Question
(a) Write down the following numbers in increasing order.
\(3.5, 1.6 \times 10^{-19}\), 60730, \(6.073 \times 10^5\), \(0.006073 \times 10^6\), \(\pi\), \(9.8 \times 10^{-18}\)
(b) State which of the numbers in part (a) is irrational.
Answer/Explanation
Ans:
(a) \(1.6 \times 10^{-19}, 9.8 \times 10^{-18}, \pi, 3.5, 0.006073 \times 10^6, 60730, 6.073 \times 10^5\)
(b) \(\pi\) is irrational.
Question
Let \(A = 4.5 \times 10^{-3}\) and \(B=6.2 \times 10^{-4}\). Find
(a) AB; (b) 2(A+B)
Give your answer in the form \(a\times 10^k\) where 1≤a<10 and \(k\epsilon \mathbb{Z}\)
Answer/Explanation
Ans:
(a) \(2.79 \times 10^{-6}\)
(b) \(1.024 \times 10^{-2}\) (Accept \(1.02 \times 10^{-2}\))
Question
Consider the following four numbers.
\(p=0.00314;\) \(q=0.00314 \times 10^2;\) \(r=\frac{\pi}{100};\) \(s=3.14 \times 10^{-2}\)
(a) One of these numbers is written in the form \(a \times 10^k\) where 1≤a<10 and \(k\epsilon \mathbb{Z}\).
Write down this number.
(b) Write down smallest of these numbers.
(c) Write down the value of q+s.
(d) Give your answer to part (c) in the form \(a \times 10^k\) where 1≤a<10 and \(k\epsilon \mathbb{Z}\).
Answer/Explanation
Ans:
(a) 3.14 × 102 or s
(b) 0.00314 or 3.14 × 10-3 or p
(c) 0.3454 (0.345)
(d) 3.454 × 10–1 (3.45 × 10–1)
Question
(a) Express each of the following in the form \(a \terms 10^k\) where 1≤a<10 and \(k\epsilon \mathbb{Z}\).
(i) mn (ii) \(\frac{m}{n}\) (iii) \(m^2\)
(b) Find the exact value of m + n
Answer/Explanation
Ans:
(a) (i) \(mn=6.0 \times 2.4 \times 10^{-2}=14.4 \times 10^{-2}\)
\(=(1.44 \times 10^1) \times 10^{-2}=1.44 \times 10^{-1}\)
(ii) \(\frac{m}{n} =\frac{6.0}{2.4} \times 10^8 = 2.5 \times 10^8\)
(b) m + n = 6000.000024
Question
(a) By using a LHS to RHS proof, prove the following identities
(i) \((x-1)^2 ≡ x^2-2x+1\) [use the fact \(a^2=a \times a]
(ii) \((x-1)^3 ≡x^3 – 3x^2 + 3x – 1\) [use the fact \(a^3=a^2 \times a]
(a) Verify the result (a)(ii)
(i) for x = 2
(ii) for x = 3
Answer/Explanation
Ans:
(a) LHS to RHS proof
(b) (i) LHS=1, RHS=1 (ii) LHS=8, RHS=8