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neco 2022 mathematics | Objective

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Question 1 View Details

A closed container consists of a right circular cylinder of height \(h\) and radius \(r\), with a hemispherical dome of radius \(r\) attached on top, and a circular base of radius \(r\) at the bottom. The total external surface area (including the curved surface of the cylinder, the base, and the dome) is \(1500\ \text{cm}^2\). It is also known that the volume of the cylinder is twice the volume of the hemispherical dome. Find the total volume of the container (cylinder plus hemisphere) in cubic centimeters, correct to the nearest whole number.

A. 4920

B. 4700

C. 4850

Correct D. 4861

Correct Answer: D

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Question 2 View Details

Find the smallest positive integer \(N\) such that when divided by \(7\) it leaves a remainder of \(5\), when divided by \(9\) it leaves a remainder of \(2\), and \(N\) is a multiple of \(5\).

A. 125

B. 95

Correct C. 110

D. 115

Correct Answer: C

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Question 3 View Details

In a class of 30 students, the average score of Exam\,1 is \(68\) and the average score of Exam\,2 is \(74\). The overall average of the 60 exam scores (both exams combined) is \(71\). Five students scored \(90\) or above on both exams, and their average combined score (the sum of their two exam scores) is \(180\). Find the average combined score of the remaining 25 students.

A. 135.6

B. 132.0

Correct C. 134.4

D. 140.8

Correct Answer: C

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Question 4 View Details

Simplify \(\left( \frac{2^{3}\,\sqrt{8}}{4^{\frac{3}{2}}} \right)^{2}\) and give your answer as an integer.

Correct A. 8

B. 2

C. 16

D. 4

Correct Answer: A

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Question 5 View Details

Solve for \(x\): \(\frac{x-2}{x+3} > \frac{2x+1}{x-4}\). Express your answer in interval notation using exact values.

Correct A. \left( \frac{-13-\sqrt{189}}{2}, -3 \right) \cup \left( \frac{-13+\sqrt{189}}{2}, 4 \right)

B. \left( \frac{-13+\sqrt{189}}{2}, -3 \right) \cup \left( \frac{-13-\sqrt{189}}{2}, 4 \right)

C. \left( -3, \frac{-13-\sqrt{189}}{2} \right) \cup \left( 4, \frac{-13+\sqrt{189}}{2} \right)

D. \left[ \frac{-13-\sqrt{189}}{2}, -3 \right) \cup \left( \frac{-13+\sqrt{189}}{2}, 4 \right]

Correct Answer: A

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Question 6 View Details

Let \(N\) be a three‑digit integer such that when \(N\) is divided by 7 the remainder is 3, and when the quotient obtained is divided by 5 the remainder is 2. Find the smallest such integer \(N\).

Correct A. 122

B. 129

C. 115

D. 124

Correct Answer: A

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Question 7 View Details

The sequence \(\{a_n\}\) satisfies \(a_1 = 5\), \(a_2 = 12\). From the third term onward the sequence is arithmetic; that is, there exists a constant difference \(d\) such that \(a_{n+1}-a_n = d\) for all \(n \ge 3\). If the sum of the first five terms equals 65, determine the value of \(a_4\).

A. 14

B. 22

Correct C. 16

D. 18

Correct Answer: C

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Question 8 View Details

The first three terms of a geometric progression have sum 26, and the sum of its first six terms is 728. Find the common ratio \(r\) (with \(r>1\)) and the first term \(a\).

Correct A. a = 2, r = 3

B. a = 3, r = 2

C. a = 4, r = 3

D. a = 2, r = 4

Correct Answer: A

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Question 9 View Details

In a class of 200 students, 120 study Mathematics, 90 study Physics, and 70 study Chemistry. The numbers of students studying both Mathematics and Physics, both Physics and Chemistry, and both Mathematics and Chemistry are 40, 30, and 35 respectively. If 10 students study none of the three subjects, how many students study all three subjects?

Correct A. 15

B. 5

C. 20

D. 25

Correct Answer: A

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Question 10 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 0.1. A ball is drawn at random, its colour noted, and then the ball is returned to the bag. The experiment is repeated independently a total of five times. Given that exactly two of the five draws resulted in a red ball, what is the probability that exactly one of the remaining three draws was blue? Express your answer as a fraction in lowest terms.

A. 9/128

B. 9/32

Correct C. 9/64

D. 3/64

Correct Answer: C

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Question 11 View Details

Find the least positive integer \(n\) such that \(\frac{n^{2}-4n-5}{\,n-2\,}\le 3\).

A. 4

B. 2

Correct C. 3

D. 5

Correct Answer: C

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Question 12 View Details

The scores of 12 students in a quiz are 45, 52, 57, 63, ?, 71, 73, 78, 82, 85, 90, 94. The mean score is 70. Find the missing score and then determine the interquartile range (IQR) of the data set.

A. 52; 27

B. 55; 24

C. 48; 31

Correct D. 50; 29

Correct Answer: D

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Question 13 View Details

In a school, 48 students study Mathematics, 36 study Physics, and 20 study both subjects. If 12 students study neither Mathematics nor Physics, how many students study exactly one of the two subjects?

Correct A. 44

B. 24

C. 64

D. 56

Correct Answer: A

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Question 14 View Details

The diagram shows the line \(y = 2x + 2\) and the parabola \(y = x^{2} - 4x + 5\). What is the sum of the \(x\)-coordinates of their points of intersection?

A. 9

Correct B. 6

C. 3

D. 5

Correct Answer: B

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Question 15 View Details

In the coordinate plane, triangle ABC has vertices A(1,2), B(5,4) and C(3,9). Find the area of triangle ABC.

A. 10

B. 16

Correct C. 12

D. 14

Correct Answer: C

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Question 16 View Details

A linear equation \((k+4)x - 7 = 3k\) has an integer solution for \(x\). If \(k\) is a positive integer less than 20, find the value of \(x\).

A. 1

B. 3

Correct C. 2

D. 5

Correct Answer: C

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Question 17 View Details

An acute angle \(\theta\) satisfies \(\sin\theta + \cos\theta = \frac{5}{4}\) and \(\theta < 45^{\circ}\). Find \(\tan\theta\) in simplest exact form.

A. \frac{5\sqrt{7}-16}{9}

B. \frac{16+5\sqrt{7}}{9}

Correct C. \frac{16-5\sqrt{7}}{9}

D. \frac{16-5\sqrt{7}}{7}

Correct Answer: C

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Question 18 View Details

The sequence is defined by \(a_n = 3n + (-1)^n\) for \(n\ge 1\). Find the smallest positive integer \(n\) such that the sum of the first \(n\) terms exceeds 500.

Correct A. 18

B. 20

C. 19

D. 17

Correct Answer: A

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Question 19 View Details

A price is first increased by 20% and then decreased by 20% of the new price. What is the overall percentage change in the price?

A. 0\\%

Correct B. -4\%

C. 4\\%

D. -20\\%

Correct Answer: B

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Question 20 View Details

A box contains red, blue and green marbles. The probability of drawing a red marble is \(\frac{1}{5}\). Given that a marble drawn is not red, the probability that it is blue is \(\frac{2}{3}\). Two marbles are drawn successively without replacement. What is the probability that the two marbles are of different colours? Express your answer as a fraction in lowest terms.

Correct A. 68/105

B. 71/105

C. 68/115

D. 64/105

Correct Answer: A

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Question 21 View Details

A quantity \(y\) varies directly as the square of \(x\) and inversely as \((x+2)\). If \(y=48\) when \(x=4\), find the value of \(x\) when \(y=96\).

Correct A. \\frac{8 + 4\\sqrt{10}}{3}

B. \\frac{8 - 4\\sqrt{10}}{3}

C. \\frac{16 + 4\\sqrt{10}}{3}

D. \\frac{4 + 2\\sqrt{10}}{3}

Correct Answer: A

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Question 22 View Details

Solve for \(x\) in the equation \(2^{\\log_{x} 8}=4\).

A. 4

Correct B. 2\\sqrt{2}

C. 2\\sqrt{3}

D. \\sqrt{2}

Correct Answer: B

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Question 23 View Details

Let \(a\) and \(b\) be non‑zero numbers such that \(\frac{a}{b}+\frac{b}{a}=3\). Find the value of \(\frac{a^{3}+b^{3}}{a^{2}b+ab^{2}}\).

A. 4

B. 3

C. 1

Correct D. 2

Correct Answer: D

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Question 24 View Details

The line through points \((0,3)\) and \((4,7)\) passes through \((2,k)\). Find \(k\). Then, a line through this point is perpendicular to \(y = 2x -5\). Determine the area of the triangle formed by this perpendicular line with the coordinate axes.

A. 30

B. 42

C. 48

Correct D. 36

Correct Answer: D

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Question 25 View Details

The quadratic equation \(x^{2}-(p+2)x+(2p-3)=0\) has real roots \(\alpha\) and \(\beta\) satisfying \(\alpha^{2}+\beta^{2}=10\). Find the value of \(p\).

A. -1

B. 2

Correct C. 0

D. 1

Correct Answer: C

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