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

कक्षा दशमा अध्ययनरत 200 जना विद्यार्थीहरूलाई गणित र अङ्ग्रेजीमध्ये कुन विषय मनपर्छ भनी सोधिएको सर्वेक्षणबाट प्राप्त विवरण तल उल्लेख गरिएको छ ।
Below is a survey of 200 students studying in class ten asking which subject do you prefer between Mathematics and English.

• 120 जना विद्यार्थीले गणित मन पराउँछन् ।
120 students like Mathematics.

• 110 जना विद्यार्थीले अङ्ग्रेजी मन पराउँछन् ।
110 students like English.

• 30 जना विद्यार्थीले यी दुई विषयहरूमध्ये कुनै पनि मन पराउँदैनन् ।
30 students like none of these two subjects.

(a) गणित मन पराउने विद्यार्थीको समूहलाई M र अङ्ग्रेजी मन पराउने विद्यार्थीको समूहलाई E मानी कुनै पनि विषय मन नपराउने विद्यार्थीको समूहलाई गणनात्मकता सङ्केतमा लेख्नुहोस् ।
Write the set of students who like none of these two subjects in cardinal notation by supposing the set of students, who like mathematics by 'M' and English by 'E'.

(b) माथिको तथ्यलाई भेनचित्रमा प्रस्तुत गर्नुहोस् ।
Present the above information in a Venn-diagram.

(c) एउटा मात्र विषय मन पराउने विद्यार्थीहरूको सङ्ख्या पत्ता लगाउनुहोस् ।
Find the number of students, who like exactly one subjects.

(d) रामले भने - "दुवै विषय मन पराउने विद्यार्थी सङ्ख्या, कुनै पनि विषय मन नपराउने विद्यार्थी सङ्ख्याको दोब्बर छ ।" तपाईं रामको भनाइसँग सहमत हुनुहुन्छ ? पुष्टि गर्नुहोस् ।
Ram said, "The number of students who like both the subjects is double of the number of students who like none of these subjects." Do you agree with Ram's statement? Justify it.

एक जना मानिसले व्यापार गर्न भनेर नजिककको आफन्तसँग प्रतिवर्ष 10% वार्षिक चक्रीय व्याजदरमा रु.1,50,000 दुई वर्षका लागि ऋण लिएको रहेछ । उक्त मानिसले ब्याज र ऋण केही कम गर्न आफन्तलाई एक वर्षको अन्त्यमा रु.85,000 तिरेछन् ।
A man took a loan of Rs.1,50,000 for two years at the rate of compound interest 10% per annum compounded annually from his close relative for the purpose of business. He paid Rs.85,000 at the end of the first year to reduce the interest and some loan.

(a) वार्षिक चक्रीय ब्याज निकाल्ने सूत्र लेख्नुहोस् ।
Write the formula for finding annual compound interest.

(b) उनले एक वर्षको अन्त्यमा ऋणमा कति रकम फिर्ता गरेछन् ? पत्ता लगाउनुहोस् ।
How much money did he return as the loan at the end of one year? Find it.

(c) उनले जम्मा कति रकम तिरेर समयमा ऋण चुक्ता गरेछन् ? पत्ता लगाउनुहोस् ।
How much total money did he pay to clear the debt on time? Find it.

एक जना बस व्यवसायीले रु. 20,00,000 मा एउटा आरामदायी बस किनेर काठमाडौँ - पोखरा रुटमा चलाउँदा 3 वर्षमा रु.7,43,124 मात्र कमाए । प्रतिवर्ष 8% का दरले बसको मूल्यमा मिश्रह्रास भएछ ।
A bus owner bought a luxury bus for Rs.20,00,000 and used it in the Kathmandu - Pokhara route for 3 years. He earned Rs.7,43,124 only in 3 years. The price of the bus is depreciated at the rate of 8% per year.

(a) मिश्रह्रासलाई परिभाषित गर्नुहोस् ।
Define the compound depreciation.

(b) 3 वर्षपछि उक्त बसको बिक्री मूल्य कति हुन्छ ? पत्ता लगाउनुहोस् ।
What will be the selling price of the bus after 3 years? Find it.

(c) समग्र कारोबारमा बस व्यावसायीलाई कति प्रतिशत नाफा वा नोक्सान हुन्छ ? पत्ता लगाउनुहोस् ।
How much profit or loss percent will bus owner make in the total transaction? Find it.

एक जना विद्यार्थीले उच्च अध्ययनका लागि अमेरिका जान नेपाली रुपैयाँ 13,00,000 वरावरको अमेरिकन डलर साटेछन् । उक्त दिनको मुद्रा विनिमय दरअनुसार 1 अमेरिकन डलर ($) को बिक्रीदर र खरिददर क्रमशः ने.रु.143.75 र ने.रु.143.15 थियो ।
A student exchanged some American dollar for Nepali rupees 13,00,000 to go to America for his/her higher study. On that day, the selling rate and buying rate of 1 American dollar ($) were NRS. 143.75 and NRS.143.15 respectively.

(a) उक्त विद्यार्थीले ने.रु.13,00,000 को कति अमेरिकन डलर साटेछन् ? पत्ता लगाउनुहोस् ।
How many American dollars was exchanged with NRS. 13,00,000 by the student? Find it.

(b) स्वास्थ्य समस्याका कारण उक्त विद्यार्थी अमेरिका जान नसकेकाले केही समयपछि आफूसँग भएको अमेरिकन डलरलाई नेपाली रुपैयाँमा साट्न गए जुन दिन नेपाली मुद्रा 0.5% ले अधिमूल्यन भएको थियो । सो दिन उक्त विद्यार्थीले कति नेपाली रुपैयाँ पाउँछन् ? पत्ता लगाउनुहोस् ।
Due to health problem of the student, the student could not go to America and after a few days he/she exchanged American dollars into Nepali rupees. On that day, Nepalese currency was revaluated by 0.5%. How many Nepalese rupees did the student get on that day? Find it.

(c) मुद्रा सटहीमा उक्त विद्यार्थीलाई लाभ वा हानी हुनाको कारण लेख्नुहोस् ।
Write the reason of gain or loss in exchanging the currency for the student.

वर्गाकार आधार भएको एउटा पिरामिडको आधारको भुजाको लम्बाइ 24 से.मी. र उचाइ 9 से.मी. छन् ।
The length of side of the base of a square based pyramid is 24 cm and height is 9 cm.

(a) वर्गाकार आधार भएको पिरामिडको आयतन निकाल्ने सूत्र लेख्नुहोस् ।
Write the formula for finding the volume of a square based pyramid.

(b) दिइएको पिरामिडको आयतन पत्ता लगाउनुहोस् ।
Find the volume of the given pyramid.

(c) सो पिरामिडको त्रिभुजाकार सतहहरूको क्षेत्रफल निकाल्नुहोस् ।
Find the area of triangular surfaces of the pyramid.

दिइएको चित्र एउटा सोली र अर्धगोला मिली बनेको खेलौना हो । सो खेलौनाको साभा अर्धव्यास 7 से.मी. र पुरा उचाइ 31 से.मी. छन् ।
The given figure is of a toy made with the combination of a cone and hemi-sphere. The common radius of the toy is 7 cm and total height is 31 cm.

[Figure: Toy made of cone and hemisphere with common radius 7cm and total height 31cm]

(a) सोलीको आधारको अर्धव्यास \((r)\), ठाडो उचाइ \((h)\) र छड्के उचाइ \((l)\) बिचको सम्बन्ध लेख्नुहोस् ।
Write the relation among the radius of the base of a cone \((r)\), vertical height \((h)\) and slant height \((l)\).

(b) सो खेलौनाको सोली भागको बक्र सतहको क्षेत्रफल निकाल्नुहोस् ।
Find the curved surface area of the conical part of the toy.

(c) सोली भागको आयतन अर्धगोलाकार भागको आयतनभन्दा कति कम वा बढी छ ? गणना गरी तुलना गर्नुहोस् ।
By how much the volume of conical part is less or more than the volume of hemi-spherical part? Compare with calculation.

एउटा भवनको बाहिर गाडी पार्किङ गर्ने ठाउँ चित्रमा देखाएजस्तै ज्यामितीय आकारको छ ।
The parking area outside of a building is in geometric shape as shown in the given figure.

[Figure: Trapezoid-shaped parking area: top AD=20ft, left AB=24ft, bottom BC=38ft, right angle at B]

(a) सो पार्किङ गर्ने ठाउँको क्षेत्रफल पत्ता लगाउनुहोस् ।
Find the area of the parking area.

(b) उक्त पार्किङमा प्रति 100 वर्गफिटको रु.2500 को दरले समथाउन जम्मा कति खर्च लाग्ला ? पत्ता लगाउनुहोस् ।
Find the total cost of leveling the parking area at the rate of Rs.2500 per 100 sq. ft.

एक जना कर्मचारीको हालको मासिक तलब रु.45,000 छ । प्रत्येक वर्ष उसको ग्रेड रकम रु.1,500 मासिक तलबमा थपिँदै जान्छ ।
The monthly salary of an employee is Rs.45,000 at present. He receives a grade of Rs.1,500 on his monthly salary every year.

(a) माथिको सन्दर्भबाट बन्ने अनुक्रमको नाम लेख्नुहोस् ।
Write the name of sequence formed from above context.

(b) यदि यही दरले बढ्दै जाने हो भने 4 वर्षपछि उसको जम्मा आम्दानी कति हुन्छ ? गणना गर्नुहोस् ।
What will be his total income in 4 years when the number of grades increases yearly at the same rate? Calculate it.

(c) यही दरले बढ्दै जाँदा कति वर्षमा उसको जम्मा आम्दानी रु.35,10,000 हुन्छ ? पत्ता लगाउनुहोस् ।
In how many years, his total income will be Rs.35,10,000 when the number of grades increases yearly by the same rate? Find it.

दाइ र भाइको हालको उमेरको योगफल 28 वर्ष छ । उनीहरूको उमेरको सङ्ख्यात्मक गुणनफल 187 छ ।
The sum of the present ages of two brothers is 28 years and the product of their ages is 187 numerically.

(a) वर्ग समीकरण \(ax^2 + bx + c = 0\) मा \(x\) को मान निकाल्ने सूत्र लेख्नुहोस् ।
In the quadratic equation of \(ax^2 + bx + c = 0\), write the formula to find the value of \(x\).

(b) तिनीहरूको हालको उमेर पत्ता लगाउनुहोस् ।
Find their present ages.

(c) कति वर्षपछि उनीहरूको उमेरको गुणनफल 247 हुन्छ ? पत्ता लगाउनुहोस् ।
After how many years will the product of their ages be 247? Find it.

(a) सरल गर्नुहोस् । (Simplify): \(\dfrac{1}{m+n} - \dfrac{1}{m-n}\)

(b) हल गर्नुहोस् । (Solve): \(3^{y+1} + 3^y = 36\)

दिइएको चित्रमा NE \(\parallel\) AW, AN \(\parallel\) ME र SN \(\parallel\) WE छन् ।
In the given figure, NE \(\parallel\) AW, AN \(\parallel\) ME and SN \(\parallel\) WE.

[Figure: Parallelogram NAME and NEWS with NE parallel to AW, AN parallel to ME, SN parallel to WE; points N, E on top, A, S, M, W on bottom]

(a) समानान्तर चतुर्भुज NEWS र त्रिभुज NWS बिचको सम्बन्ध लेख्नुहोस् ।
Write the relation between parallelogram NAME and triangle NWS.

(b) समानान्तर चतुर्भुजहरू NAME र NEWS को क्षेत्रफल बराबर हुन्छ भनी प्रमाणित गर्नुहोस् ।
Prove that the areas of parallelograms NAME and NEWS are equal.

(c) यदि AM को मध्यबिन्दु S भए प्रमाणित गर्नुहोस् : \(\triangle ANS = \dfrac{1}{4} \square NEWS\).
If 'S' is the midpoint of AM, Prove that, \(\triangle ANS = \dfrac{1}{4} \square NEWS\).

दिइएको चित्रमा केन्द्रबिन्दु O भएको वृत्तमा PARM एउटा चक्रीय चतुर्भुज हो ।
In the given figure, PARM is a cyclic quadrilateral in circle with centre O.

[Figure: Cyclic quadrilateral PARM in circle with centre O; angle A = 58 degrees, angle at O marked x]

(a) \(\angle PAR\) र \(\angle PMR\) बिचको सम्बन्ध लेख्नुहोस् ।
Write the relation between \(\angle PAR\) and \(\angle PMR\).

(b) दिएको चित्रमा \(x\) को मान पत्ता लगाउनुहोस् ।
Find the value of \(x\) from the given figure.

(c) चक्रीय चतुर्भुज PARM मा \(\angle APM + \angle ARM = 180^\circ\) हुन्छ भनी प्रयोगात्मक रूपमा परीक्षण गर्नुहोस् । (कम्तीमा अर्धव्यास 3 से.मी. भएका दुईओटा वृत्तहरू आवश्यक छ ।)
Verify experimentally that in the cyclic quadrilateral PARM, \(\angle APM + \angle ARM = 180^\circ\) (Two circles having radii at least 3 cm are necessary.)

चतुर्भुज PQRS मा PQ = 5 से.मी., QR = 5.5 से.मी., RS = 5.7 से.मी., PR = 7.1 से.मी. र PS = 6.1 से.मी. दिइएको छ ।
In a quadrilateral PQRS, PQ = 5 cm, QR = 5.5 cm, RS = 5.7 cm, PR = 7.1 cm and PS = 6.1 cm are given.

(a) चतुर्भुज PQRS को रचना गर्नुहोस् । यसको क्षेत्रफलसँग बराबर हुने गरी एउटा त्रिभुजको पनि रचना गर्नुहोस् ।
Construct quadrilateral PQRS. Also construct a triangle equal in area to the area of quadrilateral PQRS.

(b) माथिको रचनाबाट चतुर्भुज PQRS को क्षेत्रफल र त्रिभुजको क्षेत्रफल बराबर हुनुको कारण दिनुहोस् ।
From above construction, why are the area of quad. PQRS and area of triangle equal? Give reason.

एउटा 1.3 मिटर अग्लो मानिसले 41.3 मिटर उचाइ भएको स्तम्भको टुप्पामा हेर्दा 45° को उन्नतांश कोण पाएछ ।
A man 1.3 meter tall is observing at the top of a tower 41.3 meter high and found an angle of elevation of 45°.

(a) उन्नतांश कोणलाई परिभाषित गर्नुहोस् ।
Define the angle of elevation?

(b) मानिसभन्दा स्तम्भ कति मिटर अग्लो रहेछ ? पत्ता लगाउनुहोस् ।
How high is the tower than the man? Find it.

(c) स्तम्भ र मानिसबिचको दुरी पत्ता लगाउनुहोस् ।
Find the distance between the tower and man.

(d) स्तम्भको टुप्पाको उन्नतांश कोण 30° पाउन उक्त मानिस सो स्थानबाट कति मिटरपछाडि गई अवलोकन गर्नुपर्दछ ? पत्ता लगाउनुहोस् ।
How many meter should the man walk backward from that place to find the angle of elevation of the top of the tower 30°? Find it.

तलको तालिकामा 30 जना विद्यार्थीहरूले प्राप्त गरेको प्राप्ताङ्क प्रस्तुत गरिएको छ ।
The marks obtained by 30 students are presented in the table below.

(a) मध्यिका पत्ता लगाउने सूत्र \((M_d) = L + \dfrac{\frac{N}{2} - c.f.}{f} \times i\) मा 'L' ले के जनाउँछ ? लेख्नुहोस् ।
What does 'L' represent for in formula to find the median \((M_d) = L + \dfrac{\frac{N}{2} - c.f.}{f} \times i\)? Write it.

(b) दिइएको तथ्याङ्कको मध्यिका पत्ता लगाउनुहोस् ।
Find the median of the given data.

(c) उक्त तथ्याङ्कबाट औसत प्राप्ताङ्क गणना गर्नुहोस् ।
Calculate the average marks from the data.

(d) यदि दिइएको तथ्याङ्कबाट (0 - 30) प्राप्ताङ्क ल्याउने विद्यार्थी सङ्ख्या हटाइयो भने औसत प्राप्ताङ्कमा कतिको फरक पर्छ ? पत्ता लगाउनुहोस् ।
If the student securing (0 - 30) marks are avoided from the given data, what difference shows in the average marks? Find it.

राम्ररी फिटिएको 52 पत्ती तासको गड्डीबाट नहेरीकन दुईओटा तासहरू एकपछि अर्को पुनः राखेर फिटिएका छन् ।
Two cards are drawn randomly one after another with replacement from a well shuffled deck of 52 cards.

(a) अनाश्रित घटनाहरू भनेको के हो ? लेख्नुहोस् ।
What are independent events? Write it.

(b) अनुहार भएको तास पर्ने र नपर्ने सबै परिणामहरूको सम्भाव्यतालाई वृक्ष चित्रमा देखाउनुहोस् ।
Show the probability of all the possible outcomes of getting and not getting faced cards in a tree diagram.

(c) दुवै तास अनुहार भएको पत्ति पर्ने सम्भाव्यता कति हुन्छ ? पत्ता लगाउनुहोस् ।
What is the probability of getting both faced cards? Find it.

(d) दुईओटा तासहरू एकपछि अर्को पुनः राखी र नराखीकन फिक्दा दुवै तासहरू अनुहार भएको पर्ने सम्भाव्यताहरूबिच कति भिन्नता हुन्छ ? पत्ता लगाउनुहोस् ।
What is the difference between the probabilities of both cards being faced cards if two cards are drawn one after another with replacement and without replacement? Find it.