Practice gce model questions vol1 further_mathematics 2021 Past Questions

Question 1 View Details

A club has 8 members, of whom 5 are women and 3 are men. A sub‑committee of 4 members is to be formed with the condition that at least two women are on the sub‑committee. After the sub‑committee is chosen, one of its members is appointed as chairperson. In how many different ways can the sub‑committee and its chairperson be selected?

A. 65

B. 195

C. 280

Correct D. 260

Correct Answer: D

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Question 2 View Details

The word "STATISTICS" consists of the letters S, T, A, T, I, S, T, I, C, S. How many distinct arrangements of all ten letters are there in which no two S's are adjacent?

A. 282240

B. 14700

Correct C. 23520

D. 141120

Correct Answer: C

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Question 3 View Details

A bag contains 6 red, 5 blue and 4 green balls. Three balls are drawn sequentially without replacement. What is the probability that exactly two of the drawn balls are red and the remaining ball is blue, in any order? Express your answer in lowest terms.

Correct A. 15/91

B. 12/91

C. 10/91

D. 20/91

Correct Answer: A

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Question 4 View Details

Given the polynomial \(P(x)=2x^{3}-3x^{2}+x-5\), find the value of \(P(2)-P(1)\).

A. 5

B. 8

Correct C. 6

D. 7

Correct Answer: C

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Question 5 View Details

Let \(f(x)=x^{4}+ax^{3}+bx^{2}+cx+d\). When \(f(x)\) is divided by the quadratic \(x^{2}+x-2\) the remainder is \(3x+5\). Find the value of \(a+b+c+d\).

Correct A. 7

B. 9

C. 12

D. 5

Correct Answer: A

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Question 6 View Details

In three‑dimensional space, vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\) satisfy \(|\mathbf{a}|=5,\;|\mathbf{b}|=12\) and \(\mathbf{a}\cdot\mathbf{b}=0\). Define \(\mathbf{c}=2\mathbf{a}-3\mathbf{b}\). Find the magnitude \(|\mathbf{c}|\) in simplest radical form.

Correct A. 2\sqrt{349}

B. 2\sqrt{350}

C. 2\sqrt{348}

D. 2\sqrt{351}

Correct Answer: A

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Question 7 View Details

A \(2\times2\) matrix \(P\) has determinant \(3\). The product \(PQ\) equals \(\begin{pmatrix}5&2\\2&5\end{pmatrix}\), where \(Q\) is a symmetric \(2\times2\) matrix with integer entries and whose eigenvalues are positive integers. Determine the trace of \(Q\).

A. 6

B. 9

Correct C. 8

D. 10

Correct Answer: C

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Question 8 View Details

Points \(B(1,2,0)\) and \(C(0,0,3)\) are given with respect to the origin \(O\). Point \(A\) lies in the plane \(z=0\) and the line \(AB\) is perpendicular to the plane \(OBC\). If the \(y\)-coordinate of \(A\) is \(1\), find the \(x\)-coordinate of \(A\).

Correct A. 3

B. 4

C. 5

D. 2

Correct Answer: A

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Question 9 View Details

Let \(f(x)=\dfrac{x^{2}+4x+5}{x+1}\). Find all real numbers \(x\) at which \(f\) has a stationary point. Give your answers in exact form.

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

B. -1-2\sqrt{2}, -1+2\sqrt{2}

Correct C. -1-\sqrt{2}, -1+\sqrt{2}

D. -2-\sqrt{2}, -2+\sqrt{2}

Correct Answer: C

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Question 10 View Details

A particle moves on a straight line with acceleration \(a(t)=6t-8\) m s\(^{-2}\). At \(t=0\) its velocity is \(3\) m s\(^{-1}\) and its position is \(20\) m from the origin. Determine, to two decimal places, the time \(t>0\) when the particle is again \(20\) m from the origin while moving in the opposite direction to its initial motion.

Correct A. 1.00

B. 0.50

C. 2.00

D. 1.50

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. Find the equation of the tangent line that has a positive gradient. Express your answer in the form \(y = mx + c\).

A. y = ((-12+5\sqrt{15})/11) x + (63-40\sqrt{15})/11

B. y = ((12-5\sqrt{15})/11) x + (-63+40\sqrt{15})/11

C. y = ((12+5\sqrt{15})/11) x + (-63-30\sqrt{15})/11

Correct D. y = ((12+5\sqrt{15})/11) x + (-63-40\sqrt{15})/11

Correct Answer: D

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Question 12 View Details

A uniform ladder 8\,m long and weighing 30\,kg leans against a smooth vertical wall. A person weighing 600\,N stands 3\,m up the ladder from the foot. The ground is rough with a coefficient of static friction \(\mu = 0.3\). The ladder makes an angle \(\theta\) with the ground. Determine the minimum angle (to the nearest degree) for which the ladder will not slip.

Correct A. 54

B. 48

C. 45

D. 60

Correct Answer: A

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Question 13 View Details

Consider the matrix \(B = \begin{pmatrix} 2 & -1 & 0 \\ k & 3 & 1 \\ 4 & 0 & k \end{pmatrix}.\)\n(a) Find all real values of \(k\) for which \(B\) is singular.\n(b) For \(k = 2\), compute the adjugate (classical adjoint) of \(B\).

Correct A. Singular for k = -3 ± √13; adjugate at k=2 is [[6,2,-1],[0,4,-2],[-12,-4,8]]

B. Singular for k = -3 ± √13; adjugate at k=2 is [[6,2,-1],[0,4,-2],[-12,-4,7]]

C. Singular for k = -3 ± 2√13; adjugate at k=2 is [[6,2,-1],[0,4,-2],[-12,-4,8]]

D. Singular for k = -3 ± √12; adjugate at k=2 is [[6,2,-1],[0,4,-2],[-12,-4,8]]

Correct Answer: A

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Question 14 View Details

A particle moves along a straight line with acceleration \(a(t) = -6t + 4\) (in \(\text{m s}^{-2}\)), where \(t\) is in seconds. At \(t = 0\) the particle has velocity \(v(0) = 3\,\text{m s}^{-1}\) and position \(s(0) = 2\,\text{m}\).\n(a) Determine the time (to two decimal places) when the particle comes to rest.\n(b) Find the total distance travelled from \(t = 0\) up to that instant (to two decimal places).

A. t = 1.50 s (to 2 d.p.); distance = 4.80 m (to 2 d.p.)

B. t = 2.13 s (to 2 d.p.); distance = 7.20 m (to 2 d.p.)

Correct C. t = 1.87 s (to 2 d.p.); distance = 6.06 m (to 2 d.p.)

D. t = 2.00 s (to 2 d.p.); distance = 5.50 m (to 2 d.p.)

Correct Answer: C

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Question 15 View Details

A particle moves on the parametric curve \(x = t^{2}+1,\; y = t^{3}-t\).\n(a) Find \(\displaystyle\frac{dy}{dx}\) at \(t = 2\).\n(b) Write the equation of the normal line to the curve at the point corresponding to \(t = 2\).\n(c) Determine the second derivative \(\displaystyle\frac{d^{2}y}{dx^{2}}\) at \(t = 2\).

A. dy\/dx = 11\/4; normal line: y-6 = 4\/11 (x-5); d²y\/dx² = 13\/30

B. dy\/dx = 9\/4; normal line: y-6 = -4\/9 (x-5); d²y\/dx² = 12\/32

C. dy\/dx = 11\/2; normal line: y-6 = -2\/11 (x-5); d²y\/dx² = 13\/16

Correct D. dy/dx = 11/4; normal line: y-6 = -4/11 (x-5); d²y/dx² = 13/32

Correct Answer: D

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Question 16 View Details

A particle moves along a straight line with acceleration (a(t)=6t-4\) (in m\,s\(^{-2}\)). At time (t=0\) its velocity is (3\) m\,s\(^{-1}\) and its position is (2\) m. \(\) (i) Find the time (in seconds) when the particle returns to its starting position. \(\) (ii) Determine the total distance travelled by the particle up to that time.

A. Time = 1 s; Total distance = \frac{7}{27} \text{ m}

Correct B. Time = 1 s; Total distance = \frac{8}{27} \text{ m}

C. Time = 0.5 s; Total distance = \frac{8}{27} \text{ m}

D. Time = 2 s; Total distance = \frac{5}{12} \text{ m}

Correct Answer: B

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Question 17 View Details

Simplify the expression \(\frac{2^{5}\times 8^{2}}{4^{3}}\) and write the result as a power of \(2\).

A. 2^{6}

B. 2^{4}

C. 2^{3}

Correct D. 2^{5}

Correct Answer: D

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Question 18 View Details

In triangle \(ABC\) the angles satisfy \(\angle B = 2\angle A\). The side opposite \(A\) has length \(7\) and the side opposite \(B\) has length \(10\). Find \(\sin \angle C\).

A. \frac{102\sqrt{5}}{343}

Correct B. \frac{102\sqrt{6}}{343}

C. \frac{102\sqrt{6}}{287}

D. \frac{84\sqrt{6}}{343}

Correct Answer: B

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Question 19 View Details

A cubic polynomial \(y = ax^{3}+bx^{2}+cx+d\) satisfies the following conditions: \(y(0)=2\), \(y(1)=5\), the curve has a turning point at \(x=2\) (so \(y'(2)=0\)), and it passes through the origin of the x‑axis at \(x=2\) (so \(y(2)=0\)). \(\) (i) Determine the coefficient \(a\). \(\) (ii) Find the exact area enclosed between the curve and the x‑axis from \(x=0\) to \(x=2\).

A. a = \frac{5}{2}; \text{Area} = \frac{22}{5}

B. a = \frac{7}{2}; \text{Area} = \frac{20}{3}

Correct C. a = \frac{9}{2}; \text{Area} = \frac{22}{3}

D. a = \frac{9}{2}; \text{Area} = \frac{24}{3}

Correct Answer: C

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Question 20 View Details

A (2\times2) matrix (A) satisfies the relation (A^{2}=5A-6I), where (I) is the identity matrix. Express (A^{5}) in the form (pA+qI) and state the values of the integers (p) and (q).

A. p = 212, q = -391

B. p = 210, q = -389

C. p = 211, q = -389

Correct D. p = 211, q = -390

Correct Answer: D

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Question 21 View Details

On the set \(S = \{0,1,2\}\) define a binary operation \(*\) by \(a * b = (a + b) \bmod 3\). Determine whether \(*\) is associative and state the identity element of \(*\).

A. Associative; identity element = 1

Correct B. Associative; identity element = 0

C. Non-associative; identity element = 0

D. Associative; identity element = 2

Correct Answer: B

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Question 22 View Details

An urn contains red, blue and green balls. The numbers are \(2x\) red, \(x+3\) blue and \(5\) green, and the total number of balls is \(35\). A sample of \(4\) balls is drawn without replacement. (i) Find \(x\). (ii) Find the probability that exactly two of the drawn balls are red. (iii) Find the expected number of red balls in the sample.

A. x = 9; probability = 2601/6000; expected number = 72/30

Correct B. x = 9; probability = 2601/6545; expected number = 72/35

C. x = 10; probability = 2800/6545; expected number = 80/35

D. x = 8; probability = 2401/6545; expected number = 64/35

Correct Answer: B

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Question 23 View Details

Line \(L\) passes through the point \((2,\,-1)\) and is perpendicular to the line joining \(A(5,\,3)\) and \(B(k,\,-2)\). The line \(L\) meets the circle \((x-1)^2 + (y+2)^2 = 25\) in a chord of length \(2\sqrt{23}\). Find the value of \(k\).

A. -3

B. 12

C. 5

Correct D. 0

Correct Answer: D

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Question 24 View Details

Let \(A=\begin{pmatrix}2 & a\\ b & 5\end{pmatrix}\) and \(B=\begin{pmatrix}2 & 1\\ 4 & 5\end{pmatrix}\). It is given that (i) \(\operatorname{tr}(A)=\operatorname{tr}(B)\), (ii) \(\det(AB)=12\), and (iii) the element in the first row, second column of \(AB\) equals 12. Find the ordered pair \((a,b)\).

A. (3, 4)

Correct B. (2, 4)

C. (1, 4)

D. (2, 5)

Correct Answer: B

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Question 25 View Details

Simplify \(\displaystyle \frac{2^{5}\,5^{3}}{10^{2}}\) and give the result as an integer.

A. 20

B. 80

C. 400

Correct D. 40

Correct Answer: D

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