We know that 

\(tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\)

\(u=\frac{\pi}{2}-2tan^{-1}(\sqrt{tanx})\)

\(2tan^{-1}x(\sqrt{tanx})=\frac{\pi}{2}-u\)

\(tan^{-1}x(\sqrt{tanx})=\frac{\pi}{4}-\frac{u}{2}\)

\(\sqrt{tanx}=tan(\frac{\pi}{4}-\frac{u}{2})\)

Quantity I: 8 months

Quantity II:

Let the total profit be Rs. x

⇒ Akbar’s share = x/6

⇒ Shabnam’s share = (x) – (x/6) = 5x /6

Profit ratio of Akbar and Shabnam = 1 : 5

Let total capital be Rs. a and period of investment of Akbar be ‘t’ months

⇒ [(a/6) × 12] : [(5a/6) × t] = 5 : 1

⇒ [12a/6] : [5at/6] = 5 : 1

⇒ [(12a/6) × (6/5at)] = 5 : 1

⇒ t = 12/25 months

∴ Quantity I > Quantity II

Let 10 units = 1 lot

For face value 100 – income 9

For Rs. 180 - ? \(\frac{9 \times\ 180}{100} = 16.2\)

Selling price of 10 shares at 2020 per share = 20,200 

0.5 Brokerage = \(\frac{0.5}{100}\) × 20,200 = Rs. 101 

100 Amount required to purchase = 20,200 + 101 = 20,301

Income = \(\frac{5}{100}\) x 3600 = 180

Under COPRA, a three-tier quasi-judicial machinery at the district, state and national levels was setup for redressal of consumer disputes : 

(a) The district level court deals with the cases involving claims up to Rs.20 lakhs. 

(b) The state level court deals with the cases involving claims between Rs.20 lakhs and Rs.1 crore.

(c) The national level court deals with the cases involving claims exceeding Rs.1 crore.

Let amount of the bill = Rs.100 

Money deducted =Rs.10 

Money received by the holder of the bill = Rs.100-10 = Rs.90 

SI on Rs.90 for 10 months = Rs.10 

Rate =[(100*10)/(90*10/12)% = 13 1/3%

T.D. = √(P.W.*B.G) 

B.G. =(T.D.)2/ P.W. 

= Rs.[(110 x 110)/ 1100] 

= Rs. 11. 

B.D.= (T.D. + B.G.) = Rs. (110 + 11) = Rs. 121.

For 100 Rs. → Income 4 Rs. 

∴ for 8000 Rs. → \(\frac{8000 \times 4}{100} = 320 Rs.\)

Sum =[( B.D.*T.D.)/(B.D.-T.D.)] 

= Rs.[(120x110)/(120-110)] 

= Rs. 1320. 

Since B.D. is S.I. on sum due, so S.I. on Rs. 1320 for 8 months is Rs. 120. 

Rate =[(100 x 120)/( 1320 x 2/3)% = 13 7/11%

Given F = 415, n = \(\frac{9}{12}\), r = 0.15, B.D = ? 

B.D = F +r = 415 × \(\frac{9}{12}\) × 0.15 

B.D = Rs. 46.68

Let amount be Rs. x. Then, 

x*R*T/100 + (R x T) 

=T.D. 

=>x * 12*3/ 4/[100+[12*3/4]] =540 

x= 540 x 109 = Rs.6540 

Amount - Rs. 6540. P.W. = Rs. (6540 - 540) - Rs. 6000.

Let the sum be Rs. x. Then, 

T.D. = (x*25/2*1/2)/(100+(25/2*1/2)) = x*25/4*4/425 = x/17 

S.I = x*25/2*1/2*1/100 = x/16 

x/16-x/17=25 

=>17x-16x=25*16*17 

=>x=6800 

Hence, sum due = Rs. 6800.

Let the sum be Rs. x. Then, 

[x/2+(x/2*100)/100+(25/2*2)]-[(x*100)/(100+25/2*1] = 40

=>x/2+2x/5-8x/9=40 

=>x=3600 

Amount of the bill - Rs. 3600.

B.G. = S.I. on T.D. 

= Rs.(120 x 15 x 1/2 x 1/100) 

= Rs. 9. 

(B.D.) - (T.D.) = Rs. 9. 

B.D. = Rs. (120 + 9) = Rs. 129.

Ajatsatru  killed his father . Ajatasatru was a powerful and authoritative emperor of the Haryanka dynasty of the Magadha kingdom.

Let the common ratio between Rahul's age Hari's age be x

∴ Age of Rahul and Hari will be 5x years and 7x years

After 4 years the age of Rahul and Hari will be (5x+4) years and (7x+4) years

After 4 years , the sum of ages of Rahul and Hari is 56 years

∴5x+4+7x+4=56

12x+8=56

12x=56−8

12x=48

x=4

Rahul's age = 5x years

=5×4 = 20 years

Hari's age = 7x years

=7×4

=28 years

Answer : 20,28

Insulin therapy causes mostly weight gain. Long acting insulin analogues are able to induce small weight loss in Type 1 diabetes or only a small weight increase or weight loss in Type 2 diabetic patients. Procedures of bariatric surgery are very important in the treatment being able to induce remission of Type 2 diabetes. Weight reduction can be supported also using the new class of antiadiabetic drugs- SGLT inhibitors which are blocking glucose absorption in kidneys.

Concept:

The geometric mean of the numbers a₁, a₂,...,a_n is given by  ⇔ G = \(\rm (a_1\times a_2\cdots \times a_n)^{\frac 1 n}\)

Calculation:

GIven data: 2, 4, 8,16, 32?

Geometric mean = \((2 \times 4 \times8 \times 16 \times 32)^{\frac 1 5}\)

\(=(2^1 \times 2^2\times 2^3\times 2^4\times 2^5)^{\frac1 5}\)

\(=(2^{15})^{\frac1 5}\)

= 2³

= 8

Hence, option (3) is correct.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-one number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.

1. went 

2. will have won 

3. looking 

4. coming

5. will have been living 

6. reading 

7. had slept 

8. flew 

9. were reading 

10. had been exercising

It is true that Evans put up a façade to keep the governor complacent. Getting caught by the police at the hotel Golden Lion was a part of Evans’s escape plan. The governor did not have a slightest idea about Evans’s clever plan. The governor must have felt relieved after having caught Evans at the hotel; but he failed to see that it was Evans’s plan to have him caught. The real shock was to come after his capture. Evans had already made all arrangements to escape again after being caught.

He said that his father was ill and they had to go to see the doctor yesterday.

If you do not go for a walk regularly, you cannot reduce your weight.

Answer:- If you not go for a walk regularly, you cannot reduce your weight.

(a) Sindhu would not have won the world championship, if she had no single minded devotion.

(b) You will not reach your goal, if you don’t chase your dream.

(c) If we don’t plant more trees, we cannot save our planet.

(d) The rescue team would not have saved the victims if they had not received the call in time.

(e) The palace cannot be kept clean, if we don’t appoint more people.

(f) The portraits would not have been so natural if the artist had not given his best.

(g) The manager would not have selected Nithiksha if she had not exhibited good accounting skill.

(h) The policeman would not have arrested the man if he had not violated the rules.

(i) Mr Kunaal would not sponsor my higher education if I do not study well.

(j) Kavin will not stop flying kites if he does not understand the risk involved in it.

(k) Tanya would not know the answer if she had not referred to the answer key.

(l) My village cannot achieve 100 % literacy rate, if the elders of the village do not cooperate with the education department.

Tanya would not know the answer if she had not referred to the answer key.

Correct option is B) met

Correct option is C) safe

1  E) fascinated Bobby greatly

2  D) is usually a very active child.

3  A) she understood why Bobby had kept so still

Correct option is C) seriously

1  A) is not properly taken care of

2  E) we knew this wasn’t the right place to stay

3  A) wasn’t clean enough

Correct option is C) make out

New Old stock (sometimes abbreviated as NOS) is a term used in business to refer to merchandise being offered for sale that was manufactured long ago but that has never been used. Such merchandise may not be produced anymore, and the new old stock may represent the only market source of a particular item at the present time

1  C) a description of a street one Sunday

2  A) is taking his sister to the zoo

3  D) there were a lot of people waiting for the bus.

Correct answer is d) Inklewriter

A balance sheet is a statement of Assets and Liabilities of a firm which is prepared as on a particular date to show the financial position of the firm. A balance sheet shows the summary of assets and liabilities on a certain date, generally the last date of an accounting year

1  A) photography has become a popular form of art

2  D) photography today and photography fifty years ago.

3  B) people didn’t think of photography as an art.

 'A certain degree of physical harmony and comfort is necessary' is from A certain degree of physical harmony and comfort is necessary, but above a certain level it becomes a hindrance instead of a help. Therefore the ideal of creating an unlimited number of wants and satisfying them seems to be a delusion and a snare.

Correct answer is c) 1 -> iii) ; 2 -> i) ; 3 -> ii) 

Historical Sites: The sites related to events, personalities and their belongings, incidences, battles, inventions laboratories and Factories of historical importance.

Correct option is A) with

Sustainable Development Goal

1  B) enjoyed the film she saw on Friday evening

2  C) Liz and **** were talking about different films.

3  A) a television film they had seen the previous week