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A(adjA)= A(│A│ x A^-1) 

Since determinant of singular matrix is always 0 

A(adjA) = 0 

So, it is a null matrix.

A = \(\begin{pmatrix}-2&3 \\1 &1\end{pmatrix}\)

│A│= -2-3 = -5 

We know that │A^-1│= 1/|A|

=1/- 5

DNA structure was discovered by Watson and Crick. 

(d) Change in the number or arrangement of genes in the chromosome.

1. The chromosomal theory of inheritance was proposed by Sutton and Boveri in 1902.

2. Gene is the hereditary unit.

3. In mammals, the female is homozygous while the male is Heterozygous.

4. A chromosome is composed of 50% DNA and 50% histone.

5. Gregor John Mendel is the Father of Genetics.

(b) Offspring is like the parents

All points of circle of radius r are at same potential, hence work done is zero.

According to Gauss theorem, the electric flux through a closed surface depends only on the net charge enclosed by the surface and not upon the shape or size of the surface.

For any closed arbitrary shape of the surface enclosing a charge the outward flux is the same as that due to a spherical Gaussian surface enclosing the same charge.

Justification: This is due to the fact that

(i) Electric field is radial and

(ii) The electric field E ∝ \(\cfrac1{R^2}\) 

Thus, electric field at each point inside a charged thin spherical shell is zero.

Bond which are formed by sharing of a pair of electrons between two atoms is called covalent bonds.

It is a compound of hydrogen and carbon.

Hydrocarbon – Saturated and unsaturated.

Carbon occurs in free form e.g., graphite and diamond in combined form like carbon dioxide, carbonates, etc. 

In earth’s crust–0.02% and in atmosphere–0.03%.

Carbon is present in all substances of animal and vegetable origin.

Carbon always forms covalent bonds.

N has highest ionization energy because of stable exactly half filled 2p orbitals.

(iii) Assertion is correct statement but reason is wrong statement. 

Correct option is B) Magnetite

The magnetic property possessed by a magnetic substance due to the presence of a magnet near it is called ‘magnetic, induction’.

Correct option is  C) Cu

The correct answer is

(D) (IV)

(B) (i), (iv)

Due to steric hindrance and +l effect caused by two alkyl groups in propanone.

Detailed Answer:

As alkyl group is an electron releasing group (+ I inductive effect), electron releasing power of two alkyl groups in propanone is higher than that of one alkyl group in ethanal. The reduced positive charge on carbon in propanone discourages the attack of nucleophiles making propanone less reactive than ethanal. Also, presence of two alkyl groups increases the steric hindrance to the attack of nuclophile reducing the reactivity of carbonyl. Thus, propanone is less reactive than ethanal towards nucleophilic addition reactions.

Because the positive charge on carbonyl carbon of CH₃CHO decreases to a lesser extent due to one electron releasing (+I effect) CH₃ group as compared to CH₃COCH₃ (two electron releasing CH₃ group) and hence more reactive.

Because carboxylate ion (conjugate base) is more resonance stabilized than phenoxide ion.

Propanal gives silver mirror on reaction with Tollen's reagent while propanone does not give.

Benzoic acid evolves CO₂ gas with NaHCO₃ while benzaldehyde does not evolve.

(a) The delocalisation of benzene electrons contributes little towards the stability of phenoxide ion. The carboxylate ion is much more resonance stabilized than phenoxide ion. So, it is easier to lose a proton than phenol. Hence, carboxylic acid is a stronger acid than phenol. 

(b) Semicarbazide has two -NH₂ groups. One of them, which is directly attached to C=O is involved in resonance. Thus electron density on this group decreases and it does not act as a nucleophile. In contrast, the lone pair of electrons on the other-NH₂ group is available for uncleophilic attack.

Due to +I effect, CH₃ group in acetic acid increases density on carbonyl carbon atom which makes it a weak acid.

Greater resonance stabilization of acetate ion over phenoxide ion. 

The reaction will take place in solution- III and IV. Zn will react with ferrous sulphate and copper sulphate.

Zn + CuSO₄ → ZnSO₄ + Cu : Zn > Cu

Zn + FeSO₄ → ZnSO₄ + Fe : Zn > Fe

Buying a Gift
There are many occasions when we have to buy and give gifts. If we cannot choose the right gift, they will become useless. It spoils the enjoyment of presenting gifts. I got into such a dilemma when I have to choose the gift on the occasion of the birthday of my cousin. At that time he was only 17 years old. I went to the market and searched for all the shops but I failed to decide. In the meantime, I remembered one thing that could be very important to him. It was a saving set. It is the thing which a teenager needs for shaving. When I presented the set he became very happy and thanked me.

The Magi were wonderfully wise men. They brought precious gifts to the infant Jesus. The gifts were the tokens of their love and respect for Jesus. They invented the art of giving Christmas presents. They were wise, so their gifts were also wise ones. All who give and receive the wisest gifts are called Magi. The author called Jim and Della the Magi because they sacrificed for each other the greatest treasures of their house. They showed utter devotion and respect for each other. They were very poor and did not have enough money to buy gifts for each other but they loved each other so much that they did not hesitate to sell even their most valuable possessions. They were the Magi as their love for each other was limitless.

मेजाई आश्चर्यजनक रूप से बुद्धिमान व्यक्ति थे। वे शिशु ईशु के लिए कीमती उपहार लाये थे। उपहार प्रभु ईशु के प्रति उनके स्नेह तथा सम्मान के प्रतीक थे। उन्होंने क्रिसमस पर उपहार देने की कला की खोज की थी। वे बुद्धिमान थे इसलिए, उनके उपहार भी बुद्धिमत्तापूर्ण थे। वे सभी जो बुद्धिमत्तापूर्ण उपहार देते हैं या प्राप्त करते हैं मेजाई कहलाते हैं। लेखक जिम तथा डेला को मेजाई कहकर पुकारता है क्योंकि उन्होंने एक-दूसरे के लिए अपने घर की सबसे बड़ी सम्पत्ति को बलिदान कर दिया। उन्होंने एक-दूसरे के प्रति अत्यधिक समर्पण तथा सम्मान प्रदर्शित किया। वे बहुत ही गरीब थे तथा उनके पास एक-दूसरे के लिए उपहार खरीदने के लिये पर्याप्त पैसे नहीं थे लेकिन वे एक-दूसरे से इतना अधिक प्रेम करते थे कि वे अपनी सबसे मूल्यवान सम्पत्ति को बेचने से भी नहीं हिचकिचाए। वे मेजाई थे क्योंकि उनका प्रेम एक-दूसरे के प्रति असीमित था।

(i) pithy 

(ii) deceptive

(1) Della leaped up, didn’t she?

(2) You’ll have to look at the time, won’t you?

(3) They were expensive combs, weren’t they?

(4) I just had to do it, didn’t I?

(5) I am still pretty, aren’t I?

(6) Give it to me quick, won’t you?

(7) There was clearly nothing to do, was there?

(8) My hair grows so fast, doesn’t it?

(9) Let’s put our Christmas presents away, shall we?

(10) of saying a little silent prayer about the simplest everyday things.

1. expand 

2. simple 

3. adversity 

4. modest 

5. latter 

6. depreciated 

7. foolishness

Della tried to fix her chopped hair with curls before Jim got home.

(a) Della saved one dollar and eighty-seven cents for Christmas. It shows that she was very poor.

(b) because Jim thought that she was pretty because of her hair.

(c) Della sold her lovely long hair to get money to buy a Christmas present for Jim.

(d) She sold it to a woman who dealt in hair goods of all kinds. Yes, she did it willingly even though it was her prized possession.

(e) it was Christmas Eve, a time for giving gifts.

(f) it was very beautiful and her prized possession. 

(g) Jim brought out a package from his coat.

(h) The watch chain was very beautiful. It was made of platinum, and simple and chaste in design. No, I would not have liked to own it, because I do not carry a watch that needs a chain. It is old fashioned.

(i) The Magi wanted to celebrate their Christmas by exchanging gifts. They were the first people to give Christmas gifts.

(j) The moral of the story is that unselfish love is the greatest of all gifts, and those who realize this fact are the wisest.

When Della had bought the chain, she thought that it was worthy of the gold watch of Jim. It was a platinum fob chain. It was simple and chaste in design. It was properly proclaiming its value by its substance. It was not meretriciously ornamented. It was as all good things should do. When she saw it the first time, she knew that it must be Jim’s. It was like him – quietness and value. With that chain on his watch, Jim might be properly anxious about the time in any company. It was as grand as the watch. Jim had not to chain on his watch so he used the old leather strap. It was against the beauty of the watch so he looked at the watch on the sly. With the chain, he would not do so.

जब डेली ने उस चेन को खरीद लिया तो उसने सोचा कि यह जिम की सोने की घड़ी के लायक थी। यह जेब घड़ी की प्लेटिनम की चेने थी। इसकी डिजाइन सामान्य तथा पवित्र थी। इसके पदार्थ से इसके गुण का पता चल रहा था। इसे दिखावटी तरीके से नहीं सजाया गया था। यह वैसी ही थी जैसा कि सभी अच्छी चीजों को होना चाहिए। जब उसने इसे पहली बार देखा था तो वह जान गई थी कि इसे जिम की ही होना चाहिए। यह उसके जैसी ही थी- शांत तथा गुणवान। उस चेन को अपने हाथ में लेकर किसी भी संगति में वह समय देख सकेगा। यह घड़ी के जितनी ही शानदार थी। जिम की घड़ी में चेन नहीं थी इसलिए वह चमड़े का पुराना फीता उपयोग करता था। यह घड़ी की सुंदरता के खिलाफ था इसलिए वह गुपचुप तरीके से इसे देखता था। चेनं होने की वजह से वह ऐसा नहीं करेगा।

Della had no use for the combs as she had already sold her beautiful hair to buy a gift for her husband.

डेला के लिए कंघियाँ उपयोगी नहीं थीं चूँकि वह अपने सुन्दर बालों को अपने पति के लिए उपहार खरीदने के लिए पहले ही बेच चुकी थी।

Della feels that the platinum chain was like Jim. Quietness and value – the description applied to both.

The writer says that had King Solomon been the janitor, with all his treasures piled up in the basement, Jim would have pulled out his watch every time he passed, just to see him pluck at his beard with envy.

(1) True 

(2) True 

(3) False 

(4) True

Jim’s gift for Della was a set of combs, side, and back, that Della had worshipped long in a Broadway window. They were beautiful, expensive combs, made of pure tortoise shell, with jeweled rims – just the shade to wear in the beautiful brown hair of Della.

Della bought a platinum fob chain to be used for Jim’s watch in place of the old leather strap that he used.

The platinum fob chain was simple and chaste in design, properly proclaiming its value by substance alone and not by meretricious ornamentation.

(D) 4

A = Φ

∴ n(A) = 0

∴ n[P(A)] = 2^n(A) = 2⁰ = 1

∴ n[P[P(A)]] = 2^n[P(A)] = 2¹ = 2

∴ n[P[P[P(A)]]] = 2^n[P[P(A)]] = 2² = 4

R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}

(i) Here, (x, x) ∈ R, for x ∈ {1, 2, 3}

∴ R is reflexive.

(ii) Here, (1, 2) ∈ R, but (2, 1) ∉ R.

∴ R is not symmetric.

(iii) Here, (1, 2), (2, 3) ∈ R, But (1, 3) ∉ R.

∴ R is not transitive.