Practice neco further_mathematics 2020 Past Questions

Question 1 View Details

Evaluate the definite integral \( \displaystyle I = \int_{0}^{\ln 2} \frac{e^{2x}}{1+e^{x}}\,dx \) and express your answer in exact form.

A. 2+\ln\frac{2}{3}

B. 1-\ln\frac{2}{3}

C. 1+\ln\frac{3}{2}

Correct D. 1+\ln\frac{2}{3}

Correct Answer: D

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Question 2 View Details

When the polynomial \(P(x)=2x^{3}-5x^{2}+ax+6\) is divided by \(x-2\) the remainder is \(4\). Find the value of \(a\).

A. 0

Correct B. 1

C. 5

D. -3

Correct Answer: B

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Question 3 View Details

Find the coefficient of the term containing \(x^{4}\) in the expansion of \((2x-\frac{1}{x^{2}})^{7}\).

A. 448

B. -560

Correct C. -448

D. -336

Correct Answer: C

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Question 4 View Details

In three‑dimensional space vectors \(\mathbf{a}\) and \(\mathbf{b}\) satisfy \(|\mathbf{a}|=3\), \(|\mathbf{b}|=4\) and \(\mathbf{a}\cdot\mathbf{b}=6\). Find (i) the magnitude of \(2\mathbf{a}-\mathbf{b}\) and (ii) the angle between \(2\mathbf{a}-\mathbf{b}\) and \(\mathbf{a}\).

A. Magnitude = 2\\sqrt{13}, Angle = \\cos^{-1}\\left(\\frac{1}{\\sqrt{7}}\\right)

B. Magnitude = 5, Angle = \\cos^{-1}\\left(\\frac{2}{5}\\right)

C. Magnitude = \\sqrt{28}, Angle = \\cos^{-1}\\left(\\frac{3}{\\sqrt{7}}\\right)

Correct D. Magnitude = 2\sqrt{7}, Angle = \cos^{-1}\left(\frac{2}{\sqrt{7}}\right)

Correct Answer: D

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Question 5 View Details

A linear transformation \(T\) in \(\mathbb{R}^{2}\) is represented by a matrix \(A\). It is known that \(T(1,2)=(3,5)\) and \(T(2,-1)=(0,-3)\). Determine the matrix \(A\) and then find \(T(4,1)\).

A. \\left(\\frac{18}{5},-\\frac{9}{5}\\right)

B. \\left(\\frac{9}{5},\\frac{18}{5}\\right)

C. \\left(-\\frac{18}{5},\\frac{9}{5}\\right)

Correct D. \left(\frac{18}{5},\frac{9}{5}\right)

Correct Answer: D

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Question 6 View Details

The matrix \( A = \begin{pmatrix} 2 & x \\ 3 & 5 \end{pmatrix} \) is known to satisfy the equation \( A^{2} - 7A + 10I = 0 \), where \( I \) is the \(2\times2\) identity matrix. Determine the value of \( x \) and then find \( \det A \).

A. 15

B. 5

C. 12

Correct D. 10

Correct Answer: D

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Question 7 View Details

Simplify the expression \( \frac{2^{6}\, 3^{3}}{6^{4}} \times \sqrt{\frac{9}{16}} \) and give your answer as an integer.

A. 4

B. 0

Correct C. 1

D. 2

Correct Answer: C

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Question 8 View Details

For the curve \( y = \frac{x^{3}}{3} - 2x^{2} + 5x - 1 \), find the point on the curve where the tangent is parallel to the line \( 3x - y + 2 = 0 \) and \( x < 3 \). Then write the equation of the normal line at that point in the form \( y = mx + c \).

Correct A. y = -\frac{1}{3}x + \frac{13 - 6\sqrt{2}}{3}

B. y = \frac{1}{3}x + \frac{13 - 6\sqrt{2}}{3}

C. y = -\frac{1}{3}x + \frac{13 + 6\sqrt{2}}{3}

D. y = -\frac{1}{3}x + \frac{13 - 3\sqrt{2}}{3}

Correct Answer: A

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Question 9 View Details

Let \( f : \{1,2,3,4,5\} \to \{a,b,c\} \) be defined by \( f(x) = a \) if \( x \) is odd and \( f(x) = b \) if \( x \) is even. State whether \( f \) is injective, surjective, both, or neither.

A. surjective only

Correct B. neither injective nor surjective

C. injective only

D. both injective and surjective

Correct Answer: B

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Question 10 View Details

A particle moves along a straight line with acceleration \( a(t) = -6t + 4 \) (in \(\text{m s}^{-2}\)), where \( t \) is the time in seconds (\( t \ge 0 \)). At \( t = 0 \) the particle has velocity \( 3\ \text{m s}^{-1} \) and position \( 2\ \text{m} \). The particle has mass \( 2\ \text{kg} \).\n\n(a) Find the time at which the particle first comes to rest.\n(b) Determine the total distance travelled by the particle from \( t = 0 \) until that instant.\n(c) Calculate the net work done on the particle during this interval.

A. t = \\frac{2 - \\sqrt{13}}{3}\\,\\text{s}; distance = \\frac{70 - 26\\sqrt{13}}{27}\\,\\text{m}; work = 9\\,\\text{J}

Correct B. t = \frac{2 + \sqrt{13}}{3}\,\text{s}; distance = \frac{70 + 26\sqrt{13}}{27}\,\text{m}; work = -9\,\text{J}

C. t = \\frac{2 + \\sqrt{13}}{3}\\,\\text{s}; distance = \\frac{70 + 26\\sqrt{13}}{27}\\,\\text{m}; work = -12\\,\\text{J}

D. t = \\frac{2 + \\sqrt{13}}{3}\\,\\text{s}; distance = \\frac{70 + 26\\sqrt{13}}{27}\\,\\text{m}; work = 9\\,\\text{J}

Correct Answer: B

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Question 11 View Details

In the expansion of \( (2x - 3)^7 \), find the sum of the coefficients of the terms in which the power of \( x \) is even.

Correct A. -39063

B. -15120

C. -22599

D. 39063

Correct Answer: A

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Question 12 View Details

A function is defined by \( f(x) = a x^{3} - 4x^{2} + b x + 5 \). Given that \( f'(2) = 10 \) and \( f''(1) = -6 \), determine the value of \( a + b \).

Correct A. 67/3

B. 65\/3

C. 68\/3

D. 22\/3

Correct Answer: A

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Question 13 View Details

Let \( T \) be a linear transformation in \( \mathbb{R}^{2} \) represented by the matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). It is known that \( T(1,2) = (3,5) \), \( T(2,-1) = (k,4) \), and that \( \det A = 7 \). Find the value of \( k \) and then compute \( T(4,1) \).

A. k = 9/5; T(4,1) = (419/25, 58/5)

B. k = 47/5; T(4,1) = (419/25, 60/5)

Correct C. k = 47/5; T(4,1) = (419/25, 58/5)

D. k = 47/5; T(4,1) = (400/25, 58/5)

Correct Answer: C

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Question 14 View Details

Evaluate the integral \( \displaystyle \int \frac{2x+3}{(x-1)(x^{2}+4)}\,dx \).

A. \ln|x-1| + \frac{1}{2}\ln(x^{2}+4) + \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C

B. \ln|x-1| - \frac{1}{2}\ln(x^{2}+4) - \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C

Correct C. \ln|x-1| - \frac{1}{2}\ln(x^{2}+4) + \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C

D. \ln|x-1| - \frac{1}{2}\ln(x^{2}+4) + \arctan\left(\frac{x}{2}\right) + C

Correct Answer: C

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Question 15 View Details

In a workshop, three machines A, B and C produce bolts. Machine A produces 50% of the bolts, B 30% and C 20%. Their respective defect rates are 1%, 3% and 4%. Bolts are painted red or blue: A paints 80% red, 20% blue; B paints 50% red, 50% blue; C paints 30% red, 70% blue. A bolt is selected at random and is found to be non‑defective and red. What is the probability that it was produced by machine A? Give your answer correct to three decimal places.

Correct A. 0.661

B. 0.600

C. 0.714

D. 0.540

Correct Answer: A

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Question 16 View Details

From a pool of 8 teachers (4 mathematics and 4 science) and 6 students (3 senior and 3 junior), a committee of 5 members is to be formed. The committee must contain at least two teachers, at least one senior student, and the number of teachers must be greater than the number of students. How many different committees can be formed?

Correct A. 882

B. 840

C. 864

D. 900

Correct Answer: A

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Question 17 View Details

In a factory, 60% of the widgets produced are type A and 40% are type B. A type A widget is defective with probability 0.02, while a type B widget is defective with probability 0.05. If a randomly selected widget is found to be defective, (a) what is the probability that it is type B? (b) If the factory produces 10,000 widgets, what is the expected number of defective type B widgets?

Correct A. Probability = 5/8; Expected number = 200

B. Probability = 2\/3; Expected number = 300

C. Probability = 3\/5; Expected number = 250

D. Probability = 1\/2; Expected number = 150

Correct Answer: A

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Question 18 View Details

A circle has centre \(C(2,-1)\) and radius 5. Through the external point \(P(10,3)\) two tangents can be drawn to the circle. Find the equation of the tangent line with positive gradient and then give the coordinates of its point of tangency.

A. Equation: y-3 = ((32 - 5\sqrt{55})/39)(x-10). Point of tangency: ((1102 - 121\sqrt{55})/(196 + 16\sqrt{55}), -(391 + 94\sqrt{55})/(196 + 16\sqrt{55}))

B. Equation: y-3 = ((32 + 5\sqrt{55})/39)(x+10). Point of tangency: ((1102 + 121\sqrt{55})/(196 + 16\sqrt{55}), (391 + 94\sqrt{55})/(196 + 16\sqrt{55}))

Correct C. Equation: y-3 = ((32 + 5\sqrt{55})/39)(x-10). Point of tangency: ( (1102 + 121\sqrt{55})/(196 + 16\sqrt{55}), -(391 + 94\sqrt{55})/(196 + 16\sqrt{55}) )

D. Equation: y-3 = ((32 + 5\sqrt{55})/39)(x-10). Point of tangency: ((1102 + 121\sqrt{55})/(196 - 16\sqrt{55}), -(391 + 94\sqrt{55})/(196 - 16\sqrt{55}))

Correct Answer: C

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Question 19 View Details

A complex number \(z\) satisfies \(|z-(3+4i)|=5\) and \(\arg(z)=\pi/6\). Find \(z^{2}\).

A. (\frac{43}{2}+12\sqrt{3}) + i\left(\frac{41}{2}\sqrt{3}+36\right)

Correct B. (\frac{43}{2}+12\sqrt{3}) + i\left(\frac{43}{2}\sqrt{3}+36\right)

C. (\frac{43}{2}+11\sqrt{3}) + i\left(\frac{43}{2}\sqrt{3}+36\right)

D. (\frac{45}{2}+12\sqrt{3}) + i\left(\frac{43}{2}\sqrt{3}+34\right)

Correct Answer: B

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Question 20 View Details

A bag contains 5 red, 4 blue and 3 green balls (total 12). Two balls are drawn sequentially without replacement. (a) Find the probability that the first ball is red and the second is green. (b) Find the probability that exactly one blue ball is drawn in the two draws. (c) Find the expected number of red balls drawn in the two draws.

A. a) 5\/48; b) 15\/33; c) 4\/6

B. a) 5\/40; b) 16\/30; c) 1\/1

C. a) 3\/44; b) 14\/33; c) 5\/12

Correct D. a) 5/44; b) 16/33; c) 5/6

Correct Answer: D

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Question 21 View Details

If \( f:\{1,2,3,4\}\rightarrow\{a,b,c\} \) is defined by \( f(1)=a, f(2)=b, f(3)=c, f(4)=b \), state whether \( f \) is one‑to‑one and whether it is onto.

Correct A. Not one-to-one; onto

B. One-to-one; not onto

C. One-to-one; onto

D. Not one-to-one; not onto

Correct Answer: A

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Question 22 View Details

A committee of five members is to be chosen from 8 boys and 6 girls. The committee must contain at least two girls and a chairperson who must be a boy. How many different ways can the committee and its chairperson be selected?

A. 1520

B. 5720

Correct C. 3760

D. 470

Correct Answer: C

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Question 23 View Details

A block of mass \(5\,\text{kg}\) is pulled up a rough plane that makes an angle of \(30^{\circ}\) with the horizontal by a constant force of \(60\,\text{N}\) parallel to the plane. The coefficient of kinetic friction is \(0.2\). The block starts from rest at the bottom and moves \(4\,\text{m}\) up the plane. (a) Find the speed of the block at the end of the motion. (b) Find the work done by the pulling force. (c) Find the mechanical energy lost due to friction (in joules).

A. Speed = 6.57 m/s; Work by pull = 250 J; Energy lost to friction = 33.94 J

B. Speed = 5.23 m/s; Work by pull = 240 J; Energy lost to friction = 33.94 J

C. Speed = 6.57 m/s; Work by pull = 240 J; Energy lost to friction = 45.00 J

Correct D. Speed = 6.57 m/s; Work by pull = 240 J; Energy lost to friction = 33.94 J

Correct Answer: D

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Question 24 View Details

In the expansion of \((x+2)^{8}\) the coefficient of \(x^{5}\) is \(C\). Using this, determine the coefficient of \(x^{10}\) in the expansion of \((x^{2}+2x)^{8}\).

A. 896

B. 1568

Correct C. 1792

D. 2048

Correct Answer: C

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Question 25 View Details

Vectors \(\mathbf{A}=\langle 3,-4,2\rangle\) and \(\mathbf{B}=\langle k,1,-1\rangle\) satisfy \(|\mathbf{A}+\mathbf{B}|=10\). Find the value of \(k\) that gives an acute angle between \(\mathbf{A}\) and \(\mathbf{B}\), and compute that acute angle (to the nearest degree).

A. k = -3 - 3√10; acute angle ≈ 68°

B. k = -3 + 3√10; acute angle ≈ 45°

C. k = 3 + 3√10; acute angle ≈ 68°

Correct D. k = -3 + 3√10; acute angle ≈ 68°

Correct Answer: D

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