Question: Answer four questions only from this section
All questions carry equal marks.
9. a Given that $f(x) = x^2 + 2$ and $(f \circ g)(x) = 9x^2 - 6x + 3$, find the function $g(x)$.
b. Without performing long division, find the remainder when $8x^3 - 3x^2 - 5x + 2$ is divided by $(x - 1)$.
10. A cubic function is defined as: $f(x) = 2x^3 + px^2 + qx - 10$ where $p$ and $q$ are real constants. When $f(x)$ is divided by $(x - 2)$, the remainder is $14$. Also, when $f(x)$ is divided by $(x + 1)$, the remainder is $-4$.
(a) Form two linear equations in $p$ and $q$ using the given information.
(b) Hence, find the values of the constants $p$ and $q$.
(c) Using the values of $p$ and $q$ obtained, determine the remainder when $f(x)$ is divided by $(x - 3)$.
11. Resolve the expression $\frac{7x - 5}{(x - 2)(x + 3)}$ into partial fractions.
12. The 2nd term and the 5th term of a geometric progression are $12$ and $96$ respectively.
(a) Find the first term and the common ratio $r$.
(b) Find the 10th term of the progression.
(c) Calculate the sum of the first 12 terms.
13. a. Solve the equation: $2\log_{10}(x + 2) + \log_{10}(x - 1) = 1$
b. Evaluate $\sum_{n=3}^7 (n^2 - 1)$
14. A sequence of numbers $U_1, U_2, U_3, ..., U_n$ is defined by the relation $U_{n+1} + n^2 = nU_n + 2$ for all integers $n \geq 1$. Given that $U_1 = 2$, find:
(a) The values of $U_2, U_3,$ and $U_4$.
(b) An expression for the $n$th term, $U_n$, in terms of $n$.
(c) The sum of the first $n$ terms of the sequence.
15. The amount of money in a savings account after $x$ years is represented by $5^x$. The value of $x$ satisfies the equation: $5^{2x} - 12(5^x) + 20 = 0$. Find the possible values of $x$, giving your answers correct to three significant figures.
16. The universal set $R$ is given as: $R = \{0,1,2,3,4,5, ...,20\}$
Three subsets $A$, $B$, and $D$ of $R$ are defined as follows:
$A = \{\text{multiples of 3 less than 20}\}$
$B = \{x: 5 < x \leq 15, x \in R\}$
$D = \{\text{prime numbers less than 20}\}$
(a) List the members of each of the following sets:
(i) $A$
1. **Problem:** Given $f(x) = x^2 + 2$ and $(f \circ g)(x) = 9x^2 - 6x + 3$, find $g(x)$.
2. **Formula and rules:** The composition $(f \circ g)(x) = f(g(x))$. Since $f(x) = x^2 + 2$, then $f(g(x)) = (g(x))^2 + 2$.
3. **Work:**
$$(g(x))^2 + 2 = 9x^2 - 6x + 3$$
Subtract 2 from both sides:
$$(g(x))^2 = 9x^2 - 6x + 1$$
4. **Find $g(x)$:**
We want $g(x)$ such that $g(x)^2 = 9x^2 - 6x + 1$.
Notice $9x^2 - 6x + 1 = (3x - 1)^2$.
Therefore,
$$g(x) = 3x - 1$$
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1. **Problem:** Find the remainder when $8x^3 - 3x^2 - 5x + 2$ is divided by $(x - 1)$ without long division.
2. **Formula:** Remainder theorem states remainder is $f(1)$.
3. **Work:**
$$f(1) = 8(1)^3 - 3(1)^2 - 5(1) + 2 = 8 - 3 - 5 + 2 = 2$$
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1. **Problem:** Given $f(x) = 2x^3 + px^2 + qx - 10$, remainder when divided by $(x - 2)$ is 14, and by $(x + 1)$ is -4.
2. **Formula:** Remainder theorem: $f(a)$ is remainder when divided by $(x - a)$.
3. **Form equations:**
$$f(2) = 14 \Rightarrow 2(2)^3 + p(2)^2 + q(2) - 10 = 14$$
$$16 + 4p + 2q - 10 = 14$$
$$4p + 2q + 6 = 14$$
$$4p + 2q = 8$$
$$f(-1) = -4 \Rightarrow 2(-1)^3 + p(-1)^2 + q(-1) - 10 = -4$$
$$-2 + p - q - 10 = -4$$
$$p - q - 12 = -4$$
$$p - q = 8$$
4. **Solve system:**
From $4p + 2q = 8$, divide by 2:
$$2p + q = 4$$
From $p - q = 8$, add equations:
$$(2p + q) + (p - q) = 4 + 8$$
$$3p = 12 \Rightarrow p = 4$$
Substitute $p=4$ into $p - q = 8$:
$$4 - q = 8 \Rightarrow q = -4$$
5. **Find remainder when divided by $(x - 3)$:**
$$f(3) = 2(3)^3 + 4(3)^2 - 4(3) - 10 = 54 + 36 - 12 - 10 = 68$$
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1. **Problem:** Resolve $\frac{7x - 5}{(x - 2)(x + 3)}$ into partial fractions.
2. **Formula:**
$$\frac{7x - 5}{(x - 2)(x + 3)} = \frac{A}{x - 2} + \frac{B}{x + 3}$$
Multiply both sides by denominator:
$$7x - 5 = A(x + 3) + B(x - 2)$$
3. **Find A and B:**
Set $x=2$:
$$7(2) - 5 = A(2 + 3) + B(0) \Rightarrow 14 - 5 = 5A \Rightarrow 9 = 5A \Rightarrow A = \frac{9}{5}$$
Set $x = -3$:
$$7(-3) - 5 = A(0) + B(-3 - 2) \Rightarrow -21 - 5 = -5B \Rightarrow -26 = -5B \Rightarrow B = \frac{26}{5}$$
4. **Answer:**
$$\frac{7x - 5}{(x - 2)(x + 3)} = \frac{9/5}{x - 2} + \frac{26/5}{x + 3}$$