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1. Abacus: 

In 3000 BC Mesopotamians introduced this and it means calculating board or frame. It is considered as the first computer for basic arithmetical calculations and consists of beads on movable rods divided into two parts. The Chinese improved the Abacus with seven beads on each wire. Different Abacus are given below.

2. Napier’s bones: 

A Mathematician John Napier introduced this in AP 1617. 

3. Pascaline: 

A French mathematician Blaise Pascal developed this machine that can perform arithmetical operations. 

4. Leibniz’s calculator: 

In 1673, a German mathematician and Philosopher Gottfried Wilhelm Von Leibniz introduced this calculating machine. 

5. Jacquard’s loom: 

In 1801, Joseph Marie Jacquard invented a mechanical loom that simplifies the process of manufacturing textiles with complex pattern. A stored program in punched cards was used to control the machine with the help of human labour. This punched card concept was adopted by Charles Babbage to control his Analytical engine and later by Hollerith. 

6. Difference engine: 

The intervention of human beings was eliminated by Charles Babbage in calculations by using Difference engine in 1822. It could perform arithmetic operations and print results automatically. 

7. Analytical engine: 

In 1833, Charles Babbage introduced this. Charles Babbage is considered as the “Father of computer” It is considered as .the predecessor of today’s computer. This engine was controlled by programs stored in punched cards. These programs were written by Babbage’s assistant, Augusta Ada King, who was considered as the first programmer in the World.

8. Hollerith’s machine: 

In 1887, Herman Hollerith an American made first electromechanical punched cards with instructions for input and output. The card contained holes in a particular pattern with special meaning. 

The Us Census Bureau had large amount of data to tabulate, that will take nearly 10 years. By this machine this work was completed in one year. In 1896, Hollerith started a company Tabulating Machine Corporation. Now it is called International Business Machines(IBM).

9. Mark -1: 

In 1944 Howard Aiken manufactured automatic electromechanical computer in collaboration with engineers at IBM that handled 23 decimal place numbers and can perform addition, subtraction, multiplication and subtraction.

The intervention of human beings was eliminated by Charles Babbage in calculations by using Difference engine in 1822. It could perform arithmetic operations and print results automatically.

The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician Charles Babbage. The Analytical Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory. 

Charles Babbage is considered as the “Father of computer” It is considered as the predecessor of today’s computer. This engine was controlled by programs stored in punched cards. These programs were written by Babbage’s assistant, Augusta Ada King, who was considered as the first programmer in the World.

The number of transistors on IC’s doubles approximately every two years. This law is called Moore’s Law, it is named after Gordon E Moore. It is an observation and not a physical or natural law. He predicted that the trend would continue for at least ten years. It is true and the trend continued for more than half a century.

In 1887, Herman Hollerith an American made first electromechanical punched cards with instructions for input and output. The card contained holes in a particular pattern with special meaning.

The Us Census Bureau had large amount of data to tabulate, that will take nearly 10 years. By this machine this work was completed in one year. In 1896, Hollerith started a company Tabulating Machine Corporation. Now it is called International Business Machines(IBM).

(c) G.W. Von Leibniz

If the radius of a circle becomes twice then the ratio of any of the old and new circle will be 4 : 1

Find the distance between the point (–5, 7) and (–1, 3) 4√2

We have to write the complement of an angle which measures x°.

Let the other angle be y°.

We know that the sum of the complementary angles be 90°.

Therefore,

x° + y° = 90°

y° = (90 - x)°

Given:

A train takes 45 minutes to cover a certain distance at a speed of 80 km/h

His speed is increased by 125

Then how long will it take the train to cover 8/5 of the same distance 

Formula used:

\(velocity= distance/time\)

Calculation:

Distance = velocity × time

Distance = 80 × 45 × 50/3

⇒ 60000 meter

Speed is increased by 125%

⇒ 80 + 80 × (125/100)

⇒ 180 km/hour

The train to cover 8/5 of the same distance 

Distance = 60000 × (8/5)

⇒ 96000

So time is taken by train,

Time = distance/speed

⇒ (96000/180) × (3/50)

⇒ 32 minutes

Therefore, covered the distance by train in 32 minutes

arc(sinx) + arc(sin(1-x)) = arc(cosx)

⇒ sin^-1x + sin^-1(1-x) = cos^-1x

⇒ sin^-1x + sin^-1(1-x) = \(\frac{\pi}{2}\)- sin^-1x

(∵ sin^-1x + cos^-1x = \(\frac{\pi}{2}\))

⇒ 2sin^-1x = \(\frac{\pi}{2}\) - sin^-1(1-x)

⇒ sin^-1(\(2x\sqrt{1-x^2}\)) = cos^-1(1-x)

(∵ 2sin^-1x = sin^-1(\(2x\sqrt{1-x^2}\))

⇒ sin^-1(\(2x\sqrt{1-x^2}\)) = sin^-1(\(\sqrt{1-(1-x)^2}\))

(∵ cos^-1x = sin^-1\(\sqrt{1-x^2}\))

⇒ \(2x\sqrt{1-x^2}\) = \(\sqrt{1-(1-x)^2}\)

⇒ 4x²(1-x²) = 1 - (1-x²)

⇒ 4x² - 4x⁴ = 1 - (1+x²-2x)

⇒ 4x² - 4x⁴ = -x² + 2x

⇒ 4x⁴ - 5x² + 2x = 0

⇒ x (4x³ - 5x + 2) = 0

⇒ x (x - \(\frac{1}{2}\)) (4x² + 2x - 4) = 0

⇒ 2x (x - \(\frac{1}{2}\))(2x² + x - 2) = 0

⇒ x = 0,\(\frac{1}{2}\),\(\frac{-1\pm \sqrt{1+16}}{4}\) = 0,\(\frac{1}{2}\),\(\frac{-1+\sqrt{17}}{4}\), \(\frac{-1-\sqrt{17}}{4}\)

⇒ x = 0,\(\frac{1}{2}\),\(\frac{-1+\sqrt{17}}{4}\) 

(∵ Domain of f(x) is x ∈ [0,\(\frac{\pi}{2}\)] 

∴ x ≠ \(\frac{-1-\sqrt{17}}{4}\))

Given:

3, 12, 39, 120, ?

Calculation:

3 × 3 + 3 = 12

12 × 3 + 3 = 39

39 × 3 + 3 = 120

120 × 3 + 3 = 363

∴ The missing number is 363

Given :

Manufacturing price = 160

Manufacturer’s profit% = 30%

Wholesale dealer’s profit% = 20%

Selling price of Retailer = 312

Formula used:

Profit% = \(\frac{{\left( {Selling\;price - Cost\;price} \right)}}{{Cost\;price}} \times {\rm{\;}}100\)

Calculation :

SP of  Manufacture = \(160\; \times {\rm{\;}}\frac{{130}}{{100}}\)

⇒ 208

SP of wholesale dealer = ​\(208\; \times {\rm{\;}}\frac{{120}}{{100}}\)

⇒ \(208\; \times {\rm{\;}}\frac{{6}}{{5}}\)

Profit% of Retailer = \(\frac{{\left\{ {312 - \left( {208 \times {\rm{\;}}\frac{6}{5}\;} \right)} \right\} \times {\rm{\;}}100}}{{208 \times {\rm{\;}}\frac{6}{5}}}\)

⇒ 25% (Ans)

Number of ways of arranging 5 boys and 3 girls, i.e. 8 people on a round table would be 7!

We subtract the number of ways of arranging those people when B₁ and G₁ are always together to get the required answer.

When B₁ and G₁ are together, we get 4 boys +2 girls + 1(B₁ + G₁) i.e. 7 people and since B₁+G₁ can be permuted in 2 ways, these can be arranged in 6! x 2 ways.

Subtracting, we have 7! - 6! x 2 =6!(7−2)

=5×6! ways in total.

Required number of ways = Total number of ways - When A and B are always included.

=⁵C₂.⁷C₃−⁵C₁⁵C₂=300

S = {1,2,3------100}

= Total non empty subsets-subsets with product of element is odd

=2¹⁰⁰−1−1[(2⁵⁰−1)]

=2¹⁰⁰−2⁵⁰

=2⁵⁰(2⁵⁰−1)

Correct option is: (C) 2^1/2

i) Propeller shaft

ii) Universal joints fitted with propeller shaft

\(\lim\limits_{x \to 0}\)(1 + tan²x)^{\(\frac1{x^2}\)} (1^\(\infty\) = case)

 = Exp {\(\lim\limits_{x \to 0}\)(1 + tan²x - 1) \(\frac1{x^2}\)}

 = Exp {\(\lim\limits_{x \to 0}\) \(\frac{tan^2x}{x^2}\)}

 = Exp {\(\lim\limits_{x \to 0}\)\((\frac{tan y}x)^2\)}

 = e (∵ \(\lim\limits_{x \to 0}\)\(\frac{tan x}x=1\))

Answer

(a) 1- Bromopentane is most reactive towards SN2 reaction.

(b) 2-Bromo-2-methylbutane

(c) 2- Bromopentane is most reactive towards β-elimation reaction with product of pent-1-ene pent-2-ene.

Clouds are colloidal dispersion of water particles in air. These water particles carry some charge over them. On spraying oppositely charged colloidal dust or sand particles over a cloud from an aeroplane, the colloidal water particles present in the cloud will get neutralized and as a result they will come closer and will grow in size to form bigger water drops and ultimately will coagulate or precipitate causing artificial rain.

(a) Clouds are colloidal dispersion of water particles in air. These water particles carry some charge over them. On spraying oppositely charged colloidal dust or sand particles over a cloud from an aeroplane, the colloidal water particles present in the cloud will get neutralized and as a result they will come closer and will grow in size to form bigger water drops and ultimately will coagulate or precipitate causing artificial rain.

(d) Natural rubber

(d) Nylon 6, 6

 Strong covalent bonds exists between polymer chains in Thermosets. 

A. Decrease. 

(a) Metallic bond

c. ordered, repeating unitcells 

d) All of the above

d) All of the above

Answer: (c) Creep

Creep – this is the permanent deformation of a body due to persistent stress. This stress can be tensile, compressional, torsional or other mechanical stress. As mentioned, as creep is due to persistent stress, it is time dependent.

b) Increases the antimicrobial activity

d) Hypertension

The unwanted effect of dihydrotachysterol is CNS toxicity.

The Correct option is B. 38, 62

d) All of the above

b) Metronidazole

a) Nausea, vomiting, diarrhea, stomatitis

b) Inhibition of mycolic acids synthesis  

b) Inhibition of mycolic acids synthesis  

d) Hypophosphatemia

C. Composition

d) Hypercalcemia

c) Hypophosphatemia

d) All of the above

b) Headache, dizziness, insomnia

A. All three states are in equilibrium