A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems.

Yes goal for development can be varied for example, a farmer wants cheap fertilisers, a labour wants full employment and satisfactory wages like wise our parents may want good school for us etc, so everyone is free to choose their goal but cannot snatch others freedom to fulfill own need and goals

The grandmother was very old. She was slightly bent in her physique. So she could not stand and walk with her body in an upright position. She had to keep one of the two hands on her waist she did so to balance her stoop.

According to the question, we have to show that every positive integer is either even or odd.

Let us assume that there exists a smallest positive integer that is neither odd nor even, say n. Since n is the least positive integer which is neither even nor odd, n - 1 must be either odd or even.
Case 1: If n - 1 is even, n - 1 = 2k for some k.
But this implies n = 2k + 1
This implies n is odd.
Case 2: If n - 1 is odd, n - 1 = 2k + 1 for some k.
But this implies n = 2k + 2 = 2(k + 1)
This implies n is even.
Therefore,In both cases , we arrive at a contradiction.

Thus, every positive integer is either even or odd

On a graph paper, draw a horizontal line XOX' and a vertical line YOY' as the x-axis and they-axis respectively.
{tex}3x + y - 11 = 0{/tex} {tex}\Rightarrow{/tex}{tex}y = (11 -3x){/tex}. ...(i)
Putting {tex}x = 2{/tex} in (i), we get {tex}y = 5{/tex}.
Putting {tex}x = 3{/tex} in (i), we get {tex}y = 2{/tex}.
Putting {tex}x = 5{/tex} in (i), we get {tex}y = -4{/tex}.

On the graph paper, plot the points {tex}A (2, 5), B(3, 2)\ and\ C(5, -4).{/tex}
Join AB and BC to get the graph line ABC.
Thus, the line ABC is the graph of the equation {tex}3x + y - 11 = 0{/tex}.
{tex}x - y - 1 = 0{/tex} {tex}\Rightarrow{/tex}{tex}y = (x - 1){/tex} . ...(ii)
Putting {tex}x = -3{/tex} in (ii), we get {tex}y = -4{/tex}.
Putting {tex}x = 0{/tex} in (ii), we get {tex}y = -1{/tex}.
Putting {tex}x = 3{/tex} in (ii), we get {tex}y = 2{/tex}.

On the same graph paper as above, plot the points {tex}P(-3, -4)\ and\ Q(0, -1){/tex}. The third point {tex}B(3,2){/tex} is already plotted.
Join PQ and QB to get the graph line PQB.
Thus, line PQB is the graph of the equation {tex}x - y - 1 = 0{/tex}.
The two graph lines intersect at the point {tex}B(3,2){/tex}. {tex}x = 3, y = 2{/tex} is the solution of the given system of equations.
The region bounded by these lines and the y-axis has been shaded.

On extending the graph lines on both sides, we find that these graph lines intersect the y-axis at the points Q(0, -1) and R(0,11).

The Chauri Chaura incident took place on 4 February 1922 at Chauri Chaura in the Gorakhpur district of the United Provinces (modern-day Uttar Pradesh) in British India, when a large group of protesters participating in the non-cooperation movement, clashed with police who opened fire

For train A:
Initial velocity, u = 72 km/h = 20 m/s
Time, t = 50 s
Acceleration, a_I = 0 (Since it is moving with a uniform velocity)
From second equation of motion, distance (S_I) covered by train A can be obtained as:
{tex}s _ { I } = u t + \frac { 1 } { 2 } a _ { I } t ^ { 2 }{/tex}
= 20 {tex}\times{/tex} 50 + 0 = 1000 m
For train B:
Initial velocity, u = 72 km/h = 20 m/s
Acceleration, a = 1 m/s²
Time, t = 50 s
From second equation of motion, distance (S_II) covered by train A can be obtained as:
{tex}s _ { II } = u t + \frac { 1 } { 2 } a t ^ { 2 }{/tex}
{tex}= 20 \times 50 + \frac { 1 } { 2 } \times 1 \times ( 50 ) ^ { 2 } = 2250 \mathrm { m }{/tex}
Length of both trains = 2 {tex}\times{/tex} 400 m = 800 m
Hence, the original distance between the driver of train A and the guard of train B is 2250 - 1000 - 800 = 450 m.

Food which is spoilt by microorganisms contains toxic substances relased by the microbes. The spoilt food is bad in taste and emits a foul odour. If such food is eaten, it will lead to food poisoning. The major symptoms of food poisoning are vomiting, diarrhoea, pain in abdomen, fever and headache.

Vomiting and loose motion cause dehydration. Hence, the intake of fluids to maintain the electrolytes is balanced. Oral rehydration therapy (ORT) is a type of fluid replacement used to prevent and treat dehydration, especially due to diarrhoea. It involves drinking water with modest amounts of sugar and salts, specifically sodium and potassium.

The ocean sunfish is the heaviest known bony fish in the world. It has an average adult weight of 2,200 pounds and are native to warm waters all over the world.

अतीत है तथा गौरवमयी संस्कृति व सभ्यता है । हमारा देश विश्व के समस्त देशों से अद्‌भुत व निराला देश है । मुझे अपने देश की संस्कृति व सभ्यता पर गर्व है ।

मैं जब भी किसी से कहता हूँ कि मैं भारतवासी हूँ या मुझे कोई भारतीय कहकर पुकारता है तो मैं स्वयं को गौरवान्वित महसूस करता हूँ । हमारे देश के विश्व में अन्य देशों से अद्‌भुत व न्यारे होने के कई कारण हैं जिसका विस्तृत अवलोकन इस बात की पुष्टि करता है ।

हमारे देश की संस्कृति व सभ्यता विश्व की सबसे पुरानी सभ्यताओं में से एक है । यह देश ऋषियों-मुनियों का देश रहा है । भारत को इसीलिए अनेक महापुरुषों ने देवों की धरती कहा है क्योंकि यहाँ पर संस्कृति व सभ्यता पीढ़ी दर पीढ़ी चली आ रही है और हजारों वर्ष बाद भी भारतीय संस्कृति उतने ही सशक्त व जीवंत रूप में विद्‌यमान है । हमारे देश की संस्कृति त्याग, बलिदान, प्रेम, सद्‌भावना, भाईचारा, श्रद्‌धा आदि महान नैतिक, शुद्‌ध व दैवी गुणों पर आधारित है ।

विशाल हृदय वाली इस संस्कृति ने हमें अपने दुश्मनों से भी प्रेम करना सिखाया है । इसी धरती पर भगवान श्रीराम, श्रीकृष्ण, त्याग की प्रतिमूर्ति महात्मा दधीचि, दानवीर कर्ण, महाप्रतापी व सत्यवादी राजा हरिश्चंद आदि महापुरुषों ने जन्म लिया । गाँधी जी जैसे युगपुरुष यहीं पर अवतरित हुए जिन्होंने बिना शस्त्र के ‘सत्य और अहिंसा’ के मार्ग पर चलते हुए भारत को स्वतंत्र कराया । संपूर्ण विश्व युगपुरुष गाँधी जी को आज भी नमन करता है ।

हमारे देश में कश्मीर से कन्याकुमारी तक तथा गुजरात से अरुणाचल प्रदेश तक विभिन्न भाषा, जाति, वेश-भूषा व विभिन्न मतों के लोग एक साथ निवास करते हैं । इतने विभिन्न रंगों को एकीकृत रूप में पिरोना भारत जैसे महान देश में ही संभव है । भारतीय संस्कृति की उदारता व महानता का यह साक्षात् प्रमाण है ।

यहाँ विश्व के लगभग समस्त धर्मों के लोग परस्पर मेल-जोल से रहते हैं । सभी को बिना भेदभाव अपने धर्म को मानने व प्रचार-प्रसार की खुली अनुमति है । उत्तर से दक्षिण हो या फिर पूर्व से पश्चिम हम भारत के किसी भी छोर पर जाएँ हमें जो भिन्नता यहाँ देखने को मिलेगी वैसी भिन्नता विश्व के शायद ही किसी कोने में उपलब्ध हो ।

कला की दृष्टि से भी हमारा देश उत्कृष्ट है । मुगलकालीन इतिहास में मुगल शासकों द्‌वारा प्रदत्त कला, विश्व कला जगत के लिए एक महान उपलब्धि है । हम आगरा के ताजमहल को लें, या फिर दिल्ली की कुतुबमीनार को सभी कला जगत की महत्वपूर्ण उपलब्धियाँ हैं जिसे देखने के लिए हर वर्ष लाखों विदेशी पर्यटक भारत आते हैं ।

The word biology is derived from the greek words /bios/ meaning /life/ and /logos/ meaning /study/ and is defined as the science of life and living organisms. An organism is a living entity consisting of one cell e.g. bacteria, or several cells e.g. animals, plants and fungi.

It is said that Yoga has originated from Hinduism and is being practiced today in the world. People have learned about the qualities of yoga and accepted it as exercise and meditation. Originally yoga is not only a form of exercise, but it is an ancient knowledge of living in a healthy, happy and peaceful way. It helps in attaining inner peace and spiritual knowledge. People generally think that yoga is a form of exercise that involves walking parts of the body but yoga is more than exercise. Yoga is the art of living life through the mental, spiritual and physical path. It helps in achieving stability and attention in the consciousness of the inner self. It helps in learning about mind, emotion, and physical needs and how to think more about how to cope with life's challenges. Yoga works at the level of body, mind and energy; Regular practice of yoga brings positive changes in the body, including strong muscles, flexibility, patience and good health. We should be patient for yoga, generally people use shortcut like drug, steroids, or surgery to lose weight, which have a bad effect on the body over time.