Page No 259:

Question 115:

Express each of the following in standard form:

(a) The mass of a proton in gram is

Question 127:

An insect is on the 0 point of a number line, hopping towards 1. She covers half the distance from her current location to 1 with each hop. So, she will be at $\frac{1}{2}$ after one hop, $\frac{3}{4}$ after two hops, and so on.

(a) Make a table showing the insect’s location for the first 10 hops.
(b) Where will the insect be after n hops?
(c) Will the insect ever get to 1? Explain.

Answer:

Number of hops	Distance covered	Distance left	Distance covered
1.	$\frac{1}{2}$	$\frac{1}{2}$	$1 - \frac{1}{2}$
2.	$\frac{1}{2} ((\frac{1}{2}) + \frac{1}{2})$	$\frac{1}{4}$	$1 - \frac{1}{4}$
3.	$\frac{1}{2} (\frac{1}{4}) + \frac{3}{4}$	$\frac{1}{8}$	$1 - \frac{1}{8}$
4.	$\frac{1}{2} (\frac{1}{8}) + \frac{7}{8}$	$\frac{1}{16}$	$1 - \frac{1}{16}$
5.	$\frac{1}{2} (\frac{1}{16}) + \frac{15}{16}$	$\frac{1}{32}$	$1 - \frac{1}{32}$
6.	$\frac{1}{2} (\frac{1}{32}) + \frac{31}{32}$	$\frac{1}{64}$	$1 - \frac{1}{64}$
7.	$\frac{1}{2} (\frac{1}{64}) + \frac{63}{64}$	$\frac{1}{128}$	$1 - \frac{1}{128}$
8.	$\frac{1}{2} (\frac{1}{128}) + . . . .$	$\frac{1}{256}$	$1 - \frac{1}{256}$
9.	$\frac{1}{2} (\frac{1}{256}) + \frac{255}{256}$	$\frac{1}{512}$	$1 - \frac{1}{512}$
10.	$\frac{1}{2} (\frac{1}{512}) + \frac{511}{512}$	$\frac{1}{1024}$	$1 - \frac{1}{1024}$

(b) If we see the patteron.
Distance covered in 1^st hop =

1 - \frac{1}{2}

Distance covered in 2^nd hop =

1 - \frac{1}{4} = 1 - {(\frac{1}{2})}^{2}

Distance covered in 3^rd hop =

1 = \frac{1}{8} = 1 - {(\frac{1}{2})}^{3}

Thus, after observing the distance covered in the above there hops.
Distance covered in h hops =

1 - {(\frac{1}{2})}^{n}

{(\frac{1}{2})}^{n}

hops be zero. for some infinite in which is not possible.
(b) (i) 4¹² = (4²)⁶ [âˆµ o^mn = (a^m)ⁿ]
                 = (16)⁶
Hence, ones digit in 4² is 6.
(ii) 9²⁰ = (9²)¹⁰
             = (81)¹⁰
Hence, ones digit in 9²⁰ is 1.
(iii) 3¹⁷= 3⁸× 3⁸ × 3
              = 6561 × 6561 × 3
              = (6561)² × 3
Hence, ones digit in 3¹⁷ is 3.
(iv)

\underset{\leftarrow}{100} s

ones digit in 5¹⁰⁰ is 5.
(v) 10⁵⁰⁰ ones digit in 10⁵⁰⁰ is 0.
(c) (i) (31)¹⁰ = (31²)⁵
Hence, ones digit in 31¹⁰is 1.
(ii) 1210 = (12²)⁵
= (144)⁵
Hence, ones digit in 12¹⁰ is 4.
(iv) 29¹⁰ = (29)⁵
= (841)⁵
Hence, ones digit in 29¹⁰is 1.

Page No 261:

Question 128:

Predicting the ones digit, copy and complete this table and answer the questions that follow.

Powers Table
x	1^x	2^x	3^x	4^x	5^x	6^x	7^x	8^x	9^x	10^x
1	1	2
2	1	4
3	1	8
4	1	16
5	1	32
6	1	64
7	1	128
8	1	256
Ones Digits of the Powers	1	2,4,8,6

(a) Describe patterns you see in the ones digits of the powers.

(b) Predict the ones digit in the following:

1. 4¹²
2. 9²⁰
3. 3¹⁷
4. 5¹⁰⁰
5. 10⁵⁰⁰

1. 31¹⁰
2. 12¹⁰
3. 17²¹
4. 29¹⁰

Answer:

(a)

Powers Table
x	1^x	2^x	3^x	4^x	5^x	6^x	7^x	8^x	9^x	10^x
1	1	2	3	4	5	6	7	8	9	10
2	1	4	9	16	25	36	49	64	81	100
3	1	8	27	64	125	216	343	512	729	1000
4	1	16	81	256	625	1296	2401	4096	6561	10000
5	1	32	243	1024	3125	7776	16807	32768	59049	100000
6	1	64	729	4096	15625	46656	117649	262144	531441	1000000
7	1	128	2187	16384	78125	279936	823543	2097152	4782969	10000000
8	1	256	6561	65536	390625	1679616	5764801	16777216	43046721	100000000
Ones Digits of the Powers	1	2,4,8,6	3,9,7,1	4,6	5	6	7,9,3,1	8,4,2,6	9,1	0

Page No 262:

Question 129:

Astronomy The table shows the mass of the planets, the sun and the moon in our solar system.

Celestial Body	Mass (kg)	Mass (kg) Standard Notation
Sun	1,990,000,000,000,000,000,000,000,000,000	1.99 × 10³⁰
Mercury	330,000,000,000,000,000,000,000
Venus	4,870,000,000,000,000,000,000,000
Earth	5,970,000,000,000,000,000,000,000
Mars	642,000,000,000,000,000,000,000,000,000
Jupiter	1,900,000,000,000,000,000,000,000,000
Saturn	568,000,000,000,000,000,000,000,000
Uranus	86,800,000,000,000,000,000,000,000
Neptune	102,000,000,000,000,000,000,000,000
Pluto	12,700,000,000,000,000,000,000
Moon	73,500,000,000,000,000,000,000

(a) Write the mass of each planet and the Moon in scientific notation.
(b) Order the planets and the moon by mass, from least to greatest.
(c) Which planet has about the same mass as earth?

Answer:

^(b(a) (i) mass of each planet and moon in scientific relation is given below:
Using the law of exponent a^m × aⁿ = a^m⁺ⁿ
Mercury = 33 × 10²²= 3.3 × 10¹...
               = 3.3 × 10²³
Venus = 487 × 10²² = 4.87 × 10²² × 10²
           = 4.87 × 10²⁴
Earth = 597 × 10²² = 5.97 × 10²² × 10²
        = 5.97 × 10²⁴
Mars = 642 × 10²⁷ = 6.42 × 10²⁷ × 10²
         = 6.42 × 10²⁹
Jupiter = 19 × 10²⁶ = 1.9 × 10²⁶ × 10
            = 1.9 × 10²⁷
Satron = 568 × 10²⁴= 5.68 × 10²⁴× 10²
            = 5.68 × 10²⁶
Uranus = 868 × 10²³ = 8.68 × 10²³ × 10²
             = 8.68 × 10²⁵
Neptune = 102 × 10²⁴ = 1.02 × 10²⁴× 10²
              = 1.02 × 10²⁶
Pluto = 127 × 10²⁰ = 1.27 × 10²⁰ × 10²
         = 1.27 × 10²²
Moon = 735 × 10²⁰ = 7.35 × 10²⁰× 10²
          = 7.35 × 10²²
(b) Order of the planets and moon form by mass from least to greates.
Pluto < moon < mercury < venus < earth < venus < neptune < saturm < jupiter < mass.
(c) Mass of earth = 5.97 × 10²⁴
Mass of venus = 4.87 × 10²⁴
Hence, venus hop about the same mass as earth.

Page No 262:

Question 130:

Investigating Solar System The table shows the average distance from each planet in our solar system to the sun.

Planet	Distance from Sun (km)	Distance from Sun (km) Standard Notation
Earth	149,600,000	1.496 × 10⁸
Jupiter	778,300,000
Mars	227,900,000
Mercury	57,900,000
Neptune	4,497,000,000
Pluto	5,900,000,000
Saturn	1,427,000,000
Uranus	2,870,000,000
Venus	108,200,000

(a) Complete the table by expressing the distance from each planet to the Sun in scientific notation.
(b) Order the planets from closest to the sun to farthest from the sun.

Answer:

(a)

Planet	Distance from Sun (km)	Distance from Sun (km) Standard Notation
Earth	149,600,000	1.496 × 10⁸
Jupiter	778,300,000	7.783 × 10⁸
Mars	227,900,000	2.279 × 10⁸
Mercury	57,900,000	5.79 × 10⁷
Neptune	4,497,000,000	4.497 × 10⁹
Pluto	5,900,000,000	5.9 × 10⁹
Saturn	1,427,000,000	1.427 × 10⁹
Uranus	2,870,000,000	2.87 × 10⁹
Venus	108,200,000	1.082 × 10⁸

(a) Earth = 149,600,000 = 1496 × 10⁵
                                       = 1.496 × 10⁸
Jupiter = 778,300,000 = 7783 × 10⁵
                                    = 7.783 × 10⁸
Mars = 227,900,000 = 22709 × 10⁵
                                 = 2.279 × 10⁸
Mercury = 57,900,000 = 579 × 10⁵
                                     = 5.79 × 107
Neptune = 4,497,000,000 = 4457 × 10⁶
                                          = 4.497 × 10⁹
Pluto = 5,900,000,000 = 59 × 10⁸
                                     = 5.9 × 10⁹
Satuon = 1,427,000,000 = 1427 × 10⁶
                                       = 1.427 × 10⁹
Uranus = 2,870,000,000 = 287 × 10⁷
                                        = 2.87 × 10⁹
Venus = 108,200,000 = 1082 × 10⁵
                                   = 1.082 × 10⁸
(b) Order of the planets form closest to the sum to furthest to the sum:
Mercury < venus < earth < mars < jupiter < sothurn < urenus < neptune < pluto.

Page No 262:

Question 131:

This table shows the mass of one atom for five chemical elements. Use it to answer the question given.

Element	Mass of atom (kg)
Titanium	7.95 × 10⁻²⁶
Lead	3.44 × 10⁻²⁵
Silver	1.79 × 10⁻²⁵
Lithium	1.15 × 10⁻²⁶
Hydrogen	1.674 × 10⁻²⁷

While studying her family’s history. Shikha discovers records of ancestors 12 generations back. She wonders how many ancestors she has had in the past 12 generations. She starts to make a diagram to help her figure this out. The diagram soon becomes very complex

(a) Make a table and a graph showing the number of ancestors in each of the 12 generations.
(b) Write an equation for the number of ancestors in a given generation n.

Answer:

(a)

Generation	Ancestors
1st	2
2nd	2²
3rd	2³
4th	2⁴
5th	2⁵
6th	2⁶
7th	2⁷
8th	2⁸
9th	2⁹
10th	2¹⁰
11th	2¹¹
12th	2¹²

Crop	2008 Harvest (Hectare)	Increase/Decrease (Hectare) in 2009
Bajra	1.4 × 10³	−100
Jowar	1.7 × 10⁶	−440,000
Rice	3.7 × 10³	−100
Wheat	5.1 × 10⁵	+190,000

The left column of the chart lists the lengths of input pieces of ribbon. Stretching machines are listed across the top. The other entries are the outputs for sending the input ribbon from that row through the machine from that column. Copy and complete the chart.

Input Length	Machine
	×2
	1	5
3				15
	14		7

Answer:

Input Length	Machine
	×2	×10	×1	×3
0.5	1	5	0.5	1.5
3	6	30	3	15
7	14	70	7	21

Page No 273:

Question 169:

The left column of the chart lists the lengths of input chains of gold. Repeater machines are listed across the top. The other entries are the outputs you get when you send the input chain from that row through the repeater machine from that column. Copy and complete the chart.
â€‹

Input Length	Repeater Machine
	×2³
	40		125
2
		162

Answer:

Input Length	Repeater Machine
	×2³	×3⁴	×5²
5	40	245	125
2	16	162	50
2	16	162	50

Page No 273:

Question 170:

Long back in ancient times, a farmer saved the life of a king’s daughter. The king decided to reward the farmer with whatever he wished. The farmer, who was a chess champion, made an unusal request:
“I would like you to place 1 rupee on the first square of my chessboard, 2 rupees on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should have twice as many rupees as the previous square.” The king thought this to be too less and asked the farmer to think of some better reward, but the farmer didn’t agree.
How much money has the farmer earned?

Answer:

Position of square on chess board	Amount (in â‚¹)
1^st square	1	2⁰
2^nd square	2	2¹
3rd square	4	2²
4^th square	8	2³
5^th square	16	2⁴
6th square		25
7th square		26
8th square		27
9th square		2⁸
10th square		2⁹
11th square		2¹⁰
12th square		2¹¹
13th square		2¹²
14th square		2¹³
15th square		2¹⁴
16th square		2¹⁵
17th square		2¹⁶
17th square		2¹⁷
18th square		2¹⁸
19th square		2¹⁹
20^th square		2²⁰
21^th square		2²¹
22^th square		2²²
23^th square		2²³
24^th square		2²⁴
25^th square		2²⁵
26^th square		2²⁶
27^th square		2²⁷
28^th square		2²⁸
29^th square		2²⁹
30^th square		2³⁰
31^th square		2³¹
32^th square		2³²
33^th square		2³³
34^th square		2³⁴
35^th square		2³⁵
36^th square		2³⁶
37^th square		2³⁷
38^th square		2³⁸
39^th square		2³⁹
40^th square		2⁴⁰
41^th square		2⁴¹
42^th square		2⁴²
43^th square		2⁴³
44^th square		2⁴⁴
45^th square		2⁴⁵
46^th square		2⁴⁶
47^th square		2⁴⁷
48^th square		2⁴⁸
49^th square		2⁴⁹
50^th square		2⁵⁰
51^th square		2⁵¹
52^th square		2⁵²
53^thsquare		2⁵³
54^th square		2⁵⁴
55^th square		2⁵⁵
56^th square		2⁵⁶
57^th square		2⁵⁷
58^th square		2⁵⁸
59^th square		2⁵⁹
60^th square		2⁶⁰
61^thsquare		2⁶¹
62^th square		2⁶²
63^th square		2⁶³
64^th square		2⁶⁴

$\begin{array}{l} (1) [{(\frac{1}{2})}^{2} - {(\frac{1}{4})}^{3} \times 2^{- 3}] \\ = [\frac{1}{4} - \frac{1}{64}] \times \frac{1}{2^{3}} (∵ a^{- m} = \frac{1}{a^{m}}) \\ = {[\frac{15}{64}]}^{- 1} \times \frac{1}{8} \\ = \frac{64}{15} \times \frac{1}{8} (∵ a^{- m} = \frac{1}{a^{m}}) \\ Hence, {[{(\frac{1}{2})}^{2} - {(\frac{1}{4})}^{3}]}^{- 1} \times 2^{- 3} = \frac{8}{15} . \end{array}$
$\begin{array}{l} (2) {[{(\frac{4}{3})}^{- 2} - {(\frac{3}{4})}^{2}]}^{(- 2)} \\ = {[{(\frac{3}{4})}^{2} - {(\frac{3}{4})}^{2}]}^{(- 2)} (∵ a^{- m} = \frac{1}{a^{m}}) \\ = {[0]}^{(- 2)} \\ Hence, {[{(\frac{4}{3})}^{- 2} - {(\frac{3}{4})}^{2}]}^{(- 2)} = 0 \end{array}$
$\begin{array}{l} (3) {(\frac{4}{13})}^{4} \times {(\frac{15}{7})}^{2} \times {(\frac{7}{4})}^{3} \\ = \frac{4^{4}}{13^{4}} \times \frac{13^{2}}{7^{2}} \times \frac{7^{3}}{4^{3}} \\ = \frac{4^{4}}{4^{3}} \times \frac{13^{2}}{13^{4}} \times \frac{7^{3}}{7^{2}} \\ = 4^{1} \times 13^{- 2} \times 7^{1} (∵ \frac{a^{m}}{a^{n}} = a^{m - n}) \\ = \frac{28}{169} \\ Hence, {(\frac{4}{13})}^{4} \times {(\frac{13}{7})}^{2} \times {(\frac{7}{4})}^{3} = \frac{28}{169} . \end{array}$
$\begin{array}{l} (4) {(\frac{1}{5})}^{45} \times {(\frac{1}{5})}^{- 60} - {(\frac{1}{5})}^{28} \times {(\frac{1}{5})}^{- 43} \\ = {(\frac{1}{5})}^{45 - 60} - {(\frac{1}{5})}^{28 - 43} (∵ a^{m} \times a^{n} = a^{m - n}) \\ = {(\frac{1}{5})}^{- 15} - {(\frac{1}{5})}^{- 15} \\ Hence, {(\frac{1}{5})}^{45} \times {(\frac{1}{5})}^{- 60} - {(\frac{1}{5})}^{28 - 1} \times {(\frac{1}{5})}^{- 43} = 0. \end{array}$
$\begin{array}{l} (5) \frac{{(9)}^{3} \times 27 + t^{4}}{{(3)}^{- 2} \times 3^{4} \times t^{2}} = \frac{{(3^{2})}^{3} \times 3^{4} \times t^{4}}{{(3)}^{- 4} \times 3^{4} \times t^{4}} [∵ a^{m n} = {(a^{m})}^{n}] \\ = \frac{3^{6} \times 3^{3} \times t^{4}}{3^{2} \times 3^{4} \times t^{2}} \\ = \frac{3^{9}}{3^{20}} \times \frac{t^{4}}{t^{4}} (∵ a^{m} \times a^{n} = a^{m + n}) \\ = 3^{7} \times t^{2} \\ Hence, \frac{{(9)}^{3} \times 27 \times t^{4}}{{(3)}^{- 2} \times 3^{4} \times t^{2}} = 3^{7} \times t^{2} . \end{array}$
$\begin{array}{l} (6) \frac{{(9)}^{3} \times {(5^{2})}^{- 3} \times {(t^{- 3})}^{2}}{{(3^{- 2})}^{^{5}} \times {(5^{3})}^{^{2}} \times {(t^{- 4})}^{3}} = \frac{3^{- 4} \times 5^{- 6} \times t^{- 6}}{3^{^{- 10}} \times 5^{- 6} \times t^{- 12}} [∵ {(a^{m})}^{n} = a^{m n}] \\ = 3^{- 4} \times 3^{10} \times 5^{- 6} \times 5^{6} \times t^{- 6} \times t^{12} \\ = 3^{- 4 + 10} \times 5^{- 6 + 6} \times t^{- 6 + 2} (∵ a^{m} \times a^{m} = a^{m + n}) \\ = 3^{6} \times 5^{0} \times t^{6} \\ = 3^{6} \times t^{6} \\ = {(3 t)}^{6} (∵ a^{m} \times b^{m} = a b^{m}) \\ Hence, \frac{{(3^{- 2})}^{2} \times {(5^{2})}^{- 3} \times {(t^{- 3})}^{2}}{{(3^{- 2})}^{5} \times {(5^{3})}^{- 2} \times {(t^{- 4})}^{3}} = (3 t) 6. \end{array}$

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