Class 10 Compulsory Mathematicsकक्षा १० अनिवार्य गणित — SEE notes & solutions

160 जना मानिसहरूको समूहमा गरिएको सर्वेक्षणमा स्याउ मात्र र सुन्तला मात्र मन पराउने मानिसहरूको सङ्ख्या क्रमशः 75 र 45 छन् । जसमध्ये 23 जनाले दुईमध्ये कुनै पनि मन पराउदैनन् ।

A survey conducted among 160 people, it was found that the number of the people who like only apple and only orange are 75 and 45 respectively. Among them 23 people do not like any of these two.

(a) स्याउ मन पराउने मानिसको समूहलाई A र सुन्तला मन पराउने मानिसको समूहलाई O ले जनाउँदा कुनै पनि फलफुल मन नपराउने मानिसको सङ्ख्यालाई गणनात्मकता संकेतमा लेख्नुहोस् ।

'A' represents the set of people who like apple and 'O' represents the set of people who like orange then write the cardinality notation of the number of people who don't like any of these fruits.

(b) माथिको जानकारीलाई भेन चित्रमा प्रस्तुत गर्नुहोस् ।

Present the above information in a Venn diagram.

(c) स्याउ मन पराउने मानिसको सङ्ख्या पत्ता लगाउनुहोस् ।

Find the number of people who like apple.

(d) स्याउ मन पराउने र सुन्तला मन पराउने मानिसहरूको सङ्ख्याविच तुलना गर्नुहोस् ।

Compare the number of people who like apple and who like orange.

निरजले वार्षिक \(10\%\) चक्रीय व्याजदरमा एउटा वैङ्कवाट 2 वर्षका लागि रु.4,00,000 ऋण लिएछ । उनले पहिलो वर्षको अन्तमा रु.2,40,000 तिरेछन् ।

Neeraj took a loan of Rs.4,00,000 for 2 years at the rate of \(10\%\) annual compound interest. He paid Rs.2,40,000 at the end of first year.

(a) सावाँ P, वार्षिक चक्रीय व्याज दर R, समय T र चक्रीय व्याज CI विचको सम्बन्ध लेख्नुहोस् ।

Write the relation among principal P, annual compound interest rate R, time T and compound interest CI.

(b) पहिलो वर्षको चक्रीय व्याज पत्ता लगाउनुहोस् ।

Find the compound interest of the first year.

(c) निरजले दुई वर्षमा जम्मा कति व्याज तिरेको रहेछ ? पत्ता लगाउनुहोस् ।

How much total interest was paid by Niraj in two years? Find it.

निलमले एउटा मेसिन रु.40,000 मा किनिछिन् । मेसिनको मूल्यमा वार्षिक \(5\%\) का दरले मिश्रह्रास हुन्छ । केही वर्षको प्रयोगपछि उक्त मेसिन रु.36,100 मा बेचियो ।

Neelam bought a machine for Rs. 40,000. The price of machine depreciates at the rate of \(5\%\) annually. The machine is sold for Rs. 36,100 after using for some years.

(a) पहिलो वर्ष मेसिनको मूल्य कतिले घट्छ ? पत्ता लगाउनुहोस् ।

By how much does the price of machine depreciate in first year? Find it.

(b) कति वर्षपछि सो मेसिन बिक्री गरिएको थियो ? पत्ता लगाउनुहोस् ।

After how many years was the machine sold? Find it.

(c) यदि उनले उक्त मेसिन भाडामा दिई भाडा वापत रु.4,900 कमाउछिन् भने मेसिनको बिक्रीमा उनलाई कति प्रतिशत नाफा वा नोक्सान हुन्छ ? पत्ता लगाउनुहोस् ।

Find the profit or loss percentage from selling the machine if she earns Rs.4,900 from the rent of machine.

रमेशसँग ने.रु.2,07,345 थियो । उनी अमेरिकी डलर साट्न वैङ्क गएको दिन विनिमय दर निम्न अनुसार थियो ।

Ramesh had NRs. 2,07,345. When he went to bank the exchange rate was as follows.

\$1 को खरिद दर = ने.रु.138.23 (Buying rate of \$ 1 = Rs. 138.23)

\$1 को बिक्रीदर = ने.रु.138.83 (Selling rate of \$1 = Rs. 138.83)

(a) रमेशले नेपाली रुपियासँग अमेरिकी डलर साट्दा कुन विनिमय दर प्रयोग गरिन्छ ? लेख्नुहोस् ।

Which exchange rate is used when Ramesh exchange American dollar with Nepali rupees? Write it.

(b) ने.रु. 2,07,345 बाट प्राप्त हुने अमेरिकी डलर पत्ता लगाउनुहोस् ।

Find the American dollar obtained from NRs. 2,07,345.

(c) नेपाली मुद्रामा कति प्रतिशत अवमूल्यन हुँदा 1 अमेरिकी डलरको बिक्री दर रु.140.2183 हुन्छ ? पत्ता लगाउनुहोस् ।

By what percent Nepali currency is devaluated when the selling rate of 1 US dollar is Rs. 140.2183? Find it.

चित्रमा दिइएको वर्गाकार आधार भएको पिरामिडको आयतन 512 घन से.मि. र आधारको भुजाको लम्बाइ 16 से.मि. छन् ।

The volume of square based pyramid given in the figure is 512 cubic cm and length of side of base is 16 cm.

[Figure: Square based pyramid with base side 16cm and volume 512 cubic cm]

(a) वर्गाकार आधार भएको पिरामिडको पूरा सतहको क्षेत्रफल गणना गर्न, जम्मा कति ओटा समतल सतहको क्षेत्रफल गणना गरिन्छ ? लेख्नुहोस् ।

How many plane surfaces area are counted to find the total surface area of a square based pyramid? Write it.

(b) उक्त पिरामिडको ठाडो उचाइ पत्ता लगाउनुहोस् ।

Find the vertical height of the pyramid.

(c) उक्त पिरामिडको पूरा सतहको क्षेत्रफल पत्ता लगाउनुहोस् ।

Find the total surface area of the pyramid.

ज्योतिले स्थानीय बजारवाट बेलना र अर्धगोला मिलि बनेको एउटा ट्याङ्की किनेर ल्याइन । सो ट्याङ्कीको पूरा उचाइ 3.5 मिटर र आधारको अर्धव्यास 1.05 मिटर छन् ।

Jyoti bought a tank made up of a cylinder and a hemisphere from the local market. The total height of the tank is 3.5 meter and radius of base is 1.05 meter.

(a) बेलना र अर्धगोला मिलि बनेको संयुक्त ठोस वस्तुमा कतिओटा बक्र सतहहरू हुन्छन् ? लेख्नुहोस् ।

How many curved surfaces are there in a combined solid made of a cylinder and a hemisphere? Write it.

(b) उक्त ट्याङ्कीको आयतन पत्ता लगाउनुहोस् ।

Find the volume of the tank.

(c) उक्त ट्याङ्कीमा बढीमा कति लिटर पानी अटाउँछ ? पत्ता लगाउनुहोस् ।

How much maximum liters of water is contained in the tank? Find it.

एउटा आयताकार कोठाको लम्बाइ, चौडाइ र उचाइ क्रमशः 16 फिट, 12 फिट र 9 फिट छन् । उक्त कोठामा 4 फिट किनारा भएका दुईओटा वर्गाकार भ्यालहरू र एउटा 6 फिट × 2 फिटको ढोका छन् ।

The length, breadth and height of a rectangular room are 16 ft, 12 ft and 9 ft respectively. There are two square windows of dimension 4 ft and one door of dimension 6 ft × 2 ft.

(a) उक्त कोठामा प्रतिवर्ग फिट रु.300 का दरले कार्पेट विछ्याउँदा कति खर्च लाग्छ ? पत्ता लगाउनुहोस् ।

How much does it cost for carpeting the room at the rate of Rs.300 per sq. ft.? Find it.

(b) यदि भ्याल ढोका बाहेक उक्त कोठाको चारभित्ता र सिलिङमा रङ लगाउँदा रु.19,560 खर्च लाग्छ भने प्रति वर्ग फिट रङ लगाउने दर पत्ता लगाउनुहोस् ।

If the cost of coloring four walls and ceiling excluding doors and windows of the room is Rs.19,560, find the rate of coloring per square feet.

हरिले आफ्नो छोरा आशिषको पहिलो जन्मदिनमा रु.1,000, दोस्रो जन्मदिनमा रु.2,000, तेस्रो जन्मदिनमा रु.3,000 का दरले एउटा वैङ्कमा रकम जम्मा गरेछन् । यसरी हरेक जन्मदिनमा रु.1,000 का दरले वढाउँदै लगेछन् ।

Hari deposited Rs.1,000, Rs.2,000, Rs.3,000 in bank on his son Aashish's first, second and third birthday respectively. In this way, he increases the deposit by Rs. 1,000 on every birthday.

(a) समानान्तरीय श्रेणीमा मध्यमालाई परिभाषित गर्नुहोस् ।

Define mean in arithmetic series.

(b) दशौँ जन्मदिनसम्ममा कति रकम जम्मा हुन्छ ? पत्ता लगाउनुहोस् ।

How much total money is deposited upto 10th birthday? Find it.

(c) के एघारौँ जन्मदिनसम्ममा आशिषको खातामा रु.66,000 जम्मा हुन्छ ? कारण दिनुहोस् ।

Is Rs.66,000 deposited on Aashish account by 11th birthday? Give reasons.

(d) आशिषको खातामा कति वर्षसम्ममा रु.1,05,000 जम्मा होला ? पत्ता लगाउनुहोस् ।

In how many years, will Rs.1,05,000 be deposited in the account of Aashish? Find it.

कृतिमे चौडाइको दोब्बर लम्बाइ भएको आफ्नो खेतको वरिपरि तारको बार लगाउन चाहन्छिन् । उक्त खेतको क्षेत्रफल 800 वर्गफिट छ ।

Kriti wants to fence her field having length twice of breadth. The area of the field is 800 square feet.

(a) वर्ग समीकरणको स्तरीय स्वरूप लेख्नुहोस् ।

Write down the standard form of quadratic equation.

(b) खेतको वरिपरि एक पटक तारबारले घेरा हाल्न कति लामो तार चाहिएला ? पत्ता लगाउनुहोस् ।

How much the length of wire is required to fence the field once with wire? Find it.

(a) सरल गर्नुहोस् (Simplify): \(\frac{x}{xy - y^2} + \frac{y}{xy - x^2}\)

(b) हल गर्नुहोस् (Solve): \(2^x + \frac{16}{2^x} = 10\)

दिइएको चित्रमा \(\triangle PQT\), समानान्तर चतुर्भुजहरू PQRS र PQUT एउटै आधार PQ र उही समानान्तर रेखाहरू PQ र TR विच वनेका छन् ।

In the given figure, \(\triangle PQT\), parallelograms PQRS and PQUT are standing on the same base PQ and between the same parallel lines PQ and TR.

[Figure: Triangle PQT and parallelograms PQRS and PQUT on same base PQ between parallel lines PQ and TR]

(a) समानान्तर चतुर्भुजहरू PQRS र PQUT का क्षेत्रफलविचको सम्बन्ध लेख्नुहोस् ।

Write the relation between the areas of parallelograms PQRS and PQUT.

(b) \(\triangle PQT\) को क्षेत्रफल समानान्तर चतुर्भुज PQRS को क्षेत्रफलको आधा हुन्छ भनि प्रमाणित गर्नुहोस् ।

Prove that area of \(\triangle PQT\) is half of the area of parallelogram PQRS.

(c) के दिइएको चित्रमा \(\triangle APD\) र \(\triangle BPQ\) को क्षेत्रफल बराबर हुन्छन् ? कारण सहित लेख्नुहोस् ।

Are the areas of \(\triangle APD\) and \(\triangle BPQ\) equal in the given figure? Write with reason.

[Figure: Parallelogram ABCD with point P on AB and point Q, triangles APD and BPQ]

दिइएको चित्रमा O वृत्तको केन्द्रविन्दु हो । विन्दुहरू M, N, P र L वृत्तको परिधिमा छन् ।

In the given figure, O is the centre of the circle. The points M, N, P and L are on the circumference of the circle.

[Figure: Circle with centre O and points M, N, P, L on circumference, central angle (9x+2) degrees and inscribed angle (4x+5) degrees]

(a) परिधि कोणलाई परिभाषित गर्नुहोस् ।

Define the inscribed angle.

(b) यदि केन्द्रीय कोण \(LOP = (9x + 2)^\circ\) र परिधि कोण \(LMP = (4x + 5)^\circ\) भए \(x\) को मान पत्ता लगाउनुहोस् ।

If the central angle \(LOP = (9x + 2)^\circ\) and inscribed angle \(LMP = (4x + 5)^\circ\), find the value of \(x\).

(c) परिधि कोणहरू LMP र LNP बराबर हुन्छन् भनी प्रयोगात्मक विधिबाट प्रमाणित गर्नुहोस् । (कमतीमा 3 से.मि. अर्धव्यास भएका दुई वृत्तहरू आवश्यक छन् ।)

Verify experimentally that inscribed angles LMP and LNP are equal. (Two circles having at least 3cm radii are necessary.)

(a) चतुर्भुज PQRS को रचना गर्नुहोस् जहाँ \(PQ = 5.4\) से.मि., \(QR = 5.6\) से.मि., \(RS = 5.4\) से.मि., \(SP = 6.8\) से.मि. र \(\angle PQR = 75^\circ\) छन् । चतुर्भुज PQRS को क्षेत्रफलसँग बराबर हुने गरी त्रिभुज PSM को रचना गर्नुहोस् ।

Construct a quadrilateral PQRS in which \(PQ = 5.4\) cm, \(QR = 5.6\) cm, \(RS = 5.4\) cm, \(SP = 6.8\) cm and \(\angle PQR = 75^\circ\). Then construct a triangle PSM equal in area to the quadrilateral PQRS.

(b) संगै दिइएको समानान्तर चतुर्भुज ABCD मा \(AE = BE\) छ भने त्रिभुज BEC ले समानान्तर चतुर्भुज ABCD को कति प्रतिशत क्षेत्रफल ओगटेको छ ? पत्ता लगाउनुहोस् ।

In the given adjoining parallelogram ABCD, \(AE = BE\). What percentage of area of parallelogram ABCD is occupied by the triangle BEC? Find it.

[Figure: Parallelogram ABCD with E as midpoint of AB and triangle BEC]

संगै दिइएको चित्रमा PQ ले घरको उचाइ, RS ले टावरको उचाइ र QS ले घरदेखि टावरसम्मको दूरीलाई जनाउँछ ।

In the given figure alongside, PQ represents the height of house, RS represents the height of tower and QS represents the distance from the house to the tower.

[Figure: House PQ height 30m, tower RS height 72m, angle of elevation 60 degrees from roof of house to top of tower]

(a) घरको छतवाट टावरको टुप्पोमा अवलोकन गर्दा बन्ने उन्नताश कोणको नाम लेख्नुहोस् ।

Write the name of angle of elevation of top of tower as observed from the roof of the house.

(b) TR को मान पत्ता लगाउनुहोस् ।

Find the value of TR.

(c) घर र टावर विचको दूरी पत्ता लगाउनुहोस् ।

Find the distance between the house and the tower.

(d) के टावरको टुप्पोवाट 28 मिटर तलको विन्दुवाट घरको छत हेर्दा \(30^\circ\) को अवनति कोण बन्छ ? कारण लेख्नुहोस् ।

Is the angle of depression of \(30^\circ\) formed when the roof of the house is observed from a point 28 meter below the top of the tower? Give reason.

दिइएको तथ्याङ्कले गणित विषयको 50 पूर्णाङ्कको एउटा आन्तरिक परीक्षामा विद्यार्थीहरूले प्राप्त गरेको प्राप्ताङ्कलाई प्रतिनिधित्व गर्दछ । उक्त तथ्याङ्कको मध्यका 29 छ ।

The given data represents the marks obtained by the students in an internal examination of Mathematics with full marks 50. The median of the data is 29.

(a) मध्यका श्रेणी लेख्नुहोस् ।

Write the median class.

(b) \(x\) को मान पत्ता लगाउनुहोस् ।

Find the value of \(x\).

(c) दिइएको तथ्याङ्कको मध्यक प्राप्ताङ्क पत्ता लगाउनुहोस् ।

Find the mean mark of the given data.

(d) 20 अङ्क भन्दा कम र 20 वा सोभन्दा बढी अङ्क प्राप्त गर्ने विद्यार्थीहरूको अनुपात निकाल्नुहोस् ।

Find the ratio of students obtaining marks less than 20 and 20 or more than 20.

एउटा भोलामा 7 ओटा काला र 4 ओटा राता उस्तै र उत्रै बलहरू छन् । दुईओटा बलहरू एक पछि अर्को गरी पुनः नराखी भिकिएका छन् ।

A bag contains 7 black and 4 red balls of same shape and size. Two balls are drawn randomly one after another without replacement.

(a) यदि B र R दुई ओटा अनाश्रित घटना भए \(P(B \cap R)\) को सूत्र लेख्नुहोस् ।

If B and R be two independent events then write the formula of \(P(B \cap R)\).

(b) सबै सम्भावित परिणामहरूको सम्भाव्यतालाई वृक्ष चित्रमा देखाउनुहोस् ।

Show the probabilities all the possible outcomes in a tree diagram.

(c) दुवै बल कालो नै पर्ने सम्भाव्यता पत्ता लगाउनुहोस् ।

Find the probability of getting both black balls.

(d) दुवै बल रातो पर्ने सम्भाव्यता, दुवै बल कालो पर्ने सम्भाव्यताभन्दा कति बढी वा घटी छ ? पत्ता लगाउनुहोस् ।

By how much the probability of getting both red balls is more or less than the probability of getting both black balls? Find it.