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neco 2018 further_mathematics | Essay

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Question 1 View Details

Let the universal set be $U = \{1,2,3,4,5,6,7,8\}$. Define the following subsets of $U$:\n\n- $A = \{x \in U \mid x \text{ is even}\}$.\n- $B = \{x \in U \mid x > 3\}$.\n\nUsing these sets answer the following questions:

Question Parts

(a)

List the elements of $A \cup B$.

(b)

Find the complement of $A \cap B$ with respect to $U$.

(c)

Define a binary operation $*$ on the set $A$ by $a * b = (a + b) \bmod 8$.\n(i) State whether $*$ is closed on $A$.\n(ii) State whether $*$ is associative on $A$.

() Is $*$ closed on $A$?

() Is $*$ associative on $A$?

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Question 2 View Details

A certain investment grows according to the law $P = P_{0}(1+i)^{n}$, where $P_{0}$ is the initial amount, $i$ is the annual rate of increase (expressed as a surd) and $n$ is the number of years.\n\nIt is observed that after 2 years the amount is $\$5,000$ and after 5 years the amount is $\$8,000$.\n\nAnswer the following:

Question Parts

(a)

Determine the value of $i$ in its simplest surd form.

(b)

Using the value of $i$ found in part (a), evaluate $\log_{\sqrt{2}}(i+1)$ and express your answer in terms of $\log_{\sqrt{2}}5$.

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Question 3 View Details

A cubic polynomial P(x)=2x^3 - kx^2 + 12x - 8 is known to have (x-2) as a factor.

Question Parts

(a)

Determine the value of k.

(b)

Using the value of k, factor P(x) completely into linear factors.

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Question 4 View Details

The rational function R(x)= (3x^2 + 5x - 2) / ((x-1)(x+2)^2) can be expressed as A/(x-1) + B/(x+2) + C/(x+2)^2.

Question Parts

(a)

Determine the constants A, B and C.

(b)

Using the result, solve for x (x \neq 1, -2) such that R(x) = 1/(x-1).

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Question 5 View Details

The matrix $A = \begin{bmatrix} 2 & k \\ 3 & 5 \end{bmatrix}$ is used to represent a system of linear equations. The determinant of $A$ is required to be 1.

Question Parts

(a)

Determine the integer value of $k$ for which $\det(A)=1$.

(b)

Find the inverse matrix $A^{-1}$ for the value of $k$ obtained in part (a).

(c)

Using the inverse found in part (b), solve the simultaneous equations $2x + ky = 7$ and $3x + 5y = 11$.

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Question 6 View Details

A school club has eight members: four girls (G₁, G₂, G₃, G₄) and four boys (B₁, B₂, B₃, B₄). The club wishes to form sub‑committees and also arrange a banquet seating.

Question Parts

(a)

In how many ways can a sub‑committee of three members be chosen so that it contains at least one girl?

(b)

The sub‑committee will elect a chairperson.\n(i) What is the probability that a randomly selected sub‑committee of three members contains exactly two girls?\n(ii) Given that the sub‑committee contains exactly two girls, in how many different ways can a chairperson be chosen?

(c)

For a banquet the eight members are to be seated around a circular table. In how many distinct seatings will the four girls sit together as a single block?

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Question 7 View Details

Consider the binomial expansion of $(2x-3)^n$.

Question Parts

(a)

It is given that the term containing $x^3$ in the expansion is $720x^3$. Determine the positive integer value of $n$.

(b)

Using the value of $n$ found in part (a), calculate the sum of the coefficients of all terms in the expansion that contain an even power of $x$.

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Question 8 View Details

In triangle $ABC$ the following data are given: $\sin A = \dfrac{3}{5}$ where $A$ is an acute angle, and the circumradius of the triangle is $R = 5\,\text{cm}$. Also, $\angle B = 2A$.

Question Parts

(a)

Find the length of side $a$ opposite angle $A$.

(b)

Determine the length of side $b$ opposite angle $B$.

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