3 Marks Question

Question 13 Marks

What was the date 1 arab/1 billion seconds ago?

Answer

1 billion seconds $=1,000,000,000$ seconds
$1,000,000,000 \div(60 \times 60 \times 24 \times 365)=31.71$ years
31.71 years $=31$ full years and $0.71 \times 365=259$ days
Today is : 28 July 2025
Subtracting 31 years $\rightarrow 28$ July 1994
Going back 259 days from 28 July $1994 \rightarrow 11$ November 1993.
So, 1 billion seconds ago was November 11, 1993.

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Question 23 Marks

Simplify and write these in the exponential form.
(i) $10^{-2} \times 10^{-5}$
(ii) $5^7 \div 5^4$
(iii) $9^{-7} \div 9^4$
(iv) $\left(13^{-2}\right)^{-3}$
(v) $m^5 n^{12}(m n)^9$

Answer

(i) $10^{-2} \times 10^{-5}=10^{-2-5}=10^{-7}$
(ii) $5^7 \div 5^4=5^{7-4}=5^3$
(iii) $9^{-7} \div 9^4=9^{-7+4}=9^{-3}$
(iv) $\left(13^{-2}\right)^{-3}=13^{-2 \times-3}=13^6$
(v) $m^5 n^{12}(m n)^9=m^5 n^{12} \times m^9 n^9=m^{5+9} n^{12+9}=m^{14} n^{21}$

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Question 33 Marks

Find out the units digit in the value of 2224 ÷ 432 ?

Answer

$2^{224} \div 4^{32}$
$\begin{array}{l}=2^{224} \div\left(2^2\right)^{32} \\=2^{224} \div 2^{2 \times 32} \\=2^{224} \div 2^{64} \\=2^{224-64}=2^{160}\end{array}$
$2^1=2$ (units digit 2)
$2^2=4$ (units digit 4)
$2^3=8$ (units digit 8)
$2^4=16$ (units digit 6)
$2^5=32$ (units digit 2)
$2^6=64$ (units digit 4)
$2^7=128$ (units digit 8)
...........
...........
Here, the pattern repeats after every 4 steps.
So, the unit's digit of $2^{160}$ is the same as that of $2^4$, which is 6 .

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Question 43 Marks

Express the following in exponential form :
(i) 5 × 5 × 7 × 7 × 7
(ii) 2 × 2 × a × a
(iii) a × a × a × c × c × c × c × d

Answer

(i) 5 × 5 × 7 × 7 × 7 = 5² × 7³.
(ii) 2 × 2 × a × a = 2² × a².
(iii) a × a × a × c × c × c × c × d = a³ × c⁴ × d.

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Question 53 Marks

Express the following in exponential form :
(i) 6 × 6 × 6 × 6
(ii) y × y
(iii) b × b × b × b

Answer

(i) 6 × 6 × 6 × 6 = 6⁴(ii) y × y = y²(iii) b × b × b × b = b⁴

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