Question
$\begin{array}{ll}(\mathbf{a})&y~\text{is directly proportional to}\left(x-1\right)^2.\\&\text{When }x=4,\:y=3.\end{array}$
Find $y$ when $x=7.$
(b) m is inversely proportional to the square root of $p.$
Explain what happens to the value of $m$ when the value of $p$ is multiplied by 9
▶️Answer/Explanation
(a) $12$
(b) divided by $3$
(a)
$
y \propto (x – 1)^2
$
$
y = k(x – 1)^2
$
\( x = 4 \), \( y = 3 \)
$
3 = k(4 – 1)^2
$
$
3 = k(3)^2
$
$
3 = 9k
$
$
k = \frac{3}{9} = \frac{1}{3}
$
Now, we want to find the value of \( y \) when \( x = 7 \).
$
y = \frac{1}{3}(7 – 1)^2
$
$
= \frac{1}{3}(6)^2
$
$
= \frac{1}{3}(36)
$
$
= 12
$
(b)
$
m \propto \frac{1}{\sqrt{p}}
$
$
m = \frac{k}{\sqrt{p}}
$
If the value of \( p \) is multiplied by 9, the square root of \( p \) becomes 3 times larger:
$
\sqrt{9p} = 3\sqrt{p}
$
Since \( m \) is inversely proportional to the square root of \( p \), multiplying the square root by 3 means that the value of \( m \) will be divided by 3
$
m_{\text{new}} = \frac{m}{3}
$
Question
The first four terms of a sequence are
\(T_1=1^2 T_2=1^2+2^2 T_3=1^2+2^2+3^2 T_4=1^2+2^2+3^2 +4^2\)
(a) The nth term is given by \(T_n\frac{1}{6}n(n+1)(2n+1)\)
Work out the value of \(T_{23}\)
(b) A new sequence is formed as follows.
\(U_1=T_2-T_1 U_2=T_3-T_2 U_3=T_4-T_3\)
(i) Find the values of \(U_1\) and \(U_2\)
(ii) Write down a formula for the nth term, \(U_n\)
(c) The first four terms of another sequence are
\(V_1=2^2 T_2=2^2+4^2 V_3=2^2+4^2+6^2 V_4=2^2+4^2+6^2 +8^2\)
By comparing this sequence with the one in part (a), find a formula for the nth term,\( V_n\)
Answer/Explanation
(a) \(T_n=\frac{1}{6}n(n+1)(2n+1)\)
\(T_{23}=\frac{1}{6} 23(23+1)(2.23+1)\)
\(=\frac{1}{6}23(24)(46+1)\)
=4324
(b)(i) \(U_1=T_2-T_1\)
\(U_1\)=1+4-1
=4
\(U_2\)=1+4+9-(1+4)
=9
(ii) \(U_n=T_{n+1}-T_n\)
\({(n+1)}^2\) or \(n^2+2n+1\)
(c) \(V_n=\frac{2}{3}n(n+1)(2n+1)\)
Question
f(x) = 4(x + 1) \(g(x)=\frac{x^{3}}{2}-1\)
(a) Write down the value of x when f -1(x) = 2.
Answer/Explanation
Ans: 12
(b) Find fg(x). Give your answer in its simplest form.
Answer/Explanation
Ans: 2x3 cao
(c) Find g-1(x).
Answer/Explanation
Ans: \(\sqrt[3]{2(x+1)}oe\)
Question
The electrical resistance, R, of a length of cylindrical wire varies inversely as the square of the
diameter, d, of the wire.
R = 10 when d = 2.
Find R when d = 4
Answer/Explanation
Ans: 2.5