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neco 2024 mathematics | Objective

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Question 1 View Details

In a survey of 120 students, let set A be those who play football, set B those who play basketball, and set C those who play chess. The numbers are: |A| = 70, |B| = 60, |C| = 50, |A\\cap B| = 30, |A\\cap C| = 20, |B\\cap C| = 25. Ten students play none of these games. How many students play exactly two of the three games?

Correct A. 60

B. 65

C. 55

D. 45

Correct Answer: A

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Question 2 View Details

For integer values of \(k\), the quadratic equation \(x^{2}-(k+2)x+(k-3)=0\) has two distinct integer roots. Find the sum of all distinct integer roots that can occur for any such \(k\).

A. 5

B. 10

Correct C. 7

D. 8

Correct Answer: C

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Question 3 View Details

From a group of 8 boys and 6 girls, a committee of 5 members is chosen at random. Given that the committee contains at least one girl, what is the probability that it contains exactly two boys?

A. 280\/1001

Correct B. 280/973

C. 210\/973

D. 420\/973

Correct Answer: B

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Question 4 View Details

Let \(E = \frac{x^{3}-3x^{2}+4}{x^{2}-x-6}\). Find all integer values of \(x\) for which \(E\) is an integer.

A. 0, 2

B. -2, 3

Correct C. -1, 2

D. -1, 3

Correct Answer: C

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Question 5 View Details

In triangle \(ABC\) the side lengths are \(AB=5\) cm, \(AC=7\) cm and \(BC=8\) cm. The internal angle bisector of \(\angle A\) meets \(BC\) at \(D\). From \(D\) a perpendicular is drawn to \(AB\) meeting \(AB\) at \(E\). Find the length of \(DE\).

Correct A. 5\sqrt{3}/3

B. 4\\sqrt{3}\/3

C. 5\\sqrt{2}\/3

D. 5\\sqrt{3}\/2

Correct Answer: A

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Question 6 View Details

A straight line passes through the points A(2,3) and B(5,k). The line also cuts the y\-axis at point C where y = -1. (i) Find the value of k. (ii) Using the same line, determine the y\-coordinate of the point where x = 8.

A. k = 7; y = 13

B. k = 11; y = 15

C. k = 9; y = 13

Correct D. k = 9; y = 15

Correct Answer: D

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Question 7 View Details

A bag contains red, blue and green balls. The probability of drawing a red ball is twice that of drawing a blue ball, and the probability of drawing a green ball is \frac{1}{5}. A ball is drawn, its colour noted, and then the ball is replaced. Two independent draws are made. Given that exactly one of the two draws resulted in a red ball, what is the probability that the other draw was green? Express your answer as a fraction in lowest terms.

A. 4\/7

B. 2\/7

Correct C. 3/7

D. 3\/5

Correct Answer: C

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Question 8 View Details

A circle is tangent to the x\-axis and passes through the points A(2,3) and B(6,7). Find the radius of the circle that lies to the right of the y\-axis. Give your answer in simplest radical form.

A. 10-\sqrt{48}

Correct B. 10-\sqrt{42}

C. 10+\sqrt{42}

D. 10-\sqrt{30}

Correct Answer: B

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Question 9 View Details

When the integer \(7^{2023}\) is divided by 100, what is the remainder?

A. 99

Correct B. 43

C. 23

D. 57

Correct Answer: B

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Question 10 View Details

A right circular cylinder has total surface area (including top and bottom) of 1000 cm\(^2\) and volume of 500\pi cm\(^3\). Find the height of the cylinder, giving your answer correct to two decimal places.

Correct A. 43.55 cm

B. 52.10 cm

C. 47.89 cm

D. 38.27 cm

Correct Answer: A

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Question 11 View Details

The expression \( E = \frac{3x^{2} - kx + 12}{x - 2} \) is a polynomial (i.e., division leaves no remainder). Find the integer value of \( k \) and then evaluate \( E \) at \( x = 5 \).

A. 12

B. 15

C. 7

Correct D. 9

Correct Answer: D

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Question 12 View Details

Two numbers \( m \) and \( n \) satisfy: \( (m+7)n = 120 \) and \( m+n = 19 \). Determine the sum of the possible values of the larger number.

A. 26

B. 32

C. 12

Correct D. 33

Correct Answer: D

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Question 13 View Details

Find the smallest positive integer that leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7.

A. 58

B. 63

C. 47

Correct D. 52

Correct Answer: D

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Question 14 View Details

A cylindrical tank is closed at both ends. Its interior radius is \( r \) metres and its interior height is \( h \) metres. The tank holds \( 5000\pi \) cubic metres of water and its interior height is three times its radius (\( h = 3r \)). The tank walls are \(0.1\) m thick and the top and bottom are \(0.1\) m thick each, so the outer radius is \( r+0.1 \) m and the outer height is \( h+0.2 \) m. Calculate the total outer surface area (including the top and bottom) that must be painted, giving your answer rounded to the nearest square metre.

A. 3650

B. 2685

C. 3528

Correct D. 3584

Correct Answer: D

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Question 15 View Details

Express the repeating decimal \( 0.\overline{27} \) as a fraction in lowest terms.

A. 27/90

B. 2/7

C. 27/100

Correct D. 3/11

Correct Answer: D

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Question 16 View Details

If \\(x = 2^{\\frac{3}{2}}\\), evaluate \\(x^{\\frac{4}{3}}\\).

Correct A. 4

B. 8

C. 16

D. 2

Correct Answer: A

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Question 17 View Details

A cylindrical water tank of radius \\(r\\) cm and height \\(h\\) cm has a hemispherical dome on its top. The total external surface area (excluding the base) is \\(1500\\) cm\\(^2\\) and the height of the cylindrical part is twice the radius ( \\(h = 2r\\) ). Find the total volume of the tank (cylinder plus hemisphere) in cubic centimetres, giving your answer in exact form.

A. \\frac{2000}{4}\\sqrt{\\frac{250}{\\pi}}

B. \\frac{1800}{3}\\sqrt{\\frac{250}{\\pi}}

Correct C. \\frac{2000}{3}\\sqrt{\\frac{250}{\\pi}}

D. \\frac{2000}{3}\\sqrt{\\frac{200}{\\pi}}

Correct Answer: C

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Question 18 View Details

In a class of 30 students, the mean score of Exam A is \\(68\\) with a standard deviation of \\(5\\). The mean score of Exam B is \\(72\\) with a standard deviation of \\(6\\). The correlation coefficient between the two exam scores is \\(0.8\\). Assuming the scores are jointly normally distributed, find the mean and standard deviation of the total score (sum of the two exams). Give the standard deviation correct to two decimal places.

A. Mean = 140, Standard deviation = 7.81

B. Mean = 140, Standard deviation = 11.00

C. Mean = 138, Standard deviation = 10.44

Correct D. Mean = 140, Standard deviation = 10.44

Correct Answer: D

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Question 19 View Details

What is the smallest positive integer that is divisible by 8, 9 and 15?

A. 1080

B. 120

C. 72

Correct D. 360

Correct Answer: D

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Question 20 View Details

Two integers \\(x\\) and \\(y\\) satisfy the simultaneous equations \\(3x + 4y = p\\) and \\(5x - 2y = 7\\). Find the smallest positive integer value of \\(p\\) for which both \\(x\\) and \\(y\\) are integers, and give the corresponding ordered pair \\((x, y)\\).

Correct A. p = 25, (x, y) = (3, 4)

B. p = 51, (x, y) = (5, 9)

C. p = 103, (x, y) = (9, 19)

D. p = 77, (x, y) = (7, 14)

Correct Answer: A

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Question 21 View Details

Consider the system of linear equations \(3x + 2y = 7\) and \((k-1)x - y = 4\). It is known that the solution \((x,y)\) satisfies \(x = 2y\). Find the value of the parameter \(k\) and then determine \(x + y\).

Correct A. k = \frac{53}{14}; x + y = \frac{21}{8}

B. k = \frac{53}{14}; x + y = \frac{22}{8}

C. k = \frac{53}{15}; x + y = \frac{21}{9}

D. k = \frac{54}{14}; x + y = \frac{21}{8}

Correct Answer: A

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Question 22 View Details

If \(2^{x} = 5\) and \(\log_{10}\bigl( \sqrt{5}\,\cdot 10^{y}\bigr) = 2\), find the value of \(x - 2y\) (give your answer correct to three decimal places).

A. 0.979

Correct B. -0.979

C. -1.079

D. -0.978

Correct Answer: B

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Question 23 View Details

A line passes through the point \(A(2, -1)\) and is perpendicular to the line joining the points \(B(5, 3)\) and \(C(k, -2)\). The line meets the y‑axis at point \(D\) whose y‑coordinate is twice the x‑coordinate of \(B\). Find the value of \(k\) and give the equation of the required line in the form \(y = mx + c\).

Correct A. k = -\frac{45}{2}; equation: y = -\frac{11}{2}x + 10

B. k = -\frac{45}{4}; equation: y = -\frac{11}{2}x + 10

C. k = -\frac{45}{2}; equation: y = -\frac{11}{2}x - 10

D. k = -\frac{45}{2}; equation: y = -\frac{9}{2}x + 10

Correct Answer: A

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Question 24 View Details

A bag contains red, blue and green balls. The probability of drawing a red ball is twice that of drawing a blue ball, and the probability of drawing a green ball is 0.2. A ball is drawn at random, its colour noted, then the ball is replaced and an extra ball of the same colour is added to the bag. A second draw is then made. What is the probability that both draws are red?

A. 0.2

B. 0.4

Correct C. 0.3

D. 0.5

Correct Answer: C

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Question 25 View Details

The time \(T\) (in seconds) taken for a particle to travel a distance \(s\) under constant acceleration \(a\) varies directly as \(\sqrt{s}\) and inversely as \(\sqrt{a}\). If it takes 5 seconds to travel 20 m under an acceleration of 2 m/s\(^2\), find the time required to travel 45 m under an acceleration of 5 m/s\(^2\). Give your answer in exact form.

Correct A. \frac{3\sqrt{10}}{2}

B. \frac{3\sqrt{5}}{2}

C. \frac{3\sqrt{10}}{4}

D. \frac{5\sqrt{10}}{2}

Correct Answer: A

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