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The police officer told Aksionov the reason for asking so many questions from him that the merchant with whom he spent last night had been found with his throat cut in the bed.

पुलिस अधिकारी ने अक्सियोनॉव से इतने अधिक प्रश्न पूछने का कारण बताया कि जिस व्यापारी के साथ उसने पिछली रात बितायी थी, उसे बिस्तर में गला कटा हुआ पाया गया है।

The police officer asked his men to bind Aksionov and put him in the cart.

पुलिस अधिकारी ने अपने आदमियों को, अक्सियोनॉव को बाँधने तथा घोडागाड़ी में रखने के लिए कहा।

Aksionov asked his wife to petition the Czar not to let an innocent man perish.

अक्सियोनॉव ने अपनी पत्नी को जार से निवेदन करने के लिए कहा कि एक निर्दोष आदमी को बरबाद न होने दें।

Correct answer is (b) Makar

She dreamt him returned home from the town and when he took off his cap, she saw that his hair was quite grey. She was afraid of thinking that it was a bad dream for her husband.

उसने अपने पति को कस्बे से लौटते हुए देखा और जब उसने अपनी टोपी उतारी, उसने देखा कि उसके बाल पूरी तरह से सफेद हो गये। वह सोचती हुई डर गयी कि यह उसके पति के लिए एक बुरा सपना था।

1. True

2. True

3. False

4. True

5. False

6. True

When Aksionov heard all that Makar Semyonich had been telling to the other prisoners, he felt sure that this was the man who had killed the merchant. All that night he lay awake. All sorts of images rose in his mind. There was the image of his wife of that time. Then he saw his children, quite little as they were at that time. And then he remembered himself as he used to be young and merry. He saw, in his mind, the place where he was flogged, the executioner, the chain, the convicts etc. The thought of it all made him so wretched that he was ready to kill himself.

जब अक्सियोनॉव ने वह सब सुना जो मेकर सेम्योनिच दूसरे कैदियों को बता रहा था, उसे विश्वास हो गया कि यही वह आदमी था जिसने उस व्यापारी को मारा था। वह पूरी रात लेटा रहा। उसके दिमाग में सभी प्रकार की कल्पनाएँ उत्पन्न हुई। वहाँ उसकी पत्नी की कल्पना थी जैसे वह उस समय थी फिर उसने अपने बिल्कुल छोटे बच्चों को देखा जैसे वे उस समय थे। और फिर उसने स्वयं के बारे में याद किया जैसे वह हुआ करता था-युवा एवं प्रसन्नचित्त उसने अपने मस्तिष्क (मन) में उस स्थान को जहां पर उसे कोड़े मारे गये, उस जल्लाद को, हथकड़ियाँ, अपराधियों आदि को देखा। इन सभी विचारों ने उसे इतना दुःखी (घृणित) बना दिया कि वह अपने आपको मारने के लिए तैयार हो गया।

Aksionov was condemned to be flogged and sent to the mines. So he was flogged with a knot and when the wounds made by the knot were healed, he was driven to Siberia with other convicts.

अक्सियोनॉव को कोडे मारने की सजा दी गई तथा उसे खदानों में भेज दिया गया। इसलिए उसके रस्सी की गाँठ से कोड़े मारे गये और जब गाँठ से हुए घाव ठीक हो गये तो, उसे दूसरे अपराधियों के साथ साइबेरिया भेज दिया गया।

Aksionov started only praying to God because he had given up all hope. Even his wife had suspected him. He said to himself that only God could know the truth. He could expect mercy from Him.

अक्सियोनॉव ने केवल भगवान की प्रार्थना करना शुरू किया क्योंकि उसने पूरी उम्मीद छोड़ दी थी। यहाँ तक कि उसकी पत्नी ने भी उस पर संदेह किया था। उसने अपने आप से कहा कि केवल भगवान ही सत्यता को जान सकता था। वह उसी से ही दया की उम्मीद कर सकता था।

The police officer asked Aksionov so many questions because he was the only one who stayed with the merchant. The merchant was killed with a knife and it was found in his things. His voice and manner also betrayed him.

पुलिस अधिकारी ने अक्सियोनॉव से इतने अधिक प्रश्न पूछे क्योंकि यह वही आदमी था जो उस व्यापारी के साथ ठहरा था। उस व्यापारी को चाकू से मार दिया गया था जो उसकी चीजों में पाया गया। उसकी आवाज तथा व्यवहार ने भी उसे धोखा दिया।

The prison authorities liked Aksionov for his meekness. Even his fellow prisoners respected him. They called him “Grandfather’ and ‘The Saint’. When they wanted to petition the prison authorities about anything, they always made him their spokesman.

जेल अधिकारी अक्सियोनॉव को उसकी नम्रता के लिए पसंद करते थे। यहाँ तक कि उसके कैदी साथी भी उसका सम्मान करते थे। वे उसे ‘दादा’ और ‘सन्त’ कहकर पुकारते थे। जब वे किसी के भी बारे में जेल अधिकारियों को प्रार्थना करना चाहते थे, वे उसे हमेशा अपना प्रवक्ता बनाते थे।

While walking about the prison Aksionov noticed some earth that came rolling out from under one of the shelves on which the prisoners slept. He stopped to see what it was. Suddenly Makar Semyonich crept out from under the shelf and looked up at him with frightened face. Makar seized his hand and told him that he had dug a hole under the wall, getting rid of the earth by putting it into his high-boots.

जेल के आसपास टहलते हुए, अक्सियोनॉव ने कुछ मिट्टी देखी जो एक पट्टी के नीचे से जिस पर कैदी सोते थे, लाई गई थी। वह देखने के लिए रुका कि यह क्या था। अचानक मेकर सेम्योनिच रेंगकर पट्टी के नीचे से बाहर निकला और भयभीत चेहरे से उसकी ओर देखा। मेकर ने उसके हाथ को पकड़ लिया और उसे बताया कि उसने दीवार के नीचे एक गड्ढा, खोदा था और मिट्टी से छुटकारा पाने हेतु उसे अपने ऊँचे जूतों में भर लिया था

When Aksionov came to know that Semyonich was the killer of the merchant, he felt terribly unhappy and all sorts of images rose in his mind. There was the image of his wife as she was when he parted from her to go to the fair. He saw her as if she were present. Then he saw his children, quite little as they were at that time. And then he remembered himself as he used to be young and merry.

जब अक्सियोनॉव जान गया कि सेम्योनिच उस व्यापारी का हत्यारा था, उसने भयंकर रूप से दु:ख महसूस किया और सभी प्रकार की कल्पनाएँ उसके दिमाग में उत्पन्न हुईं। वहाँ उसकी पत्नी की वो छवि थी जैसी वह उस समय थी जब वह मेले में जाने के लिए उससे अलग हुआ था। उसने उसे ऐसे देखा जैसे मानो कि वह वहीं हो। फिर उसने अपने बच्चों को देखा, बिल्कुल छोटे जैसे वे उस समय थे और फिर उसने स्वयं के बारे में याद किया जैसा वह हुआ करता था – युवी और प्रसन्नचित्त।

Ivan Dmitrich Aksionov was a young merchant. He was handsome, fairheaded, curly-haired, full of fun and very fond of singing. He used to drink before his marriage. He was unjustifiably condemned with murdering of a merchant. He was a true man but the proofs were against him. He had given up all hope of life.

ईवान दमित्रच अक्सियोनॉव एक युवा व्यापारी था। वह सुंदर, साफ-मस्तिष्क, चुंघराले बालों वाला, मजाकिया तथा गानों का बड़ा शौकीन था। अपनी शादी से पहले वह शराब पीया करता था। उस पर अन्यायपूर्ण ढंग से एक व्यापारी के कत्ल का अभियोग लगाया था। वह एक सच्चा आदमी था लेकिन सबूत उसके खिलाफ थे। उसने जीवन की सभी उम्मीदें छोड़ दी थीं।।

When Aksionov came to know that Semyonich was the merchant’s killer, he . could not get peace at night because he longed for vengeance. Even he himself wanted to perish for it.

जब अक्सियोनॉव जान गया कि सेम्योनिच उस व्यापारी का हत्यारा था, वह रात में शांत नहीं रह पाता था क्योंकि उसमें बदला लेने की इच्छा थी। यहाँ तक कि वह इसके लिए स्वयं भी मरना चाहता था।

When Aksionov had travelled half way, he met a merchant whom he knew. They put up at the same inn for the night. They had some tea together and then went to bed in adjoining rooms. When Aksionov was in sound sleep, Makar came and killed the merchant. He hid the knife in Aksionov’s things. Makar stole the merchant’s twenty thousand roubles and escaped from the window.

जब अक्सियोनॉव आधा रास्ता पार कर चुका था, वह एक व्यापारी से मिला जिसको वह जानता था। उन्होंने एक ही धर्मशाला में रात बिताई उन्होंने साथ-साथ चाय पी तथा एक-दूसरे से लगे हुए कमरों के बिस्तरों पर चले गये। जब अक्सियोनॉव गहरी नींद में था, मेकर आया तथा उस व्यापारी को मार दिया। उसने चाकू को अक्सियोनॉव की चीजों में छिपा दिया। मेकर ने उसके 20,000 रूबल्स चुराये तथा खिड़की से भाग निकला।

Correct option is  D) none

(A) heterozygous

Heterozygous is not pure and is called hybrid. Heterozygote does not breed true on self fertilization. e.g. Tt.

We have,

f(x) = 2x³ - 9x² + x +12 and g(x) = 3 - 2x

In order to find g (x) = 3 – 2x = 2 \((x-\frac{3}{2})\) is a factor of f (x) or not, it is sufficient to prove that f  \((\frac{3}{2})=0\)

Now,

f(x) = 2x³ - 9x² + x +12

f \((\frac{3}{2})=2(\frac{3}{2})^{3}-9(\frac{3}{2})^{2}+\frac{3}{2}+12\)

\(=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12\)

= \(\frac{81-81}{4}\)

= 0

Hence, 

g (x) is a factor of f (x).

(i) f (x) = x² 

Put x 

= 0

Therefore, 

x = 0 is not a root of f (x) = x²

(ii) f (x) = lx + m

Put x = \(\frac{-m}{i}\)

f\((\frac{-m}{i})=\)  l x \((\frac{-m}{i})\) + m

= - m + m = 0

= 0 

Therefore, 

x = \(-\frac{-m}{i}\) is a root of f (x) = lx + m

(iii) f (x) = 2x + 1

Put x = \(\frac{1}{2}\)

f\((\frac{1}{2})\) = 2 x \(\frac{1}{2}+1\)

= 1 + 1

= 2 \(\neq0\)

Therefore, 

x = \(\frac{1}{2}\) is not a root of f (x) = 2x + 1

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0. 

g(x) = x - 3 = 0 

or x = 3

Remainder = f(3)

Now, 

f(3) = (3)³ – 6(3)² + 11 x 3 – 6 

= 27 – 54 + 33 – 6 

= 60 – 60 

= 0 

Therefore, g(x) is a factor of f(x).

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0. 

g(x) = x + 3 = 0, then x = -3

Remainder = f(-3)

Now, 

f(-3) = (-3)⁵ + 3(-3)⁴ – (-3)³ – 3(-3)² + 5(-3) + 15 

= -243 + 3 x 81 -(-27)-3 x 9 + 5(-3) + 15

= -243 + 243 + 27 - 27 - 15 + 15 

= 0

Therefore, g(x) is a factor of f(x).

Example of a binomial with degree 35 is 7x³⁵ -5 

Example of a monomial with degree 100 is 2t¹⁰⁰.

Example of a binomial with degree 35 is 7x³⁵ – 5.

Example of a monomial with degree 100 is 2t¹⁰⁰.

(i) f(x) = 3x + 1 

Put x = -1/3 

f (-1/3) = 3 x (-1/3) + 1 

= -1 + 1 

= 0 

Therefore, 

x = -1/3 is a root of f (x) = 3x + 1 

(ii) We have, 

f (x) = x² – 1 

Put x = 1 and x = -1 

f (1) = (1)² – 1 and f (-1) = (-1)² – 1 

= 1 – 1 = 1- 1 

= 0 = 0

Therefore, 

x = -1 and x = 1 are the roots of f(x) = x² – 1

(iii) g (x) = 3x² – 2

Put x = \(\frac{2}{\sqrt{3}}\) and x = \(\frac{-2}{\sqrt{3}}\)

g \((\frac{2}{\sqrt{3}}) \) = \(3(\frac{2}{\sqrt{3}})^{2}-2\) and g \((\frac{-2}{\sqrt{3}})\) = 3 \((\frac{-2}{\sqrt{3}})^{2}-2\)

= 3 x \(\frac{4}{3}\) - 2 = 3 x \(\frac{4}{3}\) - 2 

= 2\(\neq\) 0 = 2\(\neq\) 0

Therefore, 

x = \(\frac{2}{\sqrt{3}}\) and x \(\frac{-2}{\sqrt{3}}\) are not the roots of g (x) = 3x² – 2

(iv) p (x) = x³ – 6x² + 11x – 6 

Put x = 1 

p (1) = (1)³ – 6 (1)² + 11 (1) – 6 

= 1 – 6 + 11 – 6 

= 0 

Put x = 2 p (2) = (2)³ – 6 (2)² + 11 (2) – 6 

= 8 – 24 + 22 – 6 

= 0 

Put x = 3 

p (3) = (3)³ – 6 (3)² + 11 (3) – 6 

= 27 – 54 + 33 – 6 

= 0 

Therefore, 

x = 1, 2, 3 are roots of p (x) = x³ – 6x² + 11x – 6

(v) f (x) = 5x - π

Put x = \(\frac{4}{5}\)

f \((\frac{4}{5})\) = = 5 x \(\frac{4}{5}-\) π

= 4 - π \(\neq0\)

Therefore, 

x = \(\frac{4}{5}\) is not a root of f (x) = 5x - π

(i) f(x) = 3x + 1, x = \(\frac{-1}{3}\) 

f(x) = 3x + 1 

Substitute x =  \(\frac{-1}{3}\)  in f(x) 

\(f (\frac{-1}{3})\)= \(3(\frac{-1}{3}) + 1\)

= -1 + 1

= 0 

Since, the result is 0, so x =  \(\frac{-1}{3}\)  is the root of 3x + 1 

(ii) f(x) = x² – 1, x = 1,−1 

f(x) = x² – 1 

Given that x = (1 , -1) 

Substitute x = 1 in f(x) 

f(1) = 1² – 1 

= 1 – 1 

= 0 

Now, substitute x = (-1) in f(x) 

f(-1) = (−1)² – 1 

= 1 – 1 

= 0 

Since , the results when x = 1 and x = -1 are 0, so (1 , -1) are the roots of the polynomial f(x) = x² – 1 

(iii) g(x) = 3x² – 2 , \(x = \frac{2}{\sqrt3}\) ,\(\frac{-2}{\sqrt3}\) 

g(x) = 3x² – 2 

Substitute \(x = \frac{2}{\sqrt3}\) in g(x) 

g\((\frac{2}{\sqrt3})\) = \(3(\frac{2}{\sqrt3})^2 - 2\)

= \(3 (\frac{4}{3}) - 2\)

= 4 – 2 

= 2 ≠ 0 

Now, Substitute x =  \(\frac{-2}{\sqrt3}\) in g(x) 

\(g(\frac{2}{\sqrt3}) = 3 (\frac{-2}{\sqrt3})^2 - 2\)

= \(3 (\frac{4}{3}) - 2\)

= 4 – 2 

= 2 ≠ 0 

Since, the results when x = \(\frac{2}{\sqrt3}\) and x = \(\frac{-2}{\sqrt3}\) are not 0. Therefore (\(\frac{2}{\sqrt3}\) ,\(\frac{-2}{\sqrt3}\) ) are not zeros of 3x² – 2. 

(iv) p(x) = x³ – 6x² + 11x – 6 , x = 1, 2, 3 

p(1) = 1³ – 6(1)² + 11 x 1 – 6 

= 1 – 6 + 11 – 6 

= 0 

p(2) = 2³ – 6(2)² + 11 x 2 – 6 

= 8 – 24 – 22 – 6

= 0 

p(3) = 3³ – 6(3)² + 11 x 3 – 6 

= 27 – 54 + 33 – 6 

= 0 

Therefore, x = 1, 2, 3 are zeros of p(x). 

(v) f(x) = 5x – π, x = \(\frac{4}{5}\) 

f( \(\frac{4}{5}\) ) = 5 x  \(\frac{4}{5}\)  – π 

= 4 – π ≠ 0 

Therefore, x =  \(\frac{4}{5}\)  is not a zeros of f(x). 

(vi) f(x) = x² , x = 0 

f(0) = 0² = 0 

Therefore, x = 0 is a zero of f(x). 

(vii) f(x) = lx + m, x = \(\frac{-m}{l}\) 

f( \(\frac{-m}{l}\) ) = l x  \(\frac{-m}{l}\)  + m 

= -m + m 

= 0 

Therefore, x =  \(\frac{-m}{l}\)  is a zero of f(x). 

(viii) f(x) = 2x + 1, x = \(\frac{1}{2}\) 

f(\(\frac{1}{2}\)) = 2 x \(\frac{1}{2}\) + 1 

= 1 + 1 

= 2 ≠ 0 

Therefore, x = \(\frac{1}{2}\) is not a zero of f(x).

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0. 

g(x) = x + 5 = 0, then x = -5

Remainder = f(-5)

Now, f(3) = 3(-5)⁴ + 17(-5)³ + 9(-5)² – 7(-5) – 10

= 3 x 625 + 17 x (-125) + 9 x (25) – 7 x (-5) – 10

= 1875 - 2125 + 225 + 35 – 10 = 0

Therefore, g(x) is a factor of f(x).

We have, 

f(x) = 2x³ - 13x² + 17x +12 

(i) f(2) = 2 (2)³ – 13 (2)² + 17 (2) + 12 

= (2 x 8) – (13 x 4) + (17 x 2) + 12 

= 16 – 52 + 34 + 12 

= 10

(ii) f (-3) = 2 (-3)³ – 13 (-3)² + 17 (-3) + 12 

= (2 x -27) – (13 x 9) + (17 x -3) + 12 

= -54 – 117 – 51 + 12 

= - 210

(iii) f (0) = 2 (0)³ – 13 (0)² + 17 (0) + 12 

= 0 – 0 + 0 + 12 

= 12

f(x) = 2x³ – 13x² + 17x + 12 

(i) f(2) = 2(2)³ – 13(2)² + 17(2) + 12 

= 2 x 8 – 13 x 4 + 17 x 2 + 12

= 16 – 52 + 34 + 12 

= 62 – 52 

= 10 

(ii) f(-3) = 2(-3)³ – 13(-3)² + 17 x (-3) + 12 

= 2 x (-27) – 13 x 9 + 17 x (-3) + 12 

= -54 – 117 -51 + 12 

= -222 + 12 

= -210 

(iii) f(0) = 2 x (0)³ – 13(0)² + 17 x 0 + 12 

= 0 - 0 + 0 + 12

= 12

(i) 222

The given number = 222

The number formed by ten’s and unit’s digit is 22, which is not divisible by 4.

222 is not divisible by 4

(ii) 532

The given number = 532

The number formed by ten’s and unit’s digit is 32, which is divisible by 4.

532 is divisible by 4

(iii) 678

The given number = 678

The number formed by ten’s and unit’s digit is 78, which is not divisible by 4

678 is not divisible by 4

(iv) 9232

The given number = 9232

The number formed by ten’s and unit’s digit is 32, which is divisible by 4.

9232 is divisible by 4.

(i) 352

The given number = 352

Digit at unit’s place = 2

It is divisible by 2

(ii) 523

The given number = 523

Digit at unit’s place = 3

It is not divisible by 2

(iii) 496

The given number = 496

Digit at unit’s place = 6

It is divisible by 2

(iv) 649

The given number = 649

Digit at unit’s place = 9

It is not divisible by 2

(i) 11011

 = 11000+ 11

= 11 x (1000+ 1)

= 11 x 1001

Clearly, 11011 is divisible by 11.

(ii) 110011

= 110000+ 11

= 11 x (10000+ 1)

= 11 x 10001

Clearly, 110011 is divisible by 11.

(iii) 11000011

= 11000000+ 11

= 11 x (1000000+ 1)

= 11 x 1000001

Clearly, 110000 is divisible by 11.

(i) 1608

= 1600 + 8

= 8 (200 + 1)

= 8 x 201

Clearly, 1608 is divisible by 8.

(ii) 56008

= 56000 + 8

= 8 x (7000 + 1)

= 8 x 7001

Clearly, 56008 is divisible by 8.

(iii) 240008

= 240000 + 8

= 8 x (30000 + 1)

= 8 x 30001

Clearly, 240008 is divisible by 8.

2300023 = 2300000 + 23

= 23 x (100000 + 1)

= 23 x 100001

Clearly, 2300023 is divisible by 23.

7007

= 7000 + 7

= 7 x (1000+ 1)

= 7 x 1001

Clearly, 7007 is divisible by 7.

Since, 54 = 1 x 54, 2 x 27, 3 x 18, 6 x 9

Clearly, numbers are 6 and 9.

Since, 36 = 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6

Clearly, numbers are 4 and 9

Since, 48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8

Clearly, numbers are 4 and 12.

Clearly, for given condition which is, two numbers whose product is a 1-digit number and the sum is a 2-digit number, 1 and 9 satisfy.

Here, 

1×9=9 and 1+9=10 

Hence, 1 and 9 are required numbers

Here,

(A−4) = 3

∴ A = 7

Now, C ×6=36

∴ C = 6

Clearly, B = 6∴ A=7, B=C=6

48 + 96 – 24 – 6 x 18

= 48 + 4 – 6 x 18

= 48 + 4 – 108

= 52 – 108 

= -56

(i) 4

Multiples of 4 =1 x 4, 2 x 4, 3 x 4, 4 x 4, 4 x 5, 4 x 6

First six multiples of 4 are : 4, 8, 12, 16, 20, 24

(ii) 9

Multiples of 9 = 1 x 9, 2 x 9, 3 x 9, 4 x 9, 5 x 9, 6 x 9

First six multiples of 9 are : 9, 18, 27, 36, 45, 54

(iii) 11

Multiples of 11 = 1 x 11, 2 x 11, 3 x 11, 4 x11, 5 x 11, 6 x 11

First six multiples of 11 are : 11, 22, 33, 44, 55, 66

(iv) 15

Multiples of 15 = 1 x 15, 2 x 15, 3 x 15, 4 x 15, 5 x 15, 6 x 15

First six multiples of 15 are : 15, 30, 45, 60, 75, 90

(v) 18

Multiples of 18 = 1 x 18, 2 x 18,3 x 18, 4 x 18, 5 x 18, 6 x 18

First six multiples of 18 are : 18, 32, 54, 72, 90, 108

(vi) 16

Multiples of 16 = 1 x 16, 2 x 16, 3 x 16,4 x 16, 5 x 16, 6 x 16

First six multiples of 16 are : 16, 32, 48, 64, 80, 96

(i) 16

All factors of 16 are : 1, 2, 4, 8, 16

(ii) 21

All factors of 21 are : 1, 3, 7, 21.

(iii) 39

All factors of 39 are : 1, 3, 13, 39

(iv) 48

All factors of 48 are : 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

(v) 64

All factors of 64 are : 1, 2, 4, 8, 16, 32, 64

(vi) 98

All factors of 98 are : 1, 2, 7, 14, 49, 98

Given: 

Length = 24 m 

Breath = 18 m 

Thus, we have: Area of the rectangular hall = 24 x 18 = 432 sq m

Length of each carpet = 2.5 m 

Breath of each carpet = 80 cm = 0.80 m

Area of one carpet = 2.5 x 0.8 = 2 sq m

Number of carpets required = Area of the hall/Area of the carpet = 432/2 = 216

Therefore, 216 carpets will be required to cover the floor of the hall.