University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

398

Pages:

4
Author:

MAlphen

Surds (Rationalisation) 2 18 1. Show that (2 + √7) − = 𝑐 + 𝑑√7 where 𝑐 and 𝑑 are integers. 3−√7 Ans: Exponential and Logarithmic Functions 1. Solve log9[log2(4𝑥 − 16)] = log16 4 . Ans: 𝑥 = 6 2. Solve log9 729 + log√ 3(𝑥 − 3) − log3 𝑥 = log81(3 − 𝑥)4. Ans: 𝑥 = 3(𝑟𝑒𝑗𝑒𝑐𝑡) or 𝑥 = 81 . 28 1. Solve 2 log4 𝑥 − 1 = log𝑥 64 Ans: 𝑥 = 8 𝑜𝑟 𝑥 = 1 4 2. The number of people infected with a disease varies according to the formula 𝑁 = 300 − 100𝑒−0.5𝑡 where N is the number of people infected with the disease and 𝑡 is the time inyears after deduction. (i) Find the number of people who were first diagnosed with the disease. (ii) As 𝑡 approaches infinity, find the number of people who will be infected by this disease. (iii) Express 𝑡 in terms of 𝑁. (iv) Sketch the graph of 𝑁 against t. 300−𝑁 Ans: (i) 200, (ii) The disease will spread to 300 people. (iii) 𝑡 = −2 𝑙𝑛 ( ). (iv) Graph 100 3. Solve, for 𝑥 and 𝑦, the simultaneous equations 2𝑥 = 8(2𝑦), lg(2𝑥 + 𝑦) = lg 63 − lg 3. (b) Express log√2 𝑦 = 3 − log2(𝑦 − 6) as a cubic equation.