iGCSE Mathematics (0580) :E2.8 Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities.iGCSE Style Questions Paper 2

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

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