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neco 2022 further_mathematics | Objective

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Question 1 View Details

Express the rational function \(\frac{3x^{2}+5x-2}{(x-1)(x^{2}+4)}\) in the form \(\frac{A}{x-1}+\frac{Bx+C}{x^{2}+4}\) and find the value of \(A+B+C\).

A. 49/6

B. 51/5

Correct C. 49/5

D. 47/5

Correct Answer: C

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Question 2 View Details

The matrix \(M=\begin{pmatrix}a&1\\2&3\end{pmatrix}\) is invertible and its inverse has all integer entries. Find the integer value of \(a\) and give \(M^{-1}\).

Correct A. a = 1; M^{-1}=\begin{pmatrix}3&-1\\-2&1\end{pmatrix}

B. a = 1; M^{-1}=\\begin{pmatrix}1&0\\0&1\\end{pmatrix}

C. a = -1; M^{-1}=\\begin{pmatrix}-3&1\\2&-1\\end{pmatrix}

D. a = 2; M^{-1}=\\begin{pmatrix}3&-1\\-2&1\\end{pmatrix}

Correct Answer: A

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Question 3 View Details

Evaluate \(\log_{3} 27 + \sqrt{49} - \log_{5} 125\).

A. 10

B. 13

C. 4

Correct D. 7

Correct Answer: D

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Question 4 View Details

When the polynomial \(f(x)=2x^{3}-5x^{2}+kx+12\) is divided by \(x-3\) the remainder is 4. Find \(k\).

Correct A. -17/3

B. -5\/3

C. 17\/3

D. 5\/3

Correct Answer: A

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Question 5 View Details

The quadratic equation \(x^{2}-(p+4)x+(2p-3)=0\) has integer roots and one of the roots is 3. Find the value of \(p\) and the other root.

A. p = -4, other root = -2

B. p = -2, other root = -1

Correct C. p = -6, other root = -5

D. p = -8, other root = -7

Correct Answer: C

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Question 6 View Details

The equation \( \sin \theta + \sqrt{3}\cos \theta = 1\) holds for an angle \(\theta\) measured in degrees with \(0^\circ \le \theta < 360^\circ\). Determine all possible values of \(\tan \dfrac{\theta}{2}\).

A. -1, \sqrt{3} - 2

B. 1, 2 - \sqrt{3}

C. 1, \sqrt{3} + 2

Correct D. 1, \sqrt{3} - 2

Correct Answer: D

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Question 7 View Details

A linear transformation \(T\) in the plane is represented by a matrix \(M = \begin{pmatrix} p & q \\ r & s \end{pmatrix}\). It is known that \(T\) maps the vector \((1,2)\) to \((5, a)\) and maps the vector \((3,-1)\) to \((b, 7)\), where \(a\) and \(b\) are positive integers less than 20. If all entries of \(M\) are integers, find the sum of all possible values of \(\det M\).

A. 30

B. 48

C. 36

Correct D. 42

Correct Answer: D

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Question 8 View Details

Simplify the expression \(\displaystyle \frac{2^{3}\cdot 5^{2}}{10^{2}}\times\sqrt{50}\) and give your answer in simplest radical form.

Correct A. 10\sqrt{2}

B. 10\\sqrt{5}

C. 20\\sqrt{2}

D. 5\\sqrt{2}

Correct Answer: A

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Question 9 View Details

For the circle \(x^{2}+y^{2}=25\) and the external point \(P(7,1)\): (i) find the equations of the two tangents drawn from \(P\) to the circle; (ii) compute the length of the tangent segment from \(P\) to the point of tangency.

A. 4x - 3y - 25 = 0, 3x + 4y - 25 = 0; length = 7

B. 4x - 3y + 25 = 0, 3x + 4y + 25 = 0; length = 5

C. 4x + 3y - 25 = 0, -3x + 4y - 25 = 0; length = 5

Correct D. 4x - 3y - 25 = 0, 3x + 4y - 25 = 0; length = 5

Correct Answer: D

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Question 10 View Details

A factory produces two types of widgets: type A (60% of output) with a defect probability of 2%, and type B (40% of output) with a defect probability of 5%. From a large production batch, ten widgets are selected at random (independent selections). What is the probability that exactly two of the selected widgets are defective and that at least one of the defective widgets is of type B? Give your answer correct to four decimal places.

Correct A. 0.0304

B. 0.0289

C. 0.0325

D. 0.0340

Correct Answer: A

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Question 11 View Details

A circle has centre \(C(2,-1)\) and radius \(5\). A line passes through the point \(P(8,3)\) and is tangent to the circle. Determine the coordinates of the point of tangency on the circle for the tangent that has a positive slope. Give your answer correct to two decimal places.

A. (6.88, 2.07)

Correct B. (6.88, -2.07)

C. (2.89, 3.92)

D. (7.00, -1.00)

Correct Answer: B

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Question 12 View Details

A uniform horizontal beam \(AB\) of length \(8\ \text{m}\) is hinged at \(A\) to a wall. The beam is supported by two cables: cable \(AC\) is attached to the beam at point \(C\) located \(3\ \text{m}\) from \(A\) and makes an unknown angle \(\theta\) above the beam; cable \(AB\) is attached at the far end \(B\) and makes an angle of \(45^{\circ}\) with the horizontal. The beam itself weighs \(300\ \text{N}\) acting at its centre, and a load of \(1200\ \text{N}\) hangs \(5\ \text{m}\) from \(A\). If the tension in cable \(AC\) is twice the tension in cable \(AB\), find (i) the tension in each cable (to the nearest newton) and (ii) the horizontal and vertical components of the reaction at the hinge \(A\).

Correct A. Tension in AB ≈ 763 N, tension in AC ≈ 1526 N, hinge reaction ≈ (1726 N rightward, 0 N upward)

B. Tension in AB ≈ 763 N, tension in AC ≈ 1526 N, hinge reaction ≈ (1726 N rightward, 100 N upward)

C. Tension in AB ≈ 700 N, tension in AC ≈ 1400 N, hinge reaction ≈ (1700 N rightward, 0 N upward)

D. Tension in AB ≈ 820 N, tension in AC ≈ 1640 N, hinge reaction ≈ (1760 N rightward, 0 N upward)

Correct Answer: A

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Question 13 View Details

A bag contains red, blue and green marbles. The probability of drawing a green marble is \(0.20\). The probability of drawing a red marble is twice the probability of drawing a blue marble. Two marbles are drawn independently with replacement. It is observed that at least one of the two draws is green. If the experiment were instead performed three times (with replacement), what is the probability that exactly two of the three draws are red? Give your answer to three decimal places.

Correct A. 0.497

B. 0.557

C. 0.527

D. 0.467

Correct Answer: A

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Question 14 View Details

Let \(f(x) = (ax + b)\,e^{2x}\). It is known that \(\displaystyle\int_{0}^{1} f(x)\,dx = 5\) and that the derivative at zero satisfies \(f'(0)=3\). Determine the constants \(a\) and \(b\) (rounded to three decimal places).

A. a ≈ 0.415, b ≈ 1.293

B. a ≈ -0.416, b ≈ 1.292

Correct C. a ≈ 0.416, b ≈ 1.292

D. a ≈ 0.416, b ≈ -1.292

Correct Answer: C

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Question 15 View Details

A linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) is represented in the standard basis by the matrix \(M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). The transformation satisfies \(T(1,1) = (5,7)\) and the vector \((1,-1)\) is an eigenvector of \(T\) with eigenvalue \(2\). Find the matrix \(M\) and then compute the \((1,2)\) entry of \(M^{3}\).

A. 90

Correct B. 78

C. 84

D. 72

Correct Answer: B

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Question 16 View Details

If \(3^{2x}=81\), find the value of \(x\). Then evaluate \(\log_{3}\bigl( \sqrt{3^{4x}} \bigr)\).

A. 8

B. 2

C. 5

Correct D. 4

Correct Answer: D

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Question 17 View Details

Decompose \(\frac{2x+3}{x^{3}-x}\) into partial fractions of the form \(\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}\). Find the sum \(A+B+C\).

A. 2

Correct B. 0

C. 1

D. -1

Correct Answer: B

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Question 18 View Details

Evaluate the definite integral \(\displaystyle \int_{0}^{\ln 2} \frac{e^{2x}}{1+e^{x}}\,dx\).

Correct A. 1 - \ln\frac{3}{2}

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

C. 1 - \\ln\\frac{5}{2}

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

Correct Answer: A

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Question 19 View Details

The curve is defined by \(x e^{y}=5\). Find \(\frac{dy}{dx}\) at the point where \(x=5\).

A. -\frac{1}{25}

B. \frac{1}{5}

Correct C. -\frac{1}{5}

D. -\frac{5}{1}

Correct Answer: C

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Question 20 View Details

Solve for \(x\) in the interval \(0^{\circ} \le x < 360^{\circ}\) satisfying \(\sin 2x = \sqrt{3}\,\cos x\). List all solutions.

A. 30°, 90°, 150°, 270°

Correct B. 60°, 90°, 120°, 270°

C. 60°, 150°, 210°, 270°

D. 60°, 90°, 180°, 270°

Correct Answer: B

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Question 21 View Details

In the expansion of \((2x^{2} - \frac{3}{x})^{6}\), find the coefficient of the term containing \(x^{3}\).

A. 4860

Correct B. -4320

C. -2160

D. 4320

Correct Answer: B

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Question 22 View Details

Given that \(\sin\theta = \frac{3}{5}\) with \(\theta\) acute, evaluate \(\tan\left(\frac{\pi}{4} - \frac{\theta}{2}\right)\).

A. 3/5

B. 2/1

Correct C. 1/2

D. 1/3

Correct Answer: C

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Question 23 View Details

Determine the value of the parameter \(a\) such that the rational function \(\frac{x^{2}+ax+6}{(x+1)(x+2)(x+3)}\) can be expressed as \(\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{x+3}\) where the coefficients of the first and third partial fractions are equal. Also give that common coefficient.

A. a = 5, A = C = 3\/2

B. a = 4, A = C = 1

C. a = 4, A = C = 2

Correct D. a = 4, A = C = 3/2

Correct Answer: D

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Question 24 View Details

Let the linear transformation \(T:\mathbb{R}^{2}\to\mathbb{R}^{2}\) be represented by the matrix \(M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). It is known that \(T(1,2) = (3,5)\) and \(T(2,-1) = (0,k)\). Find the value of \(k\) for which the determinant of \(M\) equals \(4\), and state that determinant.

A. k = -10/3, determinant = 4

Correct B. k = -20/3, determinant = 4

C. k = -5, determinant = 4

D. k = -20/3, determinant = 5

Correct Answer: B

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Question 25 View Details

Let \(z\) be a complex number satisfying \(|z-1| = \sqrt{2}\) and \(\arg(z)=\frac{\pi}{4}\). Determine \(z\) in the form \(x+iy\) and then compute \(z^{2}\).

Correct A. z = ((1+\sqrt{3})/2)(1+i), z^2 = (2+\sqrt{3})i

B. z = ((1+\sqrt{3})\/2)(1+i), z^2 = (2+\sqrt{3})

C. z = ((1-\sqrt{3})\/2)(1+i), z^2 = (2-\sqrt{3})i

D. z = ((1+\sqrt{3})\/2)(1-i), z^2 = -(2+\sqrt{3})i

Correct Answer: A

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