Question Math 8

काठमाडौं महानगरपालिका २०८१

निम्न लिखित प्रश्नहरूको उत्तर दिनुहोस्।
(Answer the following questions.)

1. दिइएको समुह \(A = \{2, 5\}\) र \(B = \{5, 7\}\) छन।
   Given sets \(A = \{2, 5\}\) and \(B = \{5, 7\}\).
   1. \(A\) र \(B\) समुहहरू अलगिएका वा खप्टिएका कस्ता समुह हुन? लेख्नुहोस्।
      (Are sets \(A\) and \(B\) overlapping or disjoint? Write it.) [1]
   2. \(B\) समुहबाट बनाउन सकिने कुनै दुईवटा उपयुक्त उपसमुहहरु लेख्नुहोस्।
      (Write any two proper subsets that can be made from set \(B\).) [2]
   Answer (Q1)

   Given sets are
   \(A = \{2, 5\}\) and \(B = \{5, 7\}\)
   Now

   1. The sets \(A\) and \(B\) are overlapping set. Because \(\{5\}\) is common to both.
   2. The two proper subsets that can be made from set \(B\) are
      \(\{5\},\{7\}\)

   
   

   दिइएका समुहहरू छन्
   \(A = \{2, 5\}\) र \(B = \{5, 7\}\)
   अब

   1. समुह \(A\) र \(B\) खप्टिएका समुह हुन् किनभने \(\{5\}\) दुवैमा साझा छ
   2. समुह \(B\) बाट बनाउन सकिने दुईवटा उपयुक्त उपसमुहहरु यसप्रकार छन।
      \(\{5\},\{7\}\)
2. डिसेम्बर ८, २०२० मा घोषणा गरिएअनुसार संसारको सर्वोच्च शिखर सगरमाथाको उचाइ ८८४८.८६ मिटर थियो।
   As announced on December 8, 2020, the height of Mount Everest, the highest peak in the world, was 8848.86 meters.
   1. सङ्ख्या ८८४८.८६ आनुपातिक वा अनानुपातिक के हो? लेख्नुहोस्।
      ( Write whether the number 8848.86 is a rational or irrational number.) [1]
   2. सगरमाथाको उचाइ सेन्टिमिटरमा रूपान्तरण गरी वैज्ञानिक सङ्केतनमा लेख्नुहोस्।
      (Convert the height of Mt. Everest to centimeters and write it in scientific notation.) [2]
   3. प्रमाणित गर्नुहोस् Prove that \( 8848 = 240343_5 \) [2]
   Answer (Q2)

   According to the question

   1. The number 8848.86 is a rational because it is terminating in decimal.
   2. The height of Mt. Everest to centimeters is
      Height=8848.86 m
      orHeight=\(8848.86 \times 100\) cm
      orHeight=\(884886 \) cm
      The scientific notation of the height is
      Height=\(8.84886 \times 10^5\) cm
   3. We know that

      5	8848	...	3↑
      5	1769	...	4↑
      5	353	...	3↑
      5	70	...	0↑
      5	14	...	4↑
      	2	...	2↑

      Therefore
      \( 8848 = 240343_5 \)

   
   

   प्रश्नानुसार

   1. संख्या 8848.86 एक अनुपातिक संख्या हो किनभने यो दशमलवमा समाप्त भएको छ।
   2. सगरमाथाको उचाइ सेन्टिमिटरमा यसप्रकार छ।
      उचाइ = 8848.86 m
      वाउचाइ = \(8848.86 \times 100\) cm
      वाउचाइ = \(884886 \) cm
      उचाइको वैज्ञानिक संकेत निम्नाननुसार छ।
      उचाइ = \(8.84886 \times 10^5\) cm
   3. यहाँ,

      5	8848	...	3↑
      5	1769	...	4↑
      5	353	...	3↑
      5	70	...	0↑
      5	14	...	4↑
      	2	...	2↑

      तसर्थ
      \( 8848 = 240343_5 \)
3. दुई जना साथी रामनरेश र महेशले एउटा कारखानामा ३:२ को अनुपातमा रु. ५०,००,००० लगानी गरे।
   (Two friends, Ramnaresh and Mahesh, invested Rs. 50,00,000 in a factory in the ratio of 3:2.
   1. प्रत्यक्ष र अप्रत्यक्ष विचरणमा ले फरक छ? एउटा फरक लेख्नुहोस्।
      (What is the difference between direct and indirect variation? Write one difference.) [1]
   2. रामनरेशले कारखानामा कति रकम लगानी गरेको थियो? पत्ता लगाउनुहोस्।
      (How much amount has Ramnaresh invested in the factory? Find it.) [1]
   3. यदि महेशले उद्योगमा लगानी गरेको रकम बैंकमा १०\% वार्षिक ब्याजदरमा जम्मा गरेको भए २ वर्षपछि कति साधारण ब्याज पाउँथ्यो? गणना गर्नुहोस्।
      (If Mahesh had deposited the amount invested in the industry in a bank at an annual interest rate of 10\%, how much simple interest would he have received after 2 years? Calculate.) [2]
   4. यदि \(3 : 2 = x : 500\), भए \(x\) को मान पत्ता लगाउनुहोस्।
      (If \(3 : 2 = x : 500\), find the value of \(x\).) [1]
   Answer 3

   1. Difference between direct and indirect variation
      Direct variation: if one quantity increases, then other quantity also increases, or vice versa.
      Iindirect variation: if one quantity increases then other quantity decreases or vice versa.
   2. Ramnaresh's investment
      The total investment = Rs.\(50,00,000\).
      The ratio of investment between Ramnaresh and Mahesh = \(3:2\).
      Now
      Ramnaresh's share \(= \frac{3}{5} \times 50,00,000\)
      orRamnaresh's share \(= 30,00,000\)
      So, Ramnaresh invested Rs.\(30,00,000\).
   3. Simple interest for Mahesh's investment
      Mahesh's share = \(\frac{2}{5} \times 50,00,000 = 20,00,000\).
      Rate of interest = \(10\%\) per year.
      Time = 2 years.
      Now, simple interest is
      \(SI = \frac{P \times R \times T}{100}\)
      or\(SI=\frac{20,00,000 \times 10 \times 2}{100}\)
      or\(SI=4,00,000\)
      So, Mahesh would receive Rs.\(4,00,000\) as simple interest.
   4. Finding \(x\)
      We are given that
      \(3 : 2 = x : 500\)
      or\(\frac{3}{2} = \frac{x}{500}\)
      or\( 2 \times x=3 \times 500 \)
      or\(2x =1500\)
      or\(x = 750\)
      So, the value of \(x\) is \(750\).

   
   

   1. प्रत्यक्ष र अप्रत्यक्ष विचरणको फरक
      प्रत्यक्ष विचरण: यदि एउटा परिमाण बढ्छ भने अर्को परिमाण पनि बढ्छ, वा घट्छ।
      अप्रत्यक्ष विचरण: यदि एउटा परिमाण बढ्छ भने अर्को परिमाण घट्छ, वा यस्तै एउटा घट्दा अर्को बढ्छ।
   2. रामनरेशको लगानी
      कुल लगानी = रु.\(50,00,000\)।
      रामनरेश र महेशको लगानी अनुपात = \(3:2\)।
      अब
      रामनरेशको हिस्सा \(= \frac{3}{5} \times 50,00,000\)
      orरामनरेशको हिस्सा \(= 30,00,000\)
      त्यसैले, रामनरेशले रु.\(30,00,000\) लगानी गरेको छ।
   3. महेशको लगानीको साधारण ब्याज
      महेशको हिस्सा = \(\frac{2}{5} \times 50,00,000 = 20,00,000\)।
      ब्याज दर = \(10\%\) वार्षिक।
      समय = 2 वर्ष।
      अब साधारण ब्याज निम्नानुसाार हुन्छ
      \(SI = \frac{P \times R \times T}{100}\)
      or\(SI=\frac{20,00,000 \times 10 \times 2}{100}\)
      or\(SI=4,00,000\)
      त्यसैले, महेशले रु.\(4,00,000\) साधारण ब्याज पाउने थिए।
   4. \(x\) को मान
      हामीलाई दिइएको छ
      \(3 : 2 = x : 500\)
      or\(\frac{3}{2} = \frac{x}{500}\)
      or\( 2 \times x=3 \times 500 \)
      or\(2x =1500\)
      or\(x = 750\)
      त्यसैले, \(x\) को मान \(750\) हो।
4. अशिमले एउटा मेसिन रु. २५,००० मा किनेको थियो र यसको मूल्यमा २०% बढाएर बजार मूल्य तोकेको थियो। उसले केही छुट दिएर मेसिन बेचेपछि रु. १,००० को नोक्सानी भयो।
   (Ashim bought a machine for Rs. 25,000 and fixed the market price by increasing 20% on its price. He made a loss of Rs. 1,000 after selling the machine with some discount amount.)
   1. मेसिनको अङ्कित मूल्य पत्ता लगाउनुहोस्। (Find the marked price of the machine.) [1]
   2. मेसिन कति प्रतिशत छुटमा बेचिएको थियो? पत्ता लगाउनुहोस्।
      (At what discount percentage was the machine sold? Find out.) [2]
   3. यदि अशिमले मेसिन बेचेर रु. २,००० को नाफा कमाउन चाहन्छ भे, कति छुट दर कायम गर्नुपर्छ?
      (If Ashim wants to earn a profit of Rs. 2,000 by selling the machine, what discount rate should be maintained?) [1]
   Answer 4

   1. Finding the marked price of the machine
      Cost price (CP) = Rs.\(25,000\)
      Percentage increase to fix marked price (MP) = 20%
      Therefore, the marked price is
      Marked Price (MP) = 120% of CP
      orMarked Price (MP) =\( \frac{120}{100} \times 25,000 \)
      orMarked Price (MP) = 30,000
      So, the marked price of the machine is Rs.\(30,000\).
   2. Finding the discount percentage
      Ashim sold the machine at a loss of Rs.\(1,000\).
      So, Selling Price (SP) = CP - Loss = 25,000 - 1,000 = 24,000
      Now
      Discount % = \(\frac{MP - SP}{MP} \times 100\%= \frac{30,000 - 24,000}{30,000} \times 100\%\)
      or Discount% =\( \frac{6,000}{30,000} \times 100\%=20\%\)
      So, the machine was sold at 20\% discount.
   3. Finding the discount rate to earn Rs.\(2,000\) profit
      Desired profit = Rs.\(2,000\)
      So, Selling Price (SP) = CP + Profit = 25,000 + 2,000 = 27,000
      Now
      Discount% =\(\frac{MP - SP}{MP} \times 100\%= \frac{30,000 - 27,000}{30,000} \times 100\%\)
      or Discount% =\( \frac{3,000}{30,000} \times 100\%=10\%\)
      So, to earn a profit of Rs.\(2,000\), Ashim should give a 10\% discount.

   
   

   1. मेसिनको अंकित मूल्य
      क्रय मूल्य (CP) = रु.\(25,000\)
      अंकित मूल्य (MP) तोक्न प्रतिशत वृद्धि = 20%
      त्यसैले अंकित मूल्य निम्नानुसार हुन्छ
      अंकित मूल्य (MP) = CP को 120%
      वाअंकित मूल्य (MP) =\( \frac{120}{100} \times 25,000 \)
      वाअंकित मूल्य (MP) = 30,000
      त्यसैले, मेसिनको अंकित मूल्य रु.\(30,000\) हुन्छ।
   2. छूट प्रतिशत ने
      अशिमले मेसिन रु.\(1,000\) नोक्सानमा बेचेको थियो।
      त्यसैले, विक्रय मूल्य (SP) = CP - नोक्साी = 25,000 - 1,000 = 24,000
      अब
      छूट % = \(\frac{MP - SP}{MP} \times 100\%= \frac{30,000 - 24,000}{30,000} \times 100\%\)
      वा छूट % =\( \frac{6,000}{30,000} \times 100\%=20\%\)
      त्यसैले, मेसिन 20% छूटमा बेचियो।
   3. रु.\(2,000\) नाफा कमाउन आवश्यक छूट दर
      आवश्यक नाफा = रु.\(2,000\)
      त्यसैले, विक्रय मूल्य (SP) = CP + नाफा = 25,000 + 2,000 = 27,000
      अब
      छूट % =\(\frac{MP - SP}{MP} \times 100\%= \frac{30,000 - 27,000}{30,000} \times 100\%\)
      वा छूट % =\( \frac{3,000}{30,000} \times 100\%=10\%\)
      त्यसैले, रु.\(2,000\) नाफा कमाउन अशिमले 10% छूट दिनुपर्छ।
5. चित्रमा, \(ABCD\) एउटा समानान्तर चतुर्भुज हो जहाँ \(AB = 13\) सेमी व्यास भएको अर्धवृत्तमा उचाइ \(AD = 5\) सेमी भएको समकोण त्रिभुज \(ADB\) बनेको छ।
   (In the figure, \(ABCD\) is a parallelogram where a right-angled triangle \(ADB\) with height \(AD = 5\) cm is formed on the semicircle having diameter \(AB = 13\) cm.)
   1. समानान्तर चतुर्भुजको क्षेत्रफल पत्ता लगाउने सूत्र लेख्नुहोस्।
      (Write the formula to find the area of a parallelogram.) [1]
   2. अर्धवृत्तको क्षेत्रफल पत्ता लगाउनुहोस्।
      (Find the area of the semicircle.) [1]
   3. समकोण त्रिभुज \(ADB\) को क्षेत्रफल अर्धवृत्तको क्षेत्रफलभन्दा कति कम छ? गणना गर्नुहोस्।
      (By how much is the area of right-angled triangle \(ADB\) less than the area of the semicircle? Calculate it.) [2]
   4. के समानान्तर चतुर्भुज \(ABCD\) को क्षेत्रफल त्रिभुज \(ADB\) को क्षेत्रफलको दोब्बर हो? तर्कसहित उत्तर दिनुहोस्।
      (Is the area of parallelogram \(ABCD\) double the area of triangle \(ADB\)? Give a logical answer.) [1]
   Answer Q5

   1. The formula to find the area of a parallelogram is
      \(\text{Area} = \text{base} \times \text{height}\)
   2. The area of a semicircle is half the area of a circle, so it is
      \(\text{Area} = \frac{1}{2}\,\pi r^{2}\)
      or\(\text{Area} = \frac{1}{2} \times \frac{28}{7} \times (\frac{13}{2})^2\)
      or\(\text{Area} = \frac{1859}{28}=66.4\) square cm.
      
   3. The area of right-angled triangle \(ADB\) is
      \(\triangle ADB = \frac{1}{2} b \times h\)
      The height of the triangle is
      \(h = \sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12\)
      Hence, the area is
      \(\triangle ADB = \frac{1}{2} 5 \times 12\)
      or\(\triangle ADB = 30\) square cm.
      Hence, the area of right-angled triangle \(ADB\) is less than the area of the semicircle by
      Area of semicircle- Area of triangle
      or\(66.4-30=36.4\) square cm.
      
   4. Yes, the area of parallelogram \(ABCD\) is double the area of triangle \(ADB\), beacuse a diagonal of a parallelogram divides it into two equal triangles.
      Hence,
      \(\text{Area of parallelogram } = 2 \times \text{Area of one triangle}\)

   
   

   1. समानान्तर चतुर्भुजको क्षेत्रफल पत्ता लगाउने सूत्र
      क्षेत्रफल \(= \text{base} \times \text{height}\)
   2. अर्धवृत्तको क्षेत्रफल
      क्षेत्रफल \(= \frac{1}{2}\,\pi r^{2}\)
      वाक्षेत्रफल \(= \frac{1}{2} \times \frac{28}{7} \times (\frac{13}{2})^2\)
      वाक्षेत्रफल \(= \frac{1859}{28}=66.4\) वर्ग से.मी.
      
   3. समकोण त्रिभुज \(ADB\) को क्षेत्रफल
      \(\triangle ADB = \frac{1}{2} b \times h\)
      त्रिभुजको उचाइ:
      \(h = \sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12\)
      त्यसैले क्षेत्रफल:
      \(\triangle ADB = \frac{1}{2} 5 \times 12\)
      वा\(\triangle ADB = 30\) वर्ग से.मी.
      त्यसैले, समकोण त्रिभुज \(ADB\) को क्षेत्रफल अर्धवृत्तको क्षेत्रफलभन्दा निम्नानुसार कम हुन्छ।
      अर्धवृत्तको क्षेत्रफल − त्रिभुजको क्षेत्रफल
      वा\(66.4-30=36.4\) वर्ग से.मी.
      
   4. समानान्तर चतुर्भुज \(ABCD\) को क्षेत्रफल त्रिभुज \(ADB\) को क्षेत्रफलको दोब्बर हुन्छ, किनभने समानान्तर चतुर्भुजको विकर्णले यसलाई दुई बराबर त्रिभुजमा विभाजन गर्छ।
      त्यसैले
      स. च. को क्षेत्रफल= 2 x त्रिभुजको क्षेत्रफल
6. 1. \((a-b)^2\) को विस्तारित रूप लेख्नुहोस्।
      ( Write the expanded form of \((a-b)^2\).) [1]
   2. सरल गर्नुहोस्: \(x^{p(a-b)} \times x^{p(b-c)} \times x^{p(c-a)}\)
      (Simplify: \(x^{p(a-b)} \times x^{p(b-c)} \times x^{p(c-a)}\)) [2]
   Answer Q6

   1. The expanded form of \((a-b)^2\) is
      \(a^2-2ab+b^2\)
   2. The simplification is
      \(x^{p(a-b)} \times x^{p(b-c)} \times x^{p(c-a)}\)
      or\(x^{pa-pb} \times x^{pb-pc} \times x^{pc-pa}\)
      or\(x^{pa-pb+pb-pc+pc-pa}\)
      or\(x^0\)
      
   or\(1\)
   
7. एउटा आयताकार बगैंचाको लम्बाइ यसको चौडाइभन्दा ४ मिटरले बढी छ। बगैंचाको क्षेत्रफल ९६ वर्ग मिटर छ।
   (A rectangular garden has a length 4 meters more than its breadth. The area of the garden is 96 square meters.)
   1. यदि बगैंचाको चौडाइ \(x\) मिटर छ भने, लम्बाइ कति होला? \(x\) को रूपमा लेख्नुहोस्।
      (If the breadth of the garden is \(x\) meters, what is the length? Write in terms of \(x\).) [1]
   2. बगैंचाको लम्बाइ र चौडाइ कति हो? वर्ग समीकरण बनाएर पत्ता लगाउनुहोस्।
      ( What are the length and breadth of the garden? Find by making a quadratic equation.) [2]
   Answer Q7

   1. If the breadth of the garden is \(x\) meters, what is the length? Write in terms of \(x\).
      breadth (b)=\(x\)
      length (l)=\(x + 4\)
      
   2. What are the length and breadth of the garden? Find by forming a quadratic equation.
      The area is
      Area = Length × Breadth
      or\(96 = (x + 4) \cdot x\)
      or\(96 = x^2 + 4x\)
      or\(x^2 + 4x - 96 = 0\)
      or\(x^2 + (12x-8x) - 96 = 0\)
      or\(x^2 + 12x-8x- 96 = 0\)
      or\(x(x+12)-8(x+12) = 0\)
      or\((x+12)(x-8)= 0\)
      or\(x(x+12)-8(x+12) = 0\)
      or\(x=8\)
      Therefore
      Breadth = \(8\) meters
      breadth (b)=\(x=8m\)
      length (l)=\(x + 4=8+4=12m\)
      
8. 1. खण्डीकरण गर्नुहोस्: \(2x + xy - 2x - y\) (Factorize: \(2x + xy - 2x - y\)) [2]
   2. सरल गर्नुहोस्: \(\frac{1}{x-3} - \frac{1}{x^2-9}\) (Simplify: \(\frac{1}{x-3} - \frac{x-3}{x^2-9}\)) [2]
   Answer Q8

   1. The factorization is
      \(2x + xy - 2x - y\)
      or\(2x - 2x + xy - y\)
      or\(0 + xy - y\)
      or\(xy - y\)
      or\(y(x - 1)\)
      
   2. The simplification is
      \(\frac{1}{x-3} - \frac{x-3}{x^2-9}\)
      or\(\frac{1}{x-3} - \frac{x-3}{(x-3)(x+3)}\)
      or\(\frac{1}{x-3} - \frac{1}{x+3}\)
      or\(\frac{(x+3) - (x-3)}{(x-3)(x+3)}\)
      or\(\frac{x+3 - x + 3}{(x-3)(x+3)}\)
      or\(\frac{6}{(x-3)(x+3)}\)
      or\(\frac{6}{x^2 - 9}\)
      
9. दिइएको ग्राफमा, त्रिभुज \(ABC\) देखाइएको छ । (In the given graph, triangle \(ABC\) is shown.)
   1. दुई बिन्दुहरूबीचको दूरी पत्ता लगाउन कुन सूत्र प्रयोग गर्नुहुन्छ? लेख्नुहोस्।
      (Which formula do you use to find the distance between two points? Write it.) [1]
   2. \(BC\) भुजाको लम्बाइ पत्ता लगाउनुहोस्।
      (Find the length of side \(BC\).) [2]
   3. त्रिभुज \(ABC\) लाई धनात्मक दिशामा एक चौथाई परिक्रमण गर्नुहोस र प्रतिबिम्ब त्रिभुजका शीर्षबिन्दुहरूको निर्देशाङ्क लेख्नुहोस्।
      (Rotate the triangle \(ABC\) a quarter turn (\(90^\circ\)) in the positive (counter-clockwise) direction and write the coordinates of the vertices of the image triangle.) [2]
   4. त्रिभुजको आन्तरिक कोणहरूको योग \(180^\circ\) हुन्छ भन्ने प्रयोगात्मक रूपमा प्रमाणित गर्नुहोस्। (दुईवटा फरक नापका त्रिभुज आवश्यक छन्।)
      (Experimentally verify that the sum of interior angles of a triangle is \(180^\circ\). (Two triangles of different measurements are required.)) [3]
   Answer Q9

   
   
10. चित्रमा, \(PQRS\) एउटा आयत हो जसमा लम्बाइ \(QR = 6\) सेमी र चौडाइ \(RS = 4\) सेमी छ।
    (In the figure, \(PQRS\) is a rectangle in which length \(QR = 6\) cm and breadth \(RS = 4\) cm.)
    1. दिइएको नापअनुसार कम्पास प्रयोग गरी आयत \(PQRS\) को निर्माण गर्नुहोस्।
       (Construct the rectangle \(PQRS\) using a compass according to the given dimensions.) [3]
    2. के \(\triangle QPS\) र \(\triangle QRS\) एक अर्कासँग सर्वाङ्गसम छन्? कारणसहित लेख्नुहोस्।
       (Are \(\triangle QPS\) and \(\triangle QRS\) congruent to each other? Write with reason.) [1]
    Answer Q10

    
    
11. एउटा बिन्दु \(Q\), जुन बिन्दु \(P\) बाट ८ सेमी टाढा छ, १ सेमी = ५ किमी स्केलमा ११०° को बेयरिङ छ।
    (A point \(Q\), which is 8 cm away from a point \(P\), has a bearing of \(110^\circ\) on the scale 1 cm = 5 km.)
    1. माथिको सन्दर्भअनुसार दिशास्थितिको रेखाङ्कन गर्नुहोस्।
       (Draw the bearing according to the above context.) [1]
    2. बिन्दु \(Q\) बाट \(P\) को दिशास्थिति र \(P\) बाट \(Q\) को दिशास्थिति तुलना गर्नुहोस्।
       (Compare the bearing of \(P\) from point \(Q\) and the bearing of \(Q\) from \(P\).) [2]
    Answer Q11

    
    
12. कक्षा ८ का ७ विद्यार्थीहरूले प्रथम त्रैमासिक परीक्षामा गणितमा प्राप्त गरेका अङ्कहरू यी हुन्: २३, ३०, २५, २६, २३, २८, २७।
    (The following are the marks obtained by 7 students of grade eight in the first terminal examination in mathematics: 23, 30, 25, 26, 23, 28, 27.)
    1. माथिको तथ्याङकबाट रीत पत्ता लगाउनुहोस्। (Find the mode from the above data.) [1]
    2. ७ विद्यार्थीहरूले प्रथम त्रैमासिक परीक्षामा प्राप्त गरेको औसत अङ्क कति हो? पत्ता लगाउनुहोस्।
       (What is the average marks obtained by the 7 students in the first terminal examination? Find it.) [1]
    3. मध्यक, माध्य, र रीतमध्ये कुनले दिइएको तथ्याङकलाई दुई बराबर भागमा विभाजन गर्छ? कारणसहित लेख्नुहोस्।
       (Which among mean, median, and mode divides the given data into two equal parts? Write with reason.) [1]
Answer Q12