roots = c/a = −3/1 = −3.

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Question 10

Number Bases

Covert 6910 to a number in base two

  1. A.1001101
  2. B.1010001
  3. C.1000101
  4. D.100101
Show the answer and explanation

Answer: C (1000101)

Explanation

69 = 64 + 4 + 1, so in base two it is 1000101.

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Question 11

Fractions, Decimals and Approximation

The reciprocal of (3/4)/(1/4 + 1/3) is

  1. A.1 2/7
  2. B.7/9
  3. C.−1 2/7
  4. D.−7/9
Show the answer and explanation

Answer: B (7/9)

Explanation

1/4 + 1/3 = 7/12, so the fraction is 3/4 ÷ 7/12 = 9/7. Its reciprocal is 7/9.

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Question 13

Logarithms

Evaluate x if log4(x + 3)(x − 3) = 2

  1. A.3 or −3
  2. B.5 or −5
  3. C.5 or −3
  4. D.3 or −5
Show the answer and explanation

Answer: B (5 or −5)

Explanation

(x + 3)(x − 3) = 4² = 16, so x² = 25 and x = 5 or −5.

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Question 14

Surds

Given that a = 1/(2 − √3), b = 1/(2 + √3), find the value of a2 + b2

  1. A.14/37
  2. B.7
  3. C.142√3
  4. D.14
Show the answer and explanation

Answer: D (14)

Explanation

Rationalising, a = 2 + √3 and b = 2 − √3. a² + b² = (7 + 4√3) + (7 − 4√3) = 14.

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Question 15

Binary Operations

If the binary operation * is defined as x * y = 2, find 2*(4*5)

  1. A.4
  2. B.5
  3. C.−5
  4. D.2
Show the answer and explanation

Answer: D (2)

Explanation

The operation always gives 2: 4*5 = 2, then 2*2 = 2.

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Question 16

Surds

Find the value of √(6 + √(6 + √(6 + √(6 + …))))

  1. A.−2
  2. B.2
  3. C.6
  4. D.3
Show the answer and explanation

Answer: D (3)

Explanation

Let x be the value. Then x² = 6 + x, so x² − x − 6 = 0 and x = 3 or −2. A square root can't be negative, so x = 3.

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Question 17

Integration

Find the value of ∫0π/2(2π + 2cos 2x)dx

  1. A.π2 + 1
  2. B.π2
  3. C.π2 − 4
  4. D.π2 + 3
Show the answer and explanation

Answer: B (π2)

Explanation

The integral is 2πx + sin 2x. At π/2: π² + sin π = π². At 0: 0. Answer π².

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Question 18

Coordinate Geometry

The circle 2x2 + 2(y − 3/2)2 = 2 has centre and radius respectively as

  1. A.(0, 3/2) and 2
  2. B.(0, −3/2) and 1
  3. C.(3/2, 0) and 2
  4. D.(0, 3/2) and 1
Show the answer and explanation

Answer: D ((0, 3/2) and 1)

Explanation

Divide by 2: x² + (y − 3/2)² = 1. The centre is (0, 3/2) and the radius is 1.

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Question 19

Coordinate Geometry

The line perpendicular to the straight line y + (3/2)x − 1 = 0 has the gradient

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

Answer: D (2/3)

Explanation

The given line is y = −(3/2)x + 1, gradient −3/2. A perpendicular gradient is −1 ÷ (−3/2) = 2/3.

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Question 20

Indices

Find x if 2x² = 4(x+4)

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

Answer: A (−2 or 4)

Explanation

4x+4 = 22x+8, so x² − 2x − 8 = 0, (x − 4)(x + 2) = 0: x = 4 or −2.

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Question 22

Mensuration of Plane Shapes

A square has perimeter of 40cm. What is its area in cm square?

  1. A.80
  2. B.1600
  3. C.100
  4. D.160
Show the answer and explanation

Answer: C (100)

Explanation

Each side is 40 ÷ 4 = 10 cm, so the area is 10 × 10 = 100 cm².

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