Practice neco mathematics 2021 Past Questions

Question 1 View Details

In a plane, vectors \(\mathbf{u}\) and \(\mathbf{v}\) satisfy \(|\mathbf{u}| = 5\), \(|\mathbf{v}| = 7\) and the vector \(\mathbf{w}=2\mathbf{u}-\mathbf{v}\) is perpendicular to \(\mathbf{u}+\mathbf{v}\). Find (i) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) correct to one decimal place, and (ii) the exact value of \(|\mathbf{u}+\mathbf{v}|\).

A. Angle = 100.2°, |\mathbf{u}+\mathbf{v}| = 6\sqrt{3}

Correct B. Angle = 91.6°, |\mathbf{u}+\mathbf{v}| = 6\sqrt{2}

C. Angle = 85.0°, |\mathbf{u}+\mathbf{v}| = 5\sqrt{3}

D. Angle = 90.0°, |\mathbf{u}+\mathbf{v}| = 8\sqrt{2}

Correct Answer: B

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Question 2 View Details

In triangle \(ABC\), \(AB = 13\) cm, \(AC = 15\) cm and the circumradius is \(10\) cm. Point \(D\) lies on side \(BC\) such that \(BD:DC = 3:5\). Determine the length of \(AD\) (in cm).

A. 13.0

B. 9.2

Correct C. 10.5

D. 11.7

Correct Answer: C

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Question 3 View Details

The sum of the first \(n\) terms of a sequence is given by \(S_n = n^{3} + 6n^{2} + 11n\). Find the 10th term of the sequence and the sum of the terms from the 6th through the 10th inclusive.

A. 10th term = 400, sum of terms 6th to 10th = 1385

B. 10th term = 420, sum of terms 6th to 10th = 1400

Correct C. 10th term = 396, sum of terms 6th to 10th = 1380

D. 10th term = 395, sum of terms 6th to 10th = 1375

Correct Answer: C

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Question 4 View Details

An infinite geometric series has sum 12 and the sum of the squares of its terms is 20. Find the first term \(a\) and common ratio \(r\) (with \(|r|<1\)), and then give the sum of its first three terms.

A. a = 121\/41, r = 31\/41, sum of first three terms = 469560\/68921 \(\approx 6.81\)

B. a = 120\/41, r = -31\/41, sum of first three terms = 469560\/68921 \(\approx 6.81\)

C. a = 120\/41, r = 31\/41, sum of first three terms = 470000\/68921 \(\approx 6.82\)

Correct D. a = 120/41, r = 31/41, sum of first three terms = 469560/68921 \(\approx 6.81\)

Correct Answer: D

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Question 5 View Details

The Venn diagram below represents three sets \(A\), \(B\) and \(C\) within a universal set of 120 elements. The numbers in each region are: only \(A\): 10, only \(B\): 20, only \(C\): 15, \(A\cap B\) only: 20, \(A\cap C\) only: 15, \(B\cap C\) only: 10, and \(A\cap B\cap C\): 10. How many elements belong to none of the three sets?

A. 40

Correct B. 20

C. 10

D. 30

Correct Answer: B

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Question 6 View Details

In a school of 120 students, let \(A\) be the set of students who study Mathematics, \(B\) those who study Physics and \(C\) those who study Chemistry. It is known that \(|A| = 70\), \(|B| = 60\), \(|C| = 50\), \(|A \cap B| = 30\) and \(|A \cap C| = 25\). The number of students who study exactly two of the three subjects is 45. Find the number of students who study all three subjects.

A. 20

B. 10

C. 25

Correct D. 15

Correct Answer: D

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Question 7 View Details

In triangle \(ABC\) the angle at \(B\) is \(60^{\circ}\), \(AB = 8\,\text{cm}\) and \(BC = 6\,\text{cm}\). Point \(D\) lies on side \(AC\) such that \(AD : DC = 1 : 2\). Let \(\theta\) be the measure of angle \(ABD\). Find \(\sin \theta\).

A. (3\\sqrt{3})\/(\\sqrt{97})

B. (3\\sqrt{5})\/(2\\sqrt{97})

C. (\\sqrt{3})\/(2\\sqrt{97})

Correct D. (3\sqrt{3})/(2\sqrt{97})

Correct Answer: D

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Question 8 View Details

Simplify the expression \(\dfrac{\bigl(2^{3}\cdot5^{2}\bigr)^{\frac12}}{\sqrt{20}}\) and write the result in simplest surd form.

A. \sqrt{20}

B. 2\sqrt{2}

C. \sqrt{5}

Correct D. \sqrt{10}

Correct Answer: D

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Question 9 View Details

Let real numbers \(x\) and \(y\) satisfy \(x+y=6\) and \(x^{2}+y^{2}=20\). Find the value of \(x^{3}y+xy^{3}\).

A. 196

Correct B. 160

C. 48

D. 120

Correct Answer: B

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Question 10 View Details

The quadratic equation \(t^{2}-(k+2)t+(k-3)=0\) has two real roots whose sum is twice their product. Determine the value of \(k\) and then give the product of the roots.

Correct A. k = 8, product = 5

B. k = 9, product = 5

C. k = 8, product = 6

D. k = 6, product = 4

Correct Answer: A

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Question 11 View Details

Two urns, A and B, are selected at random with probabilities p and 1-p respectively. Urn A contains only white balls. Urn B contains white and black balls. The probability that a ball drawn from the chosen urn is white is \( \frac{3}{5} \). Given that the drawn ball is white, the probability that it came from Urn A is \( \frac{2}{3} \). Find the ratio of white to black balls in Urn B, expressed in simplest integer form.

A. 1:3

B. 3:5

C. 2:3

Correct D. 1:2

Correct Answer: D

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Question 12 View Details

Let \( a \) and \( b \) be non‑zero numbers satisfying \( \frac{a}{b}+\frac{b}{a}=5 \). Evaluate \( \frac{a^{3}+b^{3}}{a^{2}b+ab^{2}} \).

Correct A. 4

B. 3

C. 6

D. 5

Correct Answer: A

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Question 13 View Details

Solve the inequality \( \frac{x^{2}-6x+8}{x^{2}-9}\ge 0 \) and express the solution set in interval notation.

A. (-\\infty, -3] \\cup (2,3) \\cup (4, \\infty)

Correct B. (-\infty, -3) \cup [2,3) \cup [4, \infty)

C. (-\\infty, -3) \\cup [2,3] \\cup (4, \\infty)

D. (-\\infty, -2) \\cup [2,4)

Correct Answer: B

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Question 14 View Details

The quadratic equation \( x^{2}-4x+m=0 \) has its vertex at the point \( (2,-5) \). Determine the value of \( m \) and then give the exact roots of the equation.

A. 2 \\pm \\sqrt{3}

B. 2 \\pm \\sqrt{4}

C. 2 \\pm \\sqrt{6}

Correct D. 2 \pm \sqrt{5}

Correct Answer: D

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Question 15 View Details

Solve for \( x \): \( \frac{1}{x}+\frac{1}{x+1}=\frac{5}{6} \). List all real solutions.

Correct A. 2, -\frac{3}{5}

B. -2, -\frac{3}{5}

C. 1, -\frac{3}{5}

D. 2, -\frac{2}{5}

Correct Answer: A

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Question 16 View Details

Simplify the expression \( \frac{2^{5}\times\sqrt{2}}{8^{2}} \) and give your answer in simplest surd form.

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

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

C. \frac{4}{\sqrt{2}}

D. \frac{\sqrt{2}}{4}

Correct Answer: A

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Question 17 View Details

Group A consists of 12 students whose exam scores have a mean of 75 and a variance of 16. Group B consists of 18 students with a mean of 68 and a variance of 25. Find the overall mean and the overall standard deviation of the combined 30 students. Give your answer for the standard deviation correct to two decimal places.

Correct A. Mean = 70.8, SD = 5.74

B. Mean = 70.8, SD = 4.63

C. Mean = 70.8, SD = 5.64

D. Mean = 71.5, SD = 5.74

Correct Answer: A

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Question 18 View Details

Express the repeating decimal \(0.\overline{27}\) as a fraction in lowest terms, then find three quarters of that fraction. Give your final answer as a fraction.

A. \frac{27}{44}

B. \frac{3}{44}

Correct C. \frac{9}{44}

D. \frac{9}{33}

Correct Answer: C

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Question 19 View Details

The time \(t\) (in minutes) required to paint a wall varies directly as the area \(A\) (in square metres) and inversely as the number of painters \(p\). If 3 painters can paint a 24 m\(^2\) wall in 40 minutes, how long will it take 5 painters to paint a 30 m\(^2\) wall? Give your answer to the nearest minute.

Correct A. 30

B. 24

C. 35

D. 40

Correct Answer: A

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Question 20 View Details

A box contains 5 white, 4 black and 3 red balls. Two balls are drawn successively without replacement. Find the probability that the first ball drawn is white given that at least one of the two balls drawn is red. Express your answer as a fraction in lowest terms.

A. \frac{5}{12}

B. \frac{5}{11}

C. \frac{15}{60}

Correct D. \frac{15}{76}

Correct Answer: D

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Question 21 View Details

The quadratic equation \(x^{2}-(p+5)x+6p=0\) has two distinct positive integer roots, one of which is a divisor of 12. Find the possible values of \(p\) and the corresponding roots.

A. p=4 with roots 1 and 6; or p=10 with roots 5 and 8

Correct B. p=2 with roots 3 and 4; or p=14 with roots 7 and 12

C. p=6 with roots 2 and 8; or p=18 with roots 9 and 12

D. p=3 with roots 2 and 5; or p=12 with roots 6 and 9

Correct Answer: B

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Question 22 View Details

A rectangular garden consists of a rectangle of length \(L\) metres and width \(W\) metres, with a semicircular flower bed attached to the side of length \(L\). The total area of the garden is \(200\) m\(^2\) and the total perimeter (including the curved edge of the semicircle) is \(80\) m. Determine the dimensions \(L\) and \(W\) (give answers to two decimal places).

A. L \\approx 4.90 m, W \\approx 34.20 m

Correct B. L \approx 5.73 m, W \approx 32.64 m

C. L \\approx 5.73 m, W \\approx 33.10 m

D. L \\approx 6.12 m, W \\approx 31.45 m

Correct Answer: B

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Question 23 View Details

Find the smallest positive integer \(n\) such that \(n\) leaves a remainder of 3 when divided by 7 and a remainder of 5 when divided by 9.

A. 122

Correct B. 59

C. 53

D. 65

Correct Answer: B

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Question 24 View Details

The diagram shows a piecewise linear function \(f(x)\) defined by the points A(0,0), B(4,8), C(6,8) and D(8,0) connected consecutively. Determine (i) the maximum value of \(f\) and the smallest \(x\) at which it occurs, and (ii) the total area enclosed between the graph of \(f\) and the \(x\)-axis.

A. Maximum value = 8 at x = 4; total area = 36

Correct B. Maximum value = 8 at x = 4; total area = 40

C. Maximum value = 8 at x = 6; total area = 40

D. Maximum value = 6 at x = 4; total area = 40

Correct Answer: B

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Question 25 View Details

The graph of \(y = ax^{2}+bx+c\) is a parabola opening upwards with vertex at \((2,-3)\) and passing through the point \((0,5)\). Find the explicit equation of the parabola.

A. y = 2x^{2} - 4x + 5

Correct B. y = 2x^{2} - 8x + 5

C. y = 2x^{2} - 8x - 5

D. y = -2x^{2} - 8x + 5

Correct Answer: B

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