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neco model questions vol1 2019 mathematics | Objective

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Question 1 View Details

Find all integer values of \(p\) for which the quadratic equation \(x^{2}-(2p+3)x+p=0\) has two distinct integer roots.

Correct A. 0, -2

B. 1, -3

C. 0, -1

D. -1, 2

Correct Answer: A

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Question 2 View Details

In a school of 120 students, the numbers studying Mathematics (M), Physics (P) and Chemistry (C) are \(|M|=70\), \(|P|=55\) and \(|C|=45\). The number studying both Mathematics and Physics is twice the number studying both Physics and Chemistry. Also, 20 students study none of the three subjects and exactly 10 students study all three subjects. Find the number of students who study only Mathematics.

Correct A. 12

B. 10

C. 18

D. 14

Correct Answer: A

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Question 3 View Details

The Venn diagram shows three sets \(X\), \(Y\) and \(Z\). The numbers in the regions are: only \(X\) = 8, only \(Y\) = 5, only \(Z\) = 7, \(X\cap Y\) only = 4, \(Y\cap Z\) only = 6, \(X\cap Z\) only = ?, and \(X\cap Y\cap Z\) = 3. There are 12 elements outside the three sets, and the total number of elements belonging to \(X\) or \(Y\) (i.e., \(|X\cup Y|\)) is 30. Find the number of elements in the region \(X\cap Z\) only.

A. 14

B. 8

Correct C. 10

D. 12

Correct Answer: C

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Question 4 View Details

In triangle \(ABC\) the angles are \(\angle A = 40^{\circ}\), \(\angle B = 70^{\circ}\) and \(\angle C = 70^{\circ}\). The side \(AB\) has length 10 units. Point \(D\) lies on side \(BC\) such that \(BD:DC = 1:2\). Find the length of \(AD\) (give your answer correct to two decimal places).

A. 8.93

B. 10.12

Correct C. 9.47

D. 7.85

Correct Answer: C

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Question 5 View Details

When an integer \(N\) is divided by 8 the remainder is 5, and when \(N\) is divided by 9 the remainder is 7. What is the smallest positive integer \(N\) that satisfies both conditions?

A. 69

B. 85

C. 53

Correct D. 61

Correct Answer: D

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Question 6 View Details

A solid is formed by joining a right circular cylinder of radius \(r\) cm and height \(h\) cm to a right circular cone of the same base radius \(r\) cm and height \(\frac{h}{2}\) cm, the base of the cone being attached to one base of the cylinder. The total surface area of the solid, excluding the circular base that is hidden by the attachment, is \(600\pi\) cm\(^2\). If the height of the cylinder is three times its radius, find the total volume of the solid, giving your answer to the nearest whole cubic centimetre.

A. 6050

B. 6300

Correct C. 6189

D. 6201

Correct Answer: C

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Question 7 View Details

Vector \(\mathbf{a}=2\mathbf{i}-\mathbf{j}\). Vector \(\mathbf{b}=p\mathbf{i}+q\mathbf{j}\) where \(p\) and \(q\) are positive integers not exceeding 10. It is known that \(\mathbf{a}+\mathbf{b}\) is parallel to \(\mathbf{c}=3\mathbf{i}+4\mathbf{j}\) and that \(|\mathbf{a}+\mathbf{b}|=10\). Determine the ordered pair \((p,q)\) and then give the magnitude of \(\mathbf{b}\) in simplest radical form.

Correct A. (4,9), \sqrt{97}

B. (5,8), \sqrt{89}

C. (6,7), \sqrt{73}

D. (3,10), \sqrt{85}

Correct Answer: A

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Question 8 View Details

The total surface area (including base) of a right circular cone is \(150\pi\) cm\(^2\). A sphere is inscribed in the cone, touching the base and the lateral surface. If the volume of the cone is three times the volume of the sphere and the height of the cone is less than 20 cm, find the height of the cone, rounded to the nearest centimetre.

A. 12

Correct B. 9

C. 8

D. 10

Correct Answer: B

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Question 9 View Details

The marked price of an item is ₦12,500. The shop gives a discount of 20 % on the marked price, and then adds a sales tax of 5 % on the discounted price. What is the final price the customer pays, in naira?

A. 11250

B. 10000

Correct C. 10500

D. 12500

Correct Answer: C

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Question 10 View Details

If \(\frac{3}{8}\) of a number equals 27, what is \(\frac{5}{12}\) of the same number?

A. 24

B. 40

C. 36

Correct D. 30

Correct Answer: D

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Question 11 View Details

The quadratic equation \(x^{2}-kx+m=0\) has two distinct real roots \(r\) and \(s\). It is known that the difference between the roots is \(4\) and that the sum of the squares of the roots is \(58\). Find the value of \(k+m\).

A. 32

B. 30

Correct C. 31

D. 35

Correct Answer: C

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Question 12 View Details

If \(\log_{2}x+\log_{x}16=6\) with \(x>0,\;x\neq1\), find the product of all possible values of \(x\).

A. 128

B. 16

C. 32

Correct D. 64

Correct Answer: D

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Question 13 View Details

In a class of 20 students the mean score on a test is 68 and the median score is 70. Exactly five students scored above 80 and the highest score is 95. The sum of the scores of the students who scored below 60 is 240. Determine the minimum possible number of students who could have scored exactly 70.

A. 3

B. 4

Correct C. 2

D. 1

Correct Answer: C

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Question 14 View Details

What is the remainder when \(7^{2023}\) is divided by \(10\)?

A. 7

B. 0

Correct C. 3

D. 5

Correct Answer: C

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Question 15 View Details

In the coordinate plane shown, points \(A(1,2)\) and \(B(4,6)\) are fixed. Point \(C\) lies on the line \(y=2x+1\). If the area of triangle \(ABC\) is \(12\) square units, find the sum of the possible \(x\)-coordinates of point \(C\).

A. 0

Correct B. -1

C. -3

D. 2

Correct Answer: B

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Question 16 View Details

In a school of 50 students, the numbers studying Mathematics (\(M\)), Physics (\(P\)) and Chemistry (\(C\)) are 30, 25 and 20 respectively. The numbers studying both Mathematics and Physics (\(M\cap P\)), both Mathematics and Chemistry (\(M\cap C\)), and both Physics and Chemistry (\(P\cap C\)) are 12, 10 and 8 respectively. If the probability that a randomly selected student studies at least one of the three subjects is \(0.94\), find the probability that the student studies exactly one of the three subjects. Express your answer as a decimal to two decimal places.

A. 0.38

B. 0.46

Correct C. 0.42

D. 0.50

Correct Answer: C

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Question 17 View Details

Determine all real numbers \(k\) for which the inequality \(|2x - 3| + |x + 1| \ge k\) holds for every real \(x\).

A. k \le \frac{3}{2}

B. k \ge \frac{5}{2}

Correct C. k \le \frac{5}{2}

D. k \le 2

Correct Answer: C

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Question 18 View Details

What is the smallest positive integer \(n\) such that when divided by 7 it leaves a remainder of 3 and when divided by 9 it leaves a remainder of 5?

A. 71

B. 65

Correct C. 59

D. 53

Correct Answer: C

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Question 19 View Details

In a factory, two types of widgets are produced: type A and type B. The proportion of type A widgets is \(p\) and type B is \(1-p\). The probability that a type A widget is defective is \(0.02\) and that a type B widget is defective is \(0.05\). A widget selected at random is found to be non‑defective. Given that the probability the selected widget is of type A, conditional on it being non‑defective, is \(0.8\), determine the value of \(p\). Express your answer as a reduced fraction.

A. 190/238

Correct B. 190/239

C. 190/240

D. 191/239

Correct Answer: B

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Question 20 View Details

Let \(S_n = \displaystyle\sum_{k=1}^{n} k\cdot 2^{k}\). (a) Derive a closed‑form expression for \(S_n\) in terms of \(n\). (b) Using the expression, compute \(S_5\).

A. S_n = 2 - (n+1)2^{n} + n2^{n+1}; S_5 = 130

Correct B. S_n = 2 - (n+1)2^{n+1} + n2^{n+2}; S_5 = 258

C. S_n = 2 - (n+2)2^{n+1} + n2^{n+2}; S_5 = 254

D. S_n = (n-1)2^{n+2} - (n+2)2^{n+1} + 2; S_5 = 300

Correct Answer: B

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Question 21 View Details

Points A(0,0) and B(6,0) are given. Find the coordinates of point C such that triangle ABC is right‑angled at C and the centroid of the triangle lies on the line \(y = -x + 2\).

Correct A. (3, -3)

B. (6, -6)

C. (3, 3)

D. (-3, -3)

Correct Answer: A

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Question 22 View Details

A data set of 12 numbers consists of three distinct values a, b, c occurring respectively 5, 4 and 3 times. The mean of the data set is 24, the range (c - a) is 20, and the median equals b. Determine the values of a, b and c.

A. a = 12, b = 27, c = 32

Correct B. a = 15, b = 27, c = 35

C. a = 15, b = 25, c = 35

D. a = 10, b = 27, c = 30

Correct Answer: B

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Question 23 View Details

In a plane, vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}| = 5\), \(|\mathbf{b}| = 7\) and the angle between them is \(60^{\circ}\). Let \(\mathbf{c} = 2\mathbf{a} - 3\mathbf{b}\). Find the scalar \(k\) such that the vector \(\mathbf{d} = \mathbf{a} + k\mathbf{b}\) is perpendicular to \(\mathbf{c}\), and then compute the magnitude \(|\mathbf{d}|\) (give the magnitude to two decimal places).

A. k = -1\/224, |d| ≈ 5.10

B. k = -5\/448, |d| ≈ 4.50

C. k = -5\/112, |d| ≈ 5.00

Correct D. k = -5/224, |d| ≈ 4.92

Correct Answer: D

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Question 24 View Details

Solve the inequality \(\displaystyle \frac{|2x-5|}{x-3} \le 2\) and express the solution set in interval notation.

Correct A. (-∞, 3)

B. (-∞, 5)

C. (-∞, 3]

D. (-∞, 2)

Correct Answer: A

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Question 25 View Details

An inverted right circular cone has height \(H = 12\) m and top radius \(R = 6\) m. Initially water fills the cone to a depth of \(4\) m from the vertex. A solid right circular cylinder of radius \(2\) m and height \(6\) m is placed coaxially at the bottom of the cone, fully submerged. Assuming no water overflows, find the new water depth (from the vertex) and give the percentage increase in the water depth relative to the original depth, rounded to the nearest whole percent.

A. New depth ≈ 5.90 m, percentage increase ≈ 48 %

B. New depth ≈ 8.20 m, percentage increase ≈ 105 %

C. New depth ≈ 6.45 m, percentage increase ≈ 61 %

Correct D. New depth ≈ 7.03 m, percentage increase ≈ 76 %

Correct Answer: D

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