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पवित्र युद्ध या जिहाद उन युद्धों को कहते हैं जो मध्यकाल में फिलिस्तीन को प्राप्त करने के लिए यूरोपीय ईसाइयों ने अरबी मुसलमानों से लड़े। इन युद्धों को धर्मयुद्ध इसलिए कहा जाता है कि यह युद्ध धार्मिक स्थानों को प्राप्त करने के लिए ईसाइयों ने अरबों के विरुद्ध लड़े थे। 

धर्मयुद्ध के तीन प्रमुख कारण थे
⦁    पवित्र प्रदेशों को पुनः प्राप्त करना।
⦁    सामन्तों का वीरता प्रदर्शन का शौक।
⦁     लाडौँ तथा चर्च के नेताओं का स्वार्थ।

इस्लाम धर्म ने विश्व की सभ्यता को बड़े पैमाने पर प्रभावित किया। कला और धर्म के क्षेत्र में अपना महत्त्वपूर्ण योगदान दिया।

इस्लाम धर्म से पूर्व अरब लोगों के जीवन की मुख्य विशेषताएँ निम्नलिखित हैं

⦁    इस्लाम से पूर्व अरब लोग अनेक छोटे-छोटे कबीलों में बँटे हुए थे।
⦁    कबीले परस्पर छोटी-छोटी बातों पर लड़ते रहते थे।
⦁    अरब समाज के लोग अनेक अन्धविश्वासों के शिकार थे।
⦁    इस समय अरब के लोग अनेक देवी-देवताओं में विश्वास करते थे और मूर्तिपूजा किया करते
⦁     इस समय अरब के लोगों का मुख्य व्यवसाय पशुपालन था। कालान्तर में व्यापार भी इनकी जीविका का मुख्य साधन बन गया।

सही विकल्प है (घ) ये तीनों

बैंक ऑफ जापान की स्थापना 1882 ई० में हुई।

सन यात-सेन 

सन यात-सेन जन्म से ही क्रान्तिकारी थे। उन्होंने मंचू सरकार को हटाकर चीन में गणतन्त्र की स्थापना की थी। वे लोकतन्त्र के समर्थक थे। यह सही है कि डॉ० सेन की राजनीतिक विचारधारा रूस के साम्यवादी दर्शन से बहुत अधिक प्रभावित थी और उन्होंने अपने देश में साम्यवादी ढंग से परिवर्तन लाने का प्रयास भी किया था। फिर भी वे साम्यवादी दर्शन के अन्धभक्त नहीं थे। वे यह अच्छी तरह से जानते थे कि रूस श्रमिकों का देश है और उनका देश चीन किसानों का, इसलिए उन्होंने अपने राजनीतिक विचारों को अपने देश की परिस्थितियों के अनुकूल ही बनाया।

डॉ० सेन के तीन सिद्धान्त

डॉ० सन यात-सेन ने अपने क्रान्तिकारी जीवन के प्रारम्भ से ही अपने राजनीतिक विचारों को तीन सिद्धान्तों के रूप में रख दिया था। ये सिद्धान्त निम्नलिखित थे
⦁    राष्ट्रीयता :
चीन में सदियों से जहाँ एक ओर सांस्कृतिक एकता तो मौजूद थी, वहीं दूसरी ओर राजनीतिक एकता को पूर्ण अभाव था। यह अभाव बीसवीं शताब्दी के प्रारम्भ में भी विद्यमान था। जनता में स्थानीय तथा प्रान्तीय भावनाएँ शक्तिशाली थीं। यही कारण था कि विदेशी साम्राज्यवादी शक्तियाँ चीन में अपना प्रभाव स्थापित करने में सफल हो रही थीं। डॉ० सन यात-सेन
⦁     राजनीतिक लोकतन्त्र :
डॉ० सन यात-सेन लोकतन्त्र के पक्के समर्थक थे। इसी कारण उन्होंने चीन में सदियों से चले आ रहे मंचू राजवंश को समाप्त करके राजवंश की प्राचीन परम्परा को समाप्त कर दिया था। उन्हें जनता की शक्ति में विश्वास था और यही कारण था कि वे लोकतन्त्र के समर्थक थे। उन्होंने अपना अधिकांश जीवन विदेशों के गणतन्त्रीय वातावरण में व्यतीत किया था और स्वयं वहाँ के विकास को देखा था। यही कारण था कि उन्होंने चीन में क्रान्ति करके गणतन्त्र की स्थापना को अपने जीवन का पवित्र लक्ष्य बना लिया था। 1924 ई० में लोकतन्त्र के विषय में उनके विचार बहुत अधिक मजबूत हो गए थे। उनका विचार था “सफल लोकतन्त्र में सरकार की शासन प्रणाली, कानून, कार्य, न्याय, परीक्षा तथा नियन्त्रण के पंच शक्ति विधान पर आधारित होनी चाहिए।’ लोकतन्त्र को सफलता की अन्तिम चोटी पर पहुँचने के लिए सेन ने तीन बातों पर विशेष बल दिया था। सबसे पहले देश में सैनिक शक्ति के प्रभुत्व की स्थापना करके देश में पूरी शान्ति तथा व्यवस्था स्थापित की जाए। इसके बाद देश में राजनीतिक चेतना का प्रसार किया जाए और अन्त में वैधानिक तथा लोकतन्त्रीय सरकार को निर्माण करके देश अपने लक्ष्यों को प्राप्त करें।
⦁    जनता की आजीविका :
सन यात-सेन ने मानव जीवन में भोजन की भारी आवश्यकता का अच्छी तरह से अनुभव कर लिया था और यही कारण था कि उन्होंने कृषक वर्ग के उत्थान की ओर अधिक ध्यान दिया। उनका मत था कि ‘भूमि उसकी है जो उसे जोतता है। समाज के अन्य वर्गों की जीविका के प्रश्न का समाधान वे सामाजिक विकास के साथ करना चाहते थे। वे साम्यवादियों के समान भूमि के समान वितरण के सिद्धान्त के पक्ष में थे। इस समय उनकी नीति एक प्रबल साम्यवादी की न होकर एक समाज-सुधारक की नीति थी परन्तु वे मार्क्स के भौतिकवाद के विरोधी थे और अच्छी तरह से यह अनुभव करते थे कि मार्क्स के सिद्धान्तों को चीन में लागू नहीं किया जा सकता है। सन यात-सेन के तीन सिद्धान्तों पर विचार करने के पश्चात् यह स्पष्ट होता है कि डॉ० सेन के सिद्धान्त माक्र्सवाद से बिल्कुल भिन्न थे। सेन के राष्ट्रीयता, लोकतन्त्र और जनता की जीविका के सिद्धान्तों में मार्क्स के वर्ग संघर्ष का कोई स्थान नहीं है और न ही इन सिद्धान्तों में मार्क्स के समाजवादी अर्थतन्त्र की स्थापना के लिए कोई विशेष बल दिया गया है। इससे स्पष्ट हो जाता है। कि सन यात-सेन न तो मार्क्स की शुद्ध समाजवादी विचारधारा के समर्थक थे और न ही साम्यवादी नीति का अन्धानुकरण करने वाले थे। इस प्रकार सन यात-सेन को साम्यवाद का कट्टर समर्थक नहीं कहा जा सकता है।

चीन में तुंग-मिंग-हुई नामक क्रान्तिकारी संस्था की स्थापना सितम्बर, 1905 ई० में हुई।

मेजी संविधान की एक विशेषता जनता के मौलिक अधिकार से सम्बन्धित थी।

इतिहास की किताबों में ऑस्ट्रेलिया के मूल निवासियों को इसलिए शामिल नहीं किया गया क्योंकि वे जन्म से यहाँ नहीं थे। वे ऑस्ट्रेलिया के साथ जुड़े एक लम्बे पुल से न्यू गुयाना से आए थे। प्रारम्भिक लोग इन्हें ऐबॉरिजिनीज कंहते थे। देशी लोगों का एक और विशाल समूह उत्तर में निवास करता है। ऐसा माना जाता है कि वे किसी और स्थान से आए थे संक्षेप में निबन्ध लिखिए

ऑस्ट्रेलिया के मूल निवासियों की विशेषताएँ ऑस्ट्रेलिया के मूल निवासियों की प्रमुख विशेषताएँ निम्नलिखित हैं

⦁    प्रारम्भ में यहाँ के निवासियों को ऐबॉरीजिनीज कहा जाता था। कई भिन्न-भिन्न समाजों के लिए प्रयुक्त किया जाने वाला एक सामान्य नाम है। ये लोगे ऑस्ट्रेलिया में 40,000 वर्ष पूर्व आने प्रारम्भ हुए थे।
⦁     ये लोग ऑस्ट्रेलिया के साथ एक लम्बे पुल से जुड़े न्यू गुयाना से आए थे। मूल निवासियों की अपनी परम्परा के अनुसार वे बाहर से नहीं आए थे बल्कि हमेशा से यहीं थे।
⦁     18वीं सदी के अन्त में मूल निवासियों के 350 से 750 समुदाय थे।
⦁     प्रत्येक समुदाय की अपनी भाषा थी। इनमें से 200 भाषाएँ आज भी व्यवहार में हैं।
⦁     मूल निवासियों का एक और समूह उत्तर में रहता है। इसे टॉरस स्ट्रेट कहते हैं। वे ऑस्ट्रेलिया की वर्तमान जनसंख्या का 2.4 प्रतिशत भाग हैं।
⦁    ऑस्ट्रेलिया की जनसंख्या विरल है। अधिकांश नगर समुद्रतट के साथ-साथ बसे हैं। बीच का क्षेत्र शुष्क मरुभूमि है।

मेजी पुनस्र्थापना से जापान के सामाजिक, आर्थिक, सांस्कृतिक और राजनीतिक सभी क्षेत्रों में प्रगति हुई।

ये विशेषताएँ निम्नलिखित हैं
⦁    वे कुशल कारीगर थे और आकर्षक कपड़े बनाते थे।
⦁    लोगों का मुख्य व्यवसाय कृषि था। संयुक्त रज्य अमेरिका के लोगों ने कृषि क्षेत्र में पर्याप्त विस्तार किया। उन्होंने दूसरे देशों में भी भूमि खरीदी।
⦁    वहाँ खनन उद्योग और कारखानों का विकास किया गया।
⦁     वे मछली तथा मांस खाते थे।
⦁     खेतों में सब्जियाँ ओर मकई उगाते थे।
⦁    सत्रहवीं सदी में उन्होंने घुड़सवारी आरम्भ कर दी थी।

संयुक्त राज्य अमेरिका के मूल निवासियों को स्थायी रूप से पश्चिम में जमीन दे दी गई फिर भी यदि उनकी जमीन में तेल या सोने जैसी किसी चीज के होने का पता चलता तो उन्हें इसी में साझा करना पड़ता। ऐसे में उनके बीच झगड़े उत्पन्न हो जाते थे। मूल निवासी एक छोटे क्षेत्र में कैद कर दिए गए जिन्हें ‘आरक्षित क्षेत्र कहा जाता था। मूल निवासी अपनी जमीन के लिए जीवन भर लड़ते रहे, परन्तु आप्रवासियों ने उनके विद्रोह का दमन कर दिया। संयुक्त राज्य अमेरिका की सेना ने 1865 से 1890 के मध्य विद्रोह की एक पूरी श्रृंखला का दमन किया था। कनाडा में 1869 से 1885 के बीच मेटिसों के विद्रोह हुए किन्तु उन्हें पराजय का मुँह देखना पड़ा।

बहुसंस्कृतिवाद वह विचारधारा है जिसमें अनेक संस्कृतियों को एक साथ रहने पर जोर दिया जाता है। यह सहिष्णुता की नीति वाली संस्कृति है।

These Class 8 Maths MCQ Questions of Linear Equations in One Variable with Answers have been prepared based on the exam perspective. Linear equations in one variable are the algebraic expressions which have only one variable. These equations are easy to solve. Let us see some of the questions here to practice well. The MCQ Questions for Class 8 with answers have been prepared as per the latest 2021 syllabus, NCERT and examination pattern suggested in Standard 8 by CBSE.

Each problem consisting of four options, out of which one is the correct answer. Students have to solve the problem and select the correct answer.  If practiced MCQ Questions for Class 8 Maths properly can help you to get higher marks. Refer to more Chapter-wise Linear Equations MCQ Questions for Class 8 and also Study more maths formulas for solving equations.

Practice MCQ Question for Class 8 Maths chapter-wise

1. Which of the following is not a linear equation in one variable?

(a) 33z+ 5 = 0
(b) 33 (x+y) = 0
(c) 33x+5 = 0
(d) 33y+5 = 0

2. The solution of 2x-3 = 7 is:

(a) 5
(b) 7
(c) 12
(d) 11

3. The solution of 2y + 9 = 4 is:

(a) 9/2
(b) 4/9
(c) -2/5
(d) -5/2

4. The solution of y/5 = 10 is:

(a) 15
(b) 10
(c) 50
(d) 5

5. What should be added to -7/3 to get 3/7?

(a) 21/58
(b) 58/21
(c) 47/21
(d) 50/21

6. The perimeter of the rectangle is 20cm. If the length of the rectangle is 6cm, then its breadth will be:

(a) 4 cm
(b) 6 cm
(c) 10 cm
(d) 14 cm

7. The age of the father is three times the age of the son. If the age of the son is 15 years old, then the age of the father is:

(a) 55 years
(b) 50 years
(c) 44 years
(d) 45 years

8. The difference between two whole numbers is 66. The ratio of the two numbers is 2: 5. The two numbers are:

(a) 60 and 6
(b) 100 and 33
(c) 110 and 44
(d) 99 and 33

9. Three consecutive integers add up to 51. The integers are:

(a)  16,17,18
(b) 15,16,17
(c) 17,18,19
(d) 18,19,20

10. The solution for 3m = 5m – (8/5) is:

(a) 8/5
(b) 4/5
(c) 5/4
(d) 4/3

11. The degree of x² – 5x + 2 = x³ is:

(a) 3
(b) 2
(c) 1
(d) 0

12. The degree of x² – 9x + 2 = 4x⁴ is:

(a) 3
(b) 2
(c) 1
(d) 4

13. What is the value of x if x + 9 = 12?

(a) 2
(b) 3
(c) 8
(d) 6

14. If a number is divided by 8 it gives 6 as the value. Find the number.

(a) 36
(b) 42
(c) 48
(d) 56

15. Find the value of x if 2x + 10 = 76.

(a) 33
(b) 7.6
(c) 32
(d) 66

16. The perimeter of a rectangle is 40 cm. If its width is 10 cm, then find the length.

(a) 10
(b) 20
(c) 30
(d) 40

17. If x is an even number, then the next even number is:

(a) x+2
(b) x+1
(c) x+4
(d) x+5

18. Nitin was counting down from 32. Sumit was counting upwards, the numbers starting from 1 and he was calling out only the odd numbers. What common number will they call out at the same time if they were calling out at the same speed?

(a) 19
(b) 21
(c) 22
(d) They will not call out the same number

19. Solve 2x − 3 = x + 2

(a) 4
(b) 5
(c) 3
(d) 0

20. Amina thinks of a number and subtracts 5/2 from it. She multiplies the result by 8. The result now obtained is 3 times the same number she thought of. What is the number?

(a) 4
(b) 2
(c) 3
(d) 0

21. The sum of three consecutive multiples of 7 is 357. Find the smallest multiple.

(a) 112
(b) 126
(c) 119
(d) 116

22. The difference between the two numbers is 30. If the bigger number is x, then what is the smaller number?

(a) x – 30
(b) 30 – x
(c) 30x
(d) None of these

23. When a number is added to itself, it becomes 24. What is the number?

(a) 2
(b) 4
(c) 12
(d) 21

24. David cuts bread into two equal pieces and cuts one half into smaller pieces of equal size. Each of the small pieces is twenty grams in weight. If he has seven pieces of the bread all with him, how heavy is the original cake?

(a) 120 gm
(b) 180 gm
(c) 300 gm
(d) 240 gm

25. If x% of 75=12, then the value of x 

(a) 20
(b) 18
(c) 16
(d) 26

Answer:

1. Answer: (b) 33(x+y) = 0

Explanation: In 33(x+y) = 0, x and y are two variables.

2. Answer: (a) 5

Explanation: 2x-3=7

2x=7+3=10

x=10/2 

= 5

3. Answer: (d) -5/2

Explanation: 2y+9 = 4

2y = 4-9 

= -5

y =-5/2

4. Answer: (c) 50

Explanation: y/5 = 10

y = 5×10 

= 50

5. Answer: (b) 58/21

Explanation: Let the number be x

-7/3+x = 3/7

x=3/7+7/3 

= (9+49)/21 

= 58/21

6. Answer: (a) 4 cm

Explanation: Perimeter of rectangle = 2(Length+Breadth)

20 = 2(6+x)

6+x = 20/2

6+x = 10

x = 10-6

x = 4 cm

7. Answer: (d) 45 years

Explanation: Let the age of the father is x

Given: x = 3 × (age of son)

= 3 × (15) 

= 45 years

8. Answer: (c) 110 and 44

Explanation: Let the two numbers be 2x and 5x since they are in the ratio of 2:5.

The difference between 5x and 2x = 66

5x – 2x = 66

3x = 66

x = 22

Hence, 2x = 2(22) = 44 and 

5x = 5(22) = 110.

9. Answer: (a)  16,17,18

Explanation: Let the three consecutive integers be x, x+1, x+2

x+(x+1)+(x+2) = 51

3x+3 = 51

3x = 51 – 3

x = 48/3 

= 16

x+1 = 16+1

=17

x+2 = x+2 

= 18

10. Answer: (b) 4/5

Explanation: 3m = 5m – (8/5)

8/5 = 5m – 3m

2m = 8/5

m = 8/10 

= 4/5

11. Answer: (a) 3

Explanation: Degree is the highest power of the variable in an equation. Therefore, in the given equation, the highest degree is 3.

12. Answer: (d) 4

Explanation: Degree is the highest power of the variable in an equation. Therefore, in the given equation, the highest degree is 4.

13. Answer: (b) 3

Explanation: x + 9 = 12

x = 12 – 9

x = 3

14. Answer: (c) 48

Explanation: Let X be the number

x/8 = 6

x = 8 x 6 

= 48

15. Answer: (a) 33

Explanation: 2x + 10 = 76

2x = 76 – 10

2x = 66

x = 66/2

x = 33

16. Answer: (a) 10

Explanation: Perimeter of a rectangle = 40 cm

Width = 10 cm

Let the length be x.

Perimeter of rectangle = 2(length + width)

40 = 2 (x + 10)

40/2 = x + 10

20 = x + 10

x = 20 – 10 = 10

Thus, the length is also 10 cm.

 17. Answer: (a) x+2

Explanation:  If x = 2, then 

x + 2 = 2 + 2 

= 4

18. Answer: (d) They will not call out the same number

Explanation: Sumit is counting only 16 odd digits whereas Nitin will be counting 32 digits.
If they count at the same speed Sumit's numbers will finish faster than Nitin and they will never say the same number.

19.  Answer:(b) 5

Explanation: 2x−3 = x+2

⟹2x−x = 2+3

⟹x = 5

20.  Answer: (a) 4

Explanation: \(8(x-\frac{5}{2})=3x\)

8x−20 = 3x

8x−3x = 20

5x = 20

x = 4

Number is 4

21.  Answer: (a) 112

Explanation: Let us take three consecutive multiples of 7 as, 7x,(7x+7),(7x+14)     [x is a natural number]

7x+(7x+7)+(7x+14)=357

⇒ 21x+21=357

⇒ 21(x+1)=357

\(\frac{21(x+1)}{21}=\frac{357}{21}\)  [Diving both sides by 21]

⇒x+1=17

x=17−1

x=16

Hence, the smallest multiple of 7 is 

7 ×16

= 112

22.  Answer: (a) x – 30

Explanation: x – small number = 30

Small number = x – 30

23.  Answer: (c) 12

Explanation: Let the number be x.

x + x = 24

2x = 24

x = 24/2

x = 12

24.  Answer: (d) 240 gm

Explanation: Let the weight of original cake be x.

Then the weight of one half is x/2.

There are a total of seven pieces. 

⇒ The number of smaller pieces  = 6.
As each of the smaller piece weighs 20 grams, the weight of one half is

20 × 6 = 120 grams.

Hence the total weight of the cake 

= 2 ×120 

= 240 grams.

25.  Answer: (c) 16

Explanation: \(\frac{x}{100}\times75=12\)

\(x\times75=12\times100\)

\(x=\frac{12\times100}{75}\)

x = 16

Click here Practice MCQ Question for Linear Equations in One Variable Class 8

Correct option is  B) 6

Correct option is (B) 6

\(\frac{x}{2}+\frac{x}{3}\) = 5

\(\Rightarrow\) \(\frac{3x+2x}{6}=5\)

\(\Rightarrow\) 5x = 5 \(\times\) 6 = 30

\(\Rightarrow\) x = \(\frac{30}5\) = 6.

Correct option is  C) – 3

Correct option is (C) –3

\(\frac{x+7}{3x+16}=\frac{4}{7}\)

\(\Rightarrow\) 7(x+7) = 4(3x+16)

\(\Rightarrow\) 7x+49 = 12x+64

\(\Rightarrow\) 12x - 7x = 49 - 64 = -15

\(\Rightarrow\) 5x = -15

\(\Rightarrow\) x = \(\frac{-15}5\) = -3.

(a-8)/3 = (a-3)/2

By using cross-multiplication we get,

(a-8)2 = (a-3)3

2a – 16 = 3a – 9

2a – 3a = -9 + 16

-a = 7

a = -7

Let us verify the given equation now,

(a-8)/3 = (a-3)/2

By substituting the value of ‘a’ we get,

(-7 – 8)/3 = (-7 – 3)/2

-15/3 = -10/2

-5 = -5

Hence, the given equation is verified.

(2x+5)/3 = 3x – 10

Let us simplify,

(2x+5)/3 – 3x = – 10

By taking LCM

(2x + 5 – 9x)/3 = -10

(-7x + 5)/3 = -10

By using cross-multiplication we get,

-7x + 5 = -30

-7x = -30 – 5

-7x = -35

x = -35/-7

= 5

Let us verify the given equation now,

(2x+5)/3 = 3x – 10

By substituting the value of ‘x’ we get,

(2×5 + 5)/3 = 3(5) – 10

(10+5)/3 = 15-10

15/3 = 5

5 = 5

Hence, the given equation is verified.

Given 3x – 2 (2x -5) = 2 (x + 3) – 8

On simplifying the brackets on both sides, we get

3x – 2 × 2x + 2 × 5 = 2 × x + 2 × 3 – 8

3x – 4x + 10 = 2x + 6 – 8

-x + 10 = 2x – 2

Transposing x to RHS and 2 to LHS, we get

10 + 2 = 2x + x

3x = 12

Dividing both sides by 3, we get

3x/3 = 12/3

x = 4

Verification:

Substituting x = 4 on both sides, we get

3(4) – 2{2(4) – 5} = 2(4 + 3) – 8

12 – 2(8 – 5) = 14 – 8

12 – 6 = 6

6 = 6

Thus LHS = RHS

Hence, verified.

We have,

2/3(x – 5) – 1/4(x – 2) = 9/2

Upon expansion we get,

2x/3 – 10/3 – x/4 + 2/4 = 9/2

2x/3 – 10/3 – x/4 + 1/2 = 9/2

2x/3 – x/4 = 9/2 + 10/3 – 1/2

By taking LCM for (3 and 4 is 12) (2 and 3 is 6)

(2x×4)/12 – (x×3)/12 = (9×3)/6 + (10×2)/6 – (1×3)/6

8x/12 – 3x/12 = 27/6 + 20/6 – 3/6

(8x-3x)/12 = (27+20-3)6

5x/12 = 44/6

By using cross-multiplication we get,

5x×6 = 44×12

30x = 528

x = 528/30

= 264/15

= 88/5

Given 3 (x – 3) = 5 (2x + 1)

On simplifying the brackets we get,

3x – 9 = 10x + 5

Now transposing 10x to LHS and 9 to RHs

3x – 10x = 5 + 9

-7x = 14

Now dividing both sides by -7 we get

-7x/-7 = 14/-7

x = -2

Verification:

Substituting x = -2 in the given equation we get

3 (-2 – 3) = 5 (-4 + 1)

3 (-5) = 5 (-3)

-15 = – 15

Thus LHS = RHS

Hence, verified.

Given (3/4) (x -1) = (x – 3)

On simplifying the brackets both sides we get,

(3/4) x – (3/4) = (x – 3)

Now transposing (3/4) to RHS and (x – 3) to LHs

(3/4) x – x = (3/4) – 3

(3x – 4x)/4 = (3 – 12)/4

-x/4 = (-9/4)

Multiply both sides by -4 we get

-x/4 × -4 = (-9/4) × -4

x = 9

Verification:

Substituting x = 9 in the given equation:

(3/4) (9 – 1) = (9 -3)

(3/4) (8) = 6

3 × 2 = 6

6 = 6

Thus LHS = RHS

Hence, verified.

We have,

15(2-x) – 5(x+6) / (1-3x) = 10

(30-15x) – (5x + 30) / (1-3x) = 10

By cross-multiplying we get,

(30-15x) – (5x + 30) = 10(1- 3x)

30- 15x – 5x – 30 = 10 – 30x

30- 15x – 5x – 30 + 30x = 10

10x = 10

x = 10/10

= 1

Now let us verify the given equation,

(15(2-x) – 5(x+6)) / (1-3x) = 10

By substituting the value of ‘x’ we get,

(15(2-1) – 5(1+6)) / (1- 3) = 10

(15 – 5(7))/-2 = 10

(15-35)/-2 = 10

-20/-2 = 10

10 = 10

Hence, the given equation is verified.

We have,

\(9 \frac{1}{4} = y \,–\, 1 \frac{1}{3}\)

\(\frac{37}{4}\) = y – \(\frac{4}{3}\)

Upon solving we get,

y = \(\frac{37}{4}\) + \(\frac{4}{3}\)

By taking LCM for 4 and 3 is 12

y = \(\frac{(37×3)}{12} \)+ \(\frac{(4×4)}{12}\)

= \(\frac{111}{12} + \frac{16}{12}\)

= \(\frac{(111 + 16)}{12}\)

= \(\frac{127}{12}\)

∴ y = \(\frac{127}{12}\)

Given (x/2) + (3/2) = (2x/5) – 1

Transposing (2x/5) to LHS and (3/2) to RHS, then we get

(x/2) – (2x/5) = – 1 – (3/2)

(5x -4x)/10 = (-2 – 3)/2 [LCM of 5 and 2 is 10]

x/10 = -5/2

Multiplying both sides by 10 we get,

x/10 × 10 = (-5/2) × 10

x = (-50/2)

x = -25

Verification:

Substituting x = -25 in given equation we get

(-25/2) + (3/2) = (-50/5) – 1

(-25 + 3)/2 = -10 – 1

(-22/2) = -11

-11 = -11

Thus LHS = RHS

Hence, verified.

Given (x/2) = (x/3) + 1

Transposing (x/3) to LHS we get

(x/2) – (x/3) = 1

(3x – 2x)/6 = 1 [LCM of 3 and 2 is 6]

x/6 = 1

Multiplying 6 to both sides we get,

x = 6

Verification:

Substituting x = 6 in given equation we get

(6/2) = (6/3) + 1

3 = 2 + 1

3 = 3

Thus LHS = RHS

Hence, verified.

The chemistry of atmospheric gases is an important subject of study as it involves air pollution. O₂ in air is essential for survival of aerobic life.

Given 2 (5x – 3) – 3 (2x – 1) = 9

Simplifying the brackets, we get

2 × 5x – 2 × 3 – 3 × 2x + 3 × 1 = 9

10x – 6 – 6x + 3 = 9

10x – 6x – 6 + 3 = 9

4x – 3 = 9

Adding 3 to both sides, we get

4x – 3 + 3 = 9 + 3

4x = 12

Dividing both sides by 4, we get

4x/4 = 12/4

Therefore x = 3.

Verification:

Substituting x = 3 in LHS, we get

2(5 × 3 – 3) – 3(2 × 3 – 1) = 9

2 × 12 – 3 × 5 = 9

24 – 15 = 9

9 = 9

Thus, LHS = RHS

Hence, verified.

We have,

\(\frac{5x}{3} + \frac{2}{5} = 1\)

\(\frac{5x}{3} = 1 - \frac{2}{5}\) (by taking LCM)

= \(\frac{(5-2)}{5}\)

By using cross-multiplication we get,

\(\frac{5x}{3}\) = \(\frac{3}{5}\)

5x = \(\frac{(3×3)}{5}\)

x = \(\frac{9}{(5×5)}\)

= \(\frac{9}{25}\)

∴ x = \(\frac{9}{25}\)

We have,

13(y – 4) – 3(y – 9) – 5(y + 4) = 0

Upon expansion we get,

13y – 52 – 3y + 27 – 5y – 20 = 0

13y – 3y – 5y = 52 – 27 + 20

5y = 45

y = 45/5

= 9

∴ y = 9

We have,

x/2 + x/3 + x/4 = 13

let us take LCM for 2, 3 and 4 which is 12

(x×6)/12 + (x×4)/12 + (x×3)/12 = 13

6x/12 + 4x/12 + 3x/12 = 13

(6x+4x+3x)/12 = 13

13x/12 = 13

By using cross-multiplication we get,

13x = 12×13

x = 156/13

= 12

∴ x = 12

Correct Answer is: (D) 7

Although the pH of pure water is 7, drinking water and natural water exhibits a pH range because it contains dissolved minerals and gases. 

We have,

x/2 + x/8 = 1/8

let us take LCM for 2 and 8 which is 8

(x×4)/8 + (x×1)/8 = 1/8

4x/8 + x/8 = 1/8

5x/8 = 1/8

By using cross-multiplication we get,

5x = 8/8

5x = 1

x = 1/5

∴ x = 1/5

We have,

2x/3 – 3x/8 = 7/12

By taking LCM for 3 and 8 is 24

(2x×8)/24 – (3x×3)/24 = 7/12

16x/24 – 9x/24 = 7/12

(16x-9x)/24 = 7/12

7x/24 = 7/12

By using cross-multiplication we get,

7x×12 = 7×24

x = (7×24)/(7×12)

= 24/12

= 2

∴ x = 2

The important steps taken by promoters in the promotion of a company are as follows:

(i) Identification of Business Opportunity The first step to be taken by a promoter is to identify a business opportunity. The opportunity may be in respect of producing a new product or service or making some. product using a different process or any other opportunity having an investment potential.

(ii) Feasibility Studies. All the identified business opportunities may not be feasible or profitable as real projects. The promoters, therefore, undertake detailed feasibility studies to investigate all aspects of the business they intend to start.

Various types of feasibility have to be assessed which include 

1. Technical Feasibility 

2. Financial Feasibility 

3. Economic Feasibility

These feasibility studies are undertaken with the help of the specialists like engineers, chartered accountants etc and only when these investigations throw up positive results, the promoters may decide to actually launch a company.

(iii) Name Approval The promoters have to select a name for the company and submit an application to the registrar of companies of the state in which the registered office of the company is to be situated, for its approval. The proposed name may be approved if it is not considered undesirable.

According to the name clause the name of a company should not be identical or resembling the name of an existing company and should not violate the provisions of The Emblem and Names (Prevention of Improper Use) Act, 1950. Three names, in order of their priority are given in the application to the Registrar of Companies so that alternative; name may be allotted in case the first preference does not fulfill the name clause.

(iv) Fixing up Signatories to the Memorandum of Association Promoters have to decide about the members who will be signing the Memorandum of Association of the proposed company. Usually, the people signing memorandum are also the first Directors of the Company. Their written consent to act as Directors and to take up the qualification shares in the company is necessary.

(v) Appointment of Professionals Certain professionals such as mercantile bankers, auditors etc, are appointed by the promoters to assist them in the preparation of necessary documents which are required to be submitted with the Registrar of Companies. The names and addresses of shareholders and the number of shares allotted to each is submitted in a statement called return of allotment to the Registrar with the help of these professionals. 

(vi) Preparation of Necessary Documents The promoter takes up steps to prepare certain legal documents which include Memorandum of Association, Articles of Association and Consent of Directors These documents have to be submitted under the law, to the Registrar of the Companies for getting the company registered.

A company interacts with. SEBI (Securities and Exchange Board of India) in the third stage of formation that is, in the stage of capital subscription. SEBI is the regulatory authority of capital markets in our country which has issued guidelines for the disclosure of information and investor protection.

A company inviting funds from the general public must make adequate disclosure of all relevant information and must not conceal any material information from the potential investors as per SEBI guidelines. Prior approval from SEBI is, therefore, required before going ahead with raising funds from public. SEBI ensures that the proposed issue of securities follows all the guidelines laid down by it, no oversubscription of any issue can be retained, full underwriting of issue is important, promoters contribution must be 25% in an issue of less than Rs. 100 crore.

Promotion is the first stage in the formation of a company. It involves conceiving a business opportunity and taking an initiative to form a company so that the available business opportunity can be turned into a real business project. A promoter is said to be the one who undertakes to form a company with reference to a given project and to set it going and who takes the necessary steps to accomplish that purpose.

It is the function of promoters to analyze the prospects and bring together the men, materials, machinery, managerial abilities, financial resources and commence the business. Promoters undertake various activities to get a company registered and get it to the position of commencement of business. But they are neither the agents nor the trustees of the company. Legally they cannot be the agent of nonexisting companies.

It means that he is personally liable for the contracts, not legally. Also promoters are not the trustees they supposed to observe good faith in the promotion and must not make secret gains out of the dealings. If there is gain then it must be disclosed. Promoters are not legally entitled to claim the expenses incurred in the promotion of the company but the pre-incorporation expenses may be reimbursed.

The company may also remunerate the promoters for their efforts by paying a lump sum amount or a commission on the purchase price of property purchased through them or on the shares sold. Shares or debentures may also be allotted to the promoters or they may be given an option to purchase the securities at a future date.

Difference between Memorandum of Association and Article of Association.

(i) This statement is false because the sum of the interior angles of a quadrilateral is 360⁰.

(ii) This statement in always true. For example (-2)² = 4, then we can say x²  ≥ 0 for any real number x.

(iii) This statement is always true.

(iv) This statement is always true. For example, 2 + 2 = 4 and 6 + 4 = 10.

(v) This statement is always false. For example, 3 + 5 = 7 and 3 + 9 = 10.

(i) This statement is false because there are 12months in a year.

(ii) This statement is always ambiguous because Diwali can fall on any day.

(iii) This statement is always ambiguous because it is not fixed.

(iv) This statement is always false.

(v) This is a false statement because February has 29 days in a leap year.

Correct Answer is: (b) Registrar of Companies 

(d) 6 Alternative names

Correct Answer is: (d) In case of any of the above

Return of Allotment is a statement submitted to the Registrar which contains the names and addresses, of shareholders and the number of shares allotted to each shareholder. Return of allotment, signed by a director or secretary is filed with the Registrar of Companies within 30 days of allotment. Return of allotment shows that the company has received the minimum subscription.

Minimum subscription refers to the minimum amount required by the company for its preliminary functions. It has been provided by the Companies Act, that the company must receive applications for a certain minimum number of shares before going ahead: with the allotment of shares in order to prevent companies from commencing business with inadequate resources. This is called the ‘minimum subscription’. The limit of minimum subscription is 90% of the size of the issue.

Company formation is the term for the process of incorporation of a business in the UK. It is also sometimes referred to as company registration. Both these terms are used when incorporating a business in the Republic of Ireland. Under UK company law and most international law, a company or corporation is considered an entity that is separate from the people who own or operate the company.

Today the majority of UK companies are formed the same day electronically. Companies can be created by individuals, specialised agents, solicitors or accountants. Many solicitors and accountants subcontract incorporation out to specialised company formation agents. Most agents offer company formation packages for less than £100. The cost of carrying out paper filing directly with Companies House is £20. This fee does not include the cost of witnessing documents or preparation of memorandum, & articles of association for the company, which would usually be carried out by a 1 solicitor or accountant. Forming a company via the paper filing method can take up to 4 weeks.

Minimum number of members to form a private company is 2.